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Geometry
3D Geometry
2015-10-28 www.njctl.org
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Table of Contents Intro to 3-D Solids Views & Drawings of 3-D Solids Surface Area of a Prism Surface Area of a Cylinder Surface Area of a Pyramid Surface Area of a Cone
Click on the topic to go to that section
Volume of a Prism Volume of a Cylinder Volume of a Pyramid Volume of a Cone Surface Area & Volume of Spheres Cavaleri's Principle Similar Solids PARCC Sample Questions
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Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of
MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
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Intro to 3-Dimensional Solids
Return to Table of Contents
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2-dimensional drawings use only the x and y axes
X Y Length width
Y X Length w i d t h Y X Length w i d t h
Intro to 3-D Solids
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Y X Z height height Y X
3-dimensional drawings include the x, y and z-axis. The z-axis is the third dimension. The third dimension is the height of the figure
Intro to 3-D Solids Slide 8 / 311
Y X Z height height Y X
x
Y
Intro to 3-D Solids Slide 9 / 311
Y X Z height
Y X
X Y
r
Intro to 3-D Solids Slide 10 / 311
To give a figure more of a 3-dimensional look, lines that are not visible from the angle the figure is being viewed are drawn as dashed line segments. These are called hidden lines.
Y X Z height height
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A Polyhedron (pl. Polyhedra) is a solid that is bounded by polygons, called faces. An edge is the line segment formed by the intersection of 2 faces. A vertex is a point where 3 or more edges meet Face Edge Vertex
Intro to 3-D Solids Slide 12 / 311
The 3-Dimensional Figures discussed in this unit are: Pyramids Cylinders Prisms
Intro to 3-D Solids
SLIDE 3
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The 3-Dimensional Figures discussed in this unit are:
. C
Cones: Spheres:
Intro to 3-D Solids Slide 14 / 311 Right Vs. Oblique
Right Right In Right Prisms & Cylinders, the bases are aligned directly above one another. The edges are perpendicular with both bases.
Slide 15 / 311 Right Vs. Oblique
In Oblique Prisms & Cylinders, the bases are not aligned directly above one another. The edges are not perpendicular with the bases.
Slide 16 / 311 Right Vs. Oblique
Right Oblique Right Oblique In Right Pyramids & Cones, the vertex is aligned directly above the center of the base. In Oblique Pyramids & Cones, the vertex is not aligned directly above the center of the base.
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Prisms have 2 congruent polygonal bases. The sides of a base are called base edges. The segments connecting corresponding vertices are lateral edges. A B C X Y Z In this diagram: There are 6 vertices: A, B, C, X, Y, & Z There are 2 bases: ABC & XYZ. There are 6 base edges: AB, BC, AC, XY, YZ, & XZ. There are 3 lateral edges: AX, BY, & CZ. This prism has a total of 9 edges.
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The polygons that make up the surface of the figure are called faces. The bases are a type of face and are parallel and congruent to each other. The lateral edges are the sides of the lateral faces. A B C X Y Z In this diagram: There are 2 bases: ABC & XYZ. There are 3 lateral faces: AXBY, BYCZ, & CZAX. This prism has a total of 5 faces.
Intro to 3-D Solids
SLIDE 4
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1
A AB B DE C FS D CP E FA F CD G NP H BC I DQ
A B C D E F M N P Q R S Choose all of the base edges.
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2 A AB B CD C ER D BN E DQ F QR G MS H AM I CP Choose all of the lateral edges. A B C D E F M N P Q R S
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3 Chooses all of the bases. A AFSM B FERS C EDQR D ABCDEF E CDQP F BCPN G MNPQRS H ABNM A B C D E F M N P Q R S
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4 Chooses all of the lateral faces. A AFSM B FERS C EDQR D ABCDEF E CDQP F BCPN G MNPQRS H ABNM A B C D E F M N P Q R S
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5 Chooses all of the faces. A AFSM B FERS C EDQR D ABCDEF E CDQP F BCPN G MNPQRS H ABNM A B C D E F M N P Q R S
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A pyramid has 1 base with vertices and the lateral edges go to a single vertex. A M N P R S Q This pyramid has: 6 lateral edges, 6 base edges, 12 edges (total) 7 vertices
Intro to 3-D Solids
SLIDE 5
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A pyramid has faces that are polygons: 1 base and triangles that are the lateral faces.
A M N P Q R S
This pyramid has: 6 lateral faces, 1 base, 7 faces (total)
Intro to 3-D Solids Slide 26 / 311
6 Choose all of the base edges. A VN B KN C VL D LM E VM F VK K L M N V G KL H NM
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7 Choose all of the lateral edges. A VN B KN C VL D LM E VM F VK G KL H NM K L M N V
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8 How many edges does the pyramid have? K L M N V
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9 Choose all of the lateral faces. A KNV B NMV C KLMN D VML E KLV K L M N V
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10 Choose all of the bases. A KNV B NMV C KLMN D VML E KLV K L M N V
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11 How many faces does the pyramid have? K L M N V
Slide 32 / 311 . . A B
A cylinder has 2 bases which are congruent circles. The lateral face is a rectangle wrapped around the circles. A & B are the bases
cylinder.
Intro to 3-D Solids
A cylinder can also be formed by rotating a rectangle about an axis. Click for sample animation
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A cone, like a pyramid, has one base which is a circle.
. N V
N is the base of the cone. V is the vertex of the cone.
Intro to 3-D Solids
A cone can also be formed by rotating a right triangle about
Click for sample animation
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A sphere is a 3-dimensional circle in that every point on the sphere is the same distance from the center.
. C
Similar to a circle, a sphere is named by its center
- point. Sphere C is the solid shown above.
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12 Which solids have 2 bases? A Prism B Pyramid C Cylinder D Cone E Sphere
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13 Which solid has one vertex? A Prism B Pyramid C Cylinder D Cone E Sphere
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14 Which solid has more base edges than lateral edges? A Prism B Pyramid C Cylinder D Cone E Sphere
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15 Which solid(s) have no vertices? A Prism B Pyramid C Cylinder D Cone E Sphere
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16 Which solid is formed when rotating an isosceles triangle about its altitude? A a prism B a cylinder C a pyramid D a cone E a sphere
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Euler's Theorem states that the number of faces (F), vertices (V), and edges (E) satisfy the formula F + V = E + 2 A B C X Y Z A M N P Q R S F = 5 V = 6 E = 9 5 + 6 = 9 + 2 11 = 11 F = 7 V = 7 E = 12 7 + 7 = 12 + 2 14 = 14
Intro to 3-D Solids Slide 41 / 311
Example: A solid has 12 faces, 2 decagons and 10 trapezoids. How many vertices does the solid have? V + F = E + 2 V + 12 = 30 + 2 V + 12 = 32 V = 20 On their own, the 2 decagons & 10 trapezoids have 2(10) + 10(4) = 60 edges. In a 3-D solid, each side is shared by 2 polygons. Therefore, the number of edges in the solid is 60/2 = 30.
Intro to 3-D Solids
click click click click
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Example: A solid has 9 faces, 1 octagon and 8 triangles. How many vertices does the solid have? V + F = E + 2 V + 9 = 16 + 2 V + 9 = 18 V = 9 What information do you have? 9 faces & the 2 types of faces
Intro to 3-D Solids
click click click click
What is the problem asking? Create an equation to represent the problem. How are the number of edges in the 2-D faces, related to the number of edges in the polyhedron? Write a number sentence to describe this situation. (1(8) + 8(3))/2 (8 + 24)/2 32/2 16 edges
click click click click click
SLIDE 8
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17 A solid has 10 faces, one of them being a nonagon and 9 triangles. How many vertices does it have? A 8 B 9 C 10 D 18
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18 A solid has 12 faces, all of them being pentagons. How many vertices does it have? A 30 B 20 C 15 D 10
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19 A solid has 8 faces, all of them being triangles. How many vertices does it have? A 24 B 12 C 8 D 6
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A cross-section is the locus of points of the intersection of a plane and a 3-D solid.
Cross-Section Intro to 3-D Solids Slide 47 / 311
Think about it as if the plane were a knife and you were cutting the shape, what would the cut look like?
Cross-Section
Circle Ellipse Parabola (with the inner section shaded)
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Cross-sections of a surface are a 2-dimensional figure. Cross-sections of a solid are a 2-dimensional figure and its interior. The top can be removed to see the cross section. (Try it out)
Cross-Section
SLIDE 9
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20 What is the locus of points (cross-section) of a cube and a plane perpendicular to the base and parallel to the non-intersecting sides? A square B rectangle C trapezoid D hexagon E rhombus F parallelogram G triangle H circle
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21 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane perpendicular to the base and parallel to the non- intersecting sides? A 72 sq inches B 144 sq inches C 187.06 sq inches D 203.65 sq inches 12 in.
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22 What is the locus of points of a cube and a plane that contains the diagonal of the base and is perpendicular to the base? A square B rectangle C trapezoid D hexagon E rhombus F parallelogram G triangle H circle
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23 If the length of each edge of the cube is 12 inches, what would be the area of the cross-section of the cube and a plane that contains the diagonal of the base and is perpendicular to the base? A 72 sq inches B 144 sq inches C 187.06 sq inches D 203.65 sq inches 12 in.
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24 What is the locus of points of a cube and a plane that contains the diagonal of the base but does not intersect the opposite base? A square B rectangle C trapezoid D hexagon E rhombus F parallelogram G triangle H circle
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25 What is the locus of points of a cube and a plane that intersects all of the faces? A square B rectangle C trapezoid D hexagon E rhombus F parallelogram G triangle H circle
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Views & Drawings of 3-D Solids
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Isometric drawings are drawings that look 3-D & are created on a grid
- f dots using 3 axes that intersect to form 120° & 60° angles.
Views & Drawings Slide 57 / 311
Example: Create an Isometric drawing of a cube.
Views & Drawings Slide 58 / 311
An Orthographic projection is a 2-D drawing that shows the different viewpoints of an object, usually from the front, top & side. Each drawing depends on your position relative to the figure. Front Side Top (from front)
Views & Drawings Slide 59 / 311
Consider these three people viewing a pyramid:
Views & Drawings Slide 60 / 311
Consider these three people viewing a pyramid: The orange person is standing in front of a face, so their view is a triangle.
SLIDE 11
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Consider these three people viewing a pyramid: The green person is standing in front of a lateral edge, so from their view they can see 2 faces.
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Consider these three people viewing a pyramid: The purple person is flying over and can see the four lateral faces.
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26 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a Trapezoid
A
(front)
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27 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a Trapezoid
A (top) Slide 65 / 311
28 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a Trapezoid
A (top)
right square prism
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29 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a trapezoid
A
(front) right square prism
SLIDE 12
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30 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a Trapezoid
A
(front)
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31 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a Trapezoid
A
(above)
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32 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a Trapezoid
A (above) Slide 70 / 311
33 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a Trapezoid
A
(front)
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34 Given the surface shown, what would be the view from point A? A a Rectangle B a Square C a Circle D a Pentagon E a Triangle F a Parallelogram G a Hexagon H a Trapezoid
A
sphere
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(Looking down from above) What would the view be like from each position?
Views & Drawings
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What would the view be like from each position? From A, how many columns of blocks are visible?
How tall is each column?
- first one is 4 high
- second & third columns
are each 2 blocks high
Click to reveal Click to reveal
Views & Drawings Slide 74 / 311 B
What would the view be like from each position? From B, how many columns of blocks are visible?
How tall is each column?
- left one is 3 high
- right one is 4 high
Click to reveal Click to reveal
Views & Drawings Slide 75 / 311 C
(Looking down from above) What would the view be like from each position? From C, how many columns of blocks are visible?
How tall is each column?
high
Click to reveal Click to reveal
Views & Drawings Slide 76 / 311
Front Side Above Draw the 3 views.
Side View Top View Front View Move for Answer
Views & Drawings Slide 77 / 311
Front Side Above Draw the 3 views. Above Front Side
Views & Drawings
Move for Answer
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Here are 3 views of a solid, draw a 3-dimensional representation. Top Front Side L R F
Views & Drawings
Move for Answer
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Here are 3 views of a solid, draw a 3-dimensional representation. Top F L R Side Front
Views & Drawings
Move for Answer
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Surface Area of a Prism
Return to Table of Contents
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A Net is a 2-dimensional shape that folds into a 3-dimensional figure. The Net shows all of the faces of the surface.
Net
6 6 4 6 4 12 4 Shown is the net of a right rectangular prism. 12 6 4
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The net shown is a right triangular prism. The lateral faces are
- rectangles. The bases are on opposite sides of the rectangles,
although they do not need to be on the same rectangle.
Net Slide 83 / 311
The nets shown are for the same right triangular prism.
Net Slide 84 / 311
Nets of oblique prisms have parallelograms as lateral faces.
Nets
SLIDE 15 Slide 85 / 311 Rectangular Prisms
cube w w w H H H ℓ ℓ ℓ
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Base Base height Base height Base A prism has 2 bases. The base of a rectangular prism is a rectangle. The height of the prism is the length between the two bases.
Rectangular Prisms Slide 87 / 311
The Surface Area of a figure is the total amount of area that is needed to cover the entire figure (e.g. the amount of wrapping paper required to wrap a gift). Area Area Area Area Area Area
Top Area Side Area Front Area Bottom Area Back Area Side Area
The Surface Area of a figure is the sum of the areas of each side of the figure.
Rectangular Prisms Slide 88 / 311 Finding the Surface Area of a Rectangular Prism
H w ℓ Area of the Top = ℓ x w Area of the Bottom = ℓ x w Area of the Front = ℓ x H Area of the Back = ℓ x H Area of Left Side = w x H Area of Right Side = w x H The Surface Area is the sum of all the areas S.A. = ℓw + ℓw + ℓH + ℓH + wH + wH S.A. = 2 ℓw + 2 ℓH + 2wH
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Example: Find the surface area of the prism 7 4 3 Area of Top & Bottom Area of Right & Left Area of Front & Back A = 7(4) = 28u2 A = 3(4) = 12 u2 A = 3(7) = 21 u2
Click Click Click
Total Surface Area = 2(28) + 2(12) + 2(21) = 56 + 24 + 42 = 122 units2
Click Click
Finding the Surface Area of a Rectangular Prism Slide 90 / 311
35 What is the total surface area, in square units? 4 5 9
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36 What is the total surface area, in square units? 8 8 8
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37 Troy wants to build a cube out of straws. The cube is to have a total surface area of 96 in2, what is the total length of the straws, in inches?
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S.A. = 2B + PH The Surface Area is the sum of the areas of the 2 Bases plus the Lateral Area (Perimeter of the base, P, times the height of the prism, H) The Lateral Area is the area of the Lateral Surface. The Lateral Surface is the part that wraps around the middle of the figure (in between the two bases).
Another Way of Looking at Surface Area
Lateral Surface B a s e B a s e
Base Base
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Base Base w H ℓ Another formula for Surface Area of a right prism: S.A. = 2B + PH B = Area of the base B = ℓw P = Perimeter of the base P = 2 ℓ + 2w H = Height of the prism S.A. = 2B + PH S.A. = 2 ℓw + (2 ℓ +2w)H S.A. = 2 ℓw + 2 ℓH + 2wH
Rectangular Prisms Slide 95 / 311
Base Base w H ℓ In the surface area formula, 2B is the sum of the area of the 2 bases. What does PH represent? The area of lateral faces or Lateral Area
Click
Rectangular Prisms
Another formula for Surface Area of a right prism: S.A. = 2B + PH B = Area of the base B = ℓw P = Perimeter of the base P = 2 ℓ + 2w H = Height of the prism
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38 If the base of the prism is 12 by 6, what is the lateral area, in sq ft? 12 ft 6 ft 4 ft
SLIDE 17
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39 The surface area of the rectangular prism is : A 24 sq ft B 144 sq ft C 288 sq ft D 48 sq ft E 72 sq ft 12 ft 6 ft 4 ft
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40 If 7 by 6 is base of the prism, what is the lateral area, in sq units? 7 9 6
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41 What is the total square units of the surface area? 7 9 6
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42 Find the value of y, if the lateral area is 144 sq units, and y by 6 is the base. y 6 8
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43 What is the value of the missing variable if the surface area is 350 sq. ft. A 7 ft B 8.3 ft C 12 ft D 15 ft x ft 5 ft 10 ft
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44 Sharon was invited to Maria's birthday party. For a present, she purchased an iHome (a clock radio for an iPod or iPhone) which is contained in a box that measures 7 inches in length, 5 inches in width, and 4 inches in height. How much wrapping paper does Sharon need to wrap Maria's present?
SLIDE 18 Slide 103 / 311 Other Prisms Slide 104 / 311
base base height base base height base base height base base height A Prism has 2 Bases The Base of a Prism matches the first word in the name of the prism. e.g. the Base of a Triangular Prism is a Triangle The Height of the Prism is the length between the two bases
Other Prisms Slide 105 / 311
The Surface Area of a figure is the total amount of Area that is needed to cover the entire figure (e.g. the amount of wrapping paper required to wrap a gift). The Surface Area of a figure is the sum of the areas of each side of the figure Area Area Area Area Area Area Area
Area
Area Area
Other Prisms
Triangular Prism Net of the Triangular Prism
Slide 106 / 311 Finding the Surface Area of a Right Prism
Surface Area: S.A. = 2B + PH B = Area of the triangular base = ½bh P = Perimeter of the triangular base = a + b + c H = Height of the prism Lateral Area = PH = (a + b + c)H The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the triangular bases. base base Prism's height a b c H P = a + b + c a c b c a Lateral Surface H h b B = ½ bh Note: The formula above will work for any right prism.
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Example: Find the lateral area and surface area of the right triangular prism. 10 6 11 Since it has a base that is a right triangle, we need to find the base of the triangle using Pythagorean Theorem. 62 + b2 = 102 36 + b2 = 100 b2 = 64 b = 8 units Next, calculate the perimeter of your base. P = 6 + 8 + 10 = 24 units Use this to find the Lateral Area LA = PH = 24(11) = 264 units
2
Other Prisms Slide 108 / 311
10 6 11 Example: Find the lateral area and surface area of the right triangular prism. Then, calculate the area of your base, B B = (1/2)(8)(6) = 24 units2 Finally, calculate your Surface Area. SA = 2B + PH SA = 2(24) + (24)(11) SA = 48 + 264 = 312 units2
Other Prisms
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Example: Find the lateral area and surface area of the triangular prism. 9 9 9 12 Since it has a base that is an equilateral triangle, we need to find the height of the triangle using Pythagorean Theorem or the 30-60-90 Triangle Theorem. 4.52 + b2 = 92 20.25 + b2 = 81 b2 = 60.75 b = 4.5√3 units = 7.79 units Next, calculate the perimeter of your base. P = 9 + 9 + 9 = 27 units Use this to find the Lateral Area LA = PH = 27(12) = 324 units2
Other Prisms Slide 110 / 311
Example: Find the lateral area and surface area of the triangular prism. Then, calculate the area of your base, B B = (1/2)(9)(4.5√3) = 20.25√3 units2 = 35.07 units2 Finally, calculate your Surface Area. SA = 2B + PH SA = 2(35.07) + (27)(12) SA = 70.14 + 324 = 394.14 units2
Other Prisms
9 9 9 12
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45 The height of the triangular prism below is 11 ft, the height of the base is 3 ft, and the triangular base is an isosceles triangle. Find the surface area. A 88 sq ft B 132 sq ft C 198 sq ft D 222 sq ft
3 ft
5 ft 11 ft
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46 The height of the triangular prism below is 3, and the triangular base is an equilateral triangle. Find the surface area. A 64 sq ft B 127.43 sq ft C 72 sq ft D 55.43 sq ft 8 ft 3 ft
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47 Find the lateral area of the right prism. 5 5 6
Slide 114 / 311 Finding the Surface Area of a Right Prism
Surface Area : S.A. = 2B + PH B = Area of the regular hexagonal base = ½aP
- a is the apothem of the regular base
P = Perimeter of the base = b + c + d + e + f + g H = Height of the prism = H Lateral Area = PH = (b + c + d + e + f + g)H a B = ½ aP
g c b H e f c d e f b d P = b + c + d + e + f + g
base base Prism's height
SLIDE 20 Slide 115 / 311
a B = ½ aP
Finding the Surface Area of a Right Prism
P = b + c + d + e + f + g
base base Prism's height
g c b H e f c d e f b d
The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the prism between the triangular bases.
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8 in 7 in 30° 4 in. a Example: Find the lateral area and surface area of the regular hexagonal prism. Because the base is a regular polygon, we need to calculate the apothem. To begin, figure out the central angle & top angle in the triangle. = 60° = central angle = 30° = top angle of the triangle. 360 6 60 2
Click Click Click
Other Prisms Slide 117 / 311
Example: Find the lateral area and surface area of the regular hexagonal prism. Next find the apothem using trigonometry, or special right triangles (if it applies). tan 30 = atan30 = 4 tan30 tan30 4 a a = 4√3 = 6.93 in.
Click Click
Other Prisms
Click
8 in 7 in
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B = (1/2)aP = (1/2)(4√3)(48) = 96√3 in2 = 166.28in2 Example: Find the lateral area and surface area of the regular hexagonal prism. Next, calculate the perimeter of your base. P = 8(6) = 48 in Use this to find the Lateral Area LA = PH = 48(7) = 336 in2 Then, calculate the area of your base, B Finally, calculate your Surface Area. SA = 2B + PH SA = 2(166.28) + (48)(7) SA = 332.56 + 336 = 668.56 in2
Click Click Click Click Click Click Click Click Click
Other Prisms
8 in 7 in
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36° 3 in. a
Example: Find the lateral area and surface area of the right prism. The base is a regular pentagon. 6 ft 10 ft Because the base is a regular polygon, we need to calculate the apothem. To begin, figure out the central angle & top angle in the triangle. = 72° = central angle = 36° = top angle of the triangle. 360 5 72 2
Other Prisms
Click Click Click
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Example: Find the lateral area and surface area of the right prism. The base is a regular pentagon. Next find the apothem using trigonometry, or special right triangles (if it applies). tan 36 = 3 a atan36 = 3 tan36 tan36 a = 4.13 in. 6 ft 10 ft
Other Prisms
Click Click Click
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Example: Find the lateral area and surface area of the right prism. The base is a regular pentagon. Next, calculate the perimeter of your base. P = 5(6) = 30 in Use this to find the Lateral Area LA = PH = 30(10) = 300 in2 Then, calculate the area of your base, B B = (1/2)aP = (1/2)(4.13)(30) = 61.95 in2 Finally, calculate your Surface Area. SA = 2B + PH SA = 2(61.95) + (30)(10) SA = 123.9 + 300 = 423.9 in2 6 ft 10 ft
Other Prisms
Click Click Click Click Click Click Click Click
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Example: Find the lateral area and surface area of the right prism. 8 3 7 6 5
Angles are right angles.
First, calculate the perimeter of your base. P = 8 + 7 + 5 + 4 + 3 + 3 P = 30 units Use this to find the Lateral Area LA = PH = 30(6) = 180 units2
Other Prisms
Then, calculate the area of your base, B B = 7(5)+3(3) = 44 units2 Finally, calculate your Surface Area. SA = 2B + PH SA = 2(44) + (30)(6) SA = 88 + 180 = 268 units2
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48 Find the lateral area of the right prism. 8 11 The base is a regular hexagon.
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49 Find the total surface area of the right prism. The base is a regular hexagon. 8 11
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50 Find the total surface area of the right prism. 4 4 3 2 10 9 All angles are right angles.
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y 5 6 51 The right triangular prism has a surface area of 150 sq
- ft. Find the height of the prism.
A 5 ft B 6 ft C 7.81 ft D 6.38 ft
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Surface Area of a Cylinder
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height radius base base height radius base base
Cylinders
A Cylinder is a solid w/ 2 circular bases that lie in || planes. Because each base is a circle, it contains a radius. The remaining measurement that connects the 2 bases is the height of the cylinder.
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8
radius
The net of a right cylinder is two circles and a rectangle that forms the lateral surface. 8 x What is the length of x?
circle (base)
radius
Click to reveal
Cylinders Slide 130 / 311
Base Base height Base height Lateral Surface Base
Finding the Surface Area of a Right Cylinder
Surface Area : S.A. = 2B + PH B = Area of the circular base = πr2 C = Perimeter of the Circular base (Circumference) = 2πr H = Height of the prism Lateral Area = CH = 2πrH
Slide 131 / 311
Base Base height Base height Lateral Surface Base
Finding the Surface Area of a Right Cylinder
The Lateral Area is the area of the Lateral Surface, the rectangular area that wraps around the cylinder between the circular bases. Therefore, the Surface Area of a Cylinder can be simplified to the equation below. SA = 2πr2 + 2πrH
Slide 132 / 311
8 r = 4 Example: Find the lateral area and surface area of the right cylinder. LA = 2πrh LA = 2π(4)(8) LA = 64π units2 LA = 201.06 units2 SA = 2πr2 + 2πrh SA = 2π(4)2 + 2π(4)(8) SA = 32π + 64π SA = 96π units2 SA = 301.59 units2
Finding the Surface Area of a Right Cylinder
SLIDE 23 Slide 133 / 311
34 d = 16 SA = 2πr2 + 2πrh SA = 2π(8)2 + 2π(8)(30) SA = 128π + 480π SA = 608π units2 SA = 1,910.09 units2 Example: Find the lateral area and surface area of the right cylinder. LA = 2πrh LA = 2π(8)(30) LA = 480π units2 LA = 1507.96 units2 162 + h2 = 342 256 + h2 = 1156 h2 = 900 h = 30 Note: 16-30-34 = 2(8-15-17) Pyth. Triple
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Cylinders
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Slide 134 / 311
Example: Find the lateral area and surface area
- f the right cylinder when the base
circumference is 16π ft & the height is 10 ft. SA = 2πr2 + 2πrh SA = 2π(8)2 + 2π(8)(10) SA = 128π + 160π SA = 288π ft2 SA = 904.78 ft
2
LA = 2πrh LA = 2π(8)(10) LA = 160π ft2 LA = 502.64 ft
2
C = 2πr 16π = 2πr 2π 2π 8 ft = r
Cylinders
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h = 12 r = 7 52 Find the lateral area of the right cylinder.
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h = 12 r = 7 53 Find the surface area of the right cylinder. Use 3.14 as your value of π & round to two decimal places. A 1200 sq in. B 307.72 sq in. C 835.24 sq in. D 1670.48 sq in.
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54 Find the lateral area of the right cylinder. 13 r = 5
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h = 12 55 Find the lateral area of the right cylinder. Base area is 36π units2
SLIDE 24 Slide 139 / 311
h = 12 56 Find the surface area of the right cylinder. Base area is 36π units2
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r = 8 in. h 57 The surface area of the right cylinder is 653.12 sq in. Find the height of the cylinder. Use 3.14 as your value
A 7 in. B 8 in. C 5 in. D 6 in.
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58 A food company packages soup in aluminum cans that have a diameter of 2 1/2 inches and a height of 4 inches. Before shipping the cans off to the stores, they add their company label to the can which does not cover the top and bottom. If the company is shipping 200 cans of soup to one store, how much paper material is required to make the labels?
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59 Maria's mom baked a cake for her daughter's birthday
- party. The diameter of the cake is 9 inches and the
height is 2 inches. How much base frosting (pink in the picture below) was required to cover the cake?
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Surface Area of a Pyramid
Return to Table of Contents
Slide 144 / 311
A Pyramid is a polyhedron in which the base is a polygon & the lateral faces are triangles with a common vertex. Lateral Edges are the intersection
Vertex Lateral Face Lateral Edge Base
Pyramids
SLIDE 25 Slide 145 / 311 Net
This is a right square pyramid. Another name for it is pentahedron. Hedron is a suffix that means face. Why is this a pentahedron?
Slide 146 / 311
Slant Height
ℓ
The Pyramid has a square base and 4 triangular faces The triangular faces are all isosceles triangles if its a right pyramid. The Height of each triangular face is the Slant Height of the pyramid if it is a regular pyramid (labeled as , or a cursive lower case L).
Surface Area = Sum of the Areas of all the sides ℓ
Height
Triangle
Slide 147 / 311
Square Base (B) Slant Height ( )
ℓ
Pyramid's Height (h)
Segment Lengths in a Pyramid Slide 148 / 311
Example: Find the value of x. a2 + 122 = 132 a2 + 144 = 169 a2 = 25 a = 5 Note: 5-12-13 Right Triangle Therefore x = 2(5) = 10 x 13 12
Segment Lengths in a Pyramid Slide 149 / 311
Example: Find the value of x. Base Area of the right square pyramid is 64 u2. x 8 Square Base has an area
64 = y2 y = 8, so a = 4 of the right triangle. 42 + 82 = x2 16 + 64 = x2 x2 = 80 x = 8.94 units
Segment Lengths in a Pyramid Slide 150 / 311
Example: Find the length of the slant height.
r
This is a regular hexagonal pyramid. r = 6 lateral edge = 12
Segment Lengths in a Pyramid
SLIDE 26 Slide 151 / 311
First, find the height of the pyramid using Pythagorean Theorem. h 12 6 62 + h2 = 122 36 + h2 = 144 h2 = 108 h = 6√3 = 10.39 Note: 30-60-90 triangle
r Segment Lengths in a Pyramid
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Slide 152 / 311
Second, find the apothem of the hexagonal base. a 6 6 3 3 32 + a2 = 62 9 + a2 = 36 a2 = 27 a = 3√3 = 5.20 Note: 30-60-90 triangle = 60° = central Note: equilateral = 30° = top of the . 360 6 60 2 r
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Segment Lengths in a Pyramid Slide 153 / 311
(3√3)2 + (6√3)2 = 2 27 + 108 = 2 2 = 135 = 3√15 = 11.62
ℓ ℓ ℓ ℓ
Last, find the slant height of your pyramid w/ the apothem & height. a = 3√3
ℓ
h = 6√3
r
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Segment Lengths in a Pyramid Slide 154 / 311
60 Find the value of the variable. 16 x 6
Slide 155 / 311
61 Find the value of the variable. 12 11 x
Slide 156 / 311
62 Find the value of the variable. x 6
area of the base is 36 u2
SLIDE 27 Slide 157 / 311
63 Find the value of the slant height.
r r = 8 lateral edge = 17 Regular Hexagonal Pyramid
Slide 158 / 311
64 Find the value of the slant height.
a a = 9 lateral edge = 12 Regular Hexagonal Pyramid
Slide 159 / 311
Square Base (B) Slant Height ( ) Pyramid's Height (h)
ℓ
Surface Area = B + ½P and Lateral Area = ½P = Slant Height P = Perimeter of Base B = Area of Base
Surface Area of a Regular Pyramid
ℓ ℓ
ℓ Slide 160 / 311
Square Base (B) Slant Height ( ) Pyramid's Height (h) Why is the Surface Area SA = B + P ? 1 2 Surface Area is the sum of all of the areas that make up the solid. In
- ur diagram, these are 4 triangles & 1 square.
Asquare = s s = s2 = B A∆ = s 1 2
ℓ ℓ ℓ
Surface Area of a Regular Pyramid Slide 161 / 311
Why is the Surface Area SA = B + P ? 1 2 Since there are 4 ∆s, we can multiply the area of each ∆ by 4. Therefore, our Surface Area for the Pyramid above is SA = s
2 + 4(1/2)s
SA = s
2 + (1/2)(4s)
SA = B +
1/2 P
s
ℓ
Net of Pyramid
ℓℓ ℓ
Surface Area of a Regular Pyramid
ℓ
Slide 162 / 311
ℓ = 7 s = 6
Example: Find the lateral area and the surface area of the pyramid. LA =
1/2 P ℓ
LA =
1/2 (24)(7)
LA = 12(7) LA = 84 units
2
SA = B +
1/2 P ℓ
SA = 6
2 + 1/2 (24)(7)
SA = 36 + 84 SA = 120 units
2
Surface Area of a Regular Pyramid
SLIDE 28 Slide 163 / 311
Example: Find the lateral area and the surface area of the pyramid. First, calculate the slant height. 32 + 82 = ℓ 2 9 + 64 = ℓ2 73 = ℓ2 ℓ = 8.54 Next, calculate the LA & SA LA = 1/2 P ℓ LA = 1/2 (24)(8.54) LA = 12(8.54) LA = 102.48 units2 SA = B + 1/2 P ℓ SA = 62 + 1/2 (24)(8.54) SA = 36 + 102.48 SA = 138.48 units2
h = 8 s = 6
Surface Area of a Regular Pyramid Slide 164 / 311
Example: Find the lateral area and the surface area of the pyramid. 10 8
ℓ
First, calculate the slant height. 82 + ℓ 2 = 102 64 + ℓ 2 = 100 ℓ 2 = 36 ℓ = 6 s = 16 e = 10
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Surface Area of a Regular Pyramid Slide 165 / 311
Example: Find the lateral area and the surface area of the pyramid. LA = 1/2 P ℓ LA = 1/2 (64)(6) LA = 32(6) LA = 192 units2 SA = B + 1/2 P ℓ SA = 162 + 1/2 (64)(6) SA = 256 + 192 SA = 448 units2 Next, calculate the LA & SA
click click click click click click click click
s = 16 e = 10
Surface Area of a Regular Pyramid Slide 166 / 311
Example: Find the lateral area and the surface area of the pyramid.
a
a = 4 lateral edge = 8 Regular Pentagonal Pyramid
Surface Area of a Regular Pyramid Slide 167 / 311
72° 36°36°
4
x r Example: Find the lateral area and the surface area of the pyramid. First, find the radius & side length of the regular pentagon using the apothem & trigonometric ratios tan36 = x = 4tan36 = 2.91 Therefore, s = 2(2.91) = 5.82 = 36° = top
360 5 72 2 x 4 4 r cos36 = rcos36 = 4 cos36 cos36 r = 4.94 = 72° = central
Click Click Click Click Click Click Click Click
Click Click Click
Surface Area of a Regular Pyramid Slide 168 / 311
Next, find the slant height of the pyramid using the lateral edge, the value of x from the previous slide & Pythagorean Theorem.
8 2.91
2.912 + ℓ 2 = 82 8.4681 + ℓ 2 = 64 ℓ 2 = 55.5319 ℓ = 7.45
ℓ
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Surface Area of a Regular Pyramid
SLIDE 29 Slide 169 / 311
Last, find the lateral area & surface area of the pyramid. LA =
1/2 P ℓ
LA =
1/2 (29.1)(7.45)
LA = 108.40 units
2
SA = B +
1/2 P ℓ
SA =
1/2 (4)(29.1) + 1/2 (29.1)(7.45)
SA = 58.2 + 108.40 SA = 166.6 units
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Surface Area of a Regular Pyramid Slide 170 / 311
65 Find the lateral area of the right pyramid.
s = 10
ℓ = 9
Slide 171 / 311
66 Find the surface area of the right pyramid.
s = 10 ℓ = 9
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67 Find the lateral area of the right pyramid. base e = 10 area = 16
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68 Find the surface area of the right pyramid. base e = 10 area = 16
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a
a = 5 h = 12 Regular Octagonal Pyramid 69 Find the lateral area of the right pyramid.
SLIDE 30 Slide 175 / 311
a
a = 5 h = 12 Regular Octagonal Pyramid 70 Find the surface area of the right pyramid.
Slide 176 / 311
Hint: The pyramid is NOT regular. So, B + 1/2 P ℓ doesn't work. Instead, draw a net of the pyramid & find each area.
71 Find the surface area of the right pyramid.
30 12 8
Hint
Slide 177 / 311
Surface Area of a Cone
Return to Table of Contents
Slide 178 / 311
r height S l a n t H e i g h t
ℓ
Lateral Surface Slant Height
ℓ
Base The Base of the cone is a circle The length of the circular portion of the Lateral Surface is the same as the Circumference of the Circlular Base. The Slant Height is the length of the diagonal slant of the cone from the top to the edge of the base. The Height of the cone is the length from the top to the center of the circular base.
Cones Slide 179 / 311
Surface Area = Area of the Base + Lateral Area Lateral Area= ½P ℓ S.A. = B + ½P ℓ ℓ = Slant Height P = Perimeter of Circular Base B = Area of Circular Base Because the base is a circle. P = Circumference = 2πr L.A. = ½(2πr) ℓ
= πr ℓ
S.A. = πr2 + πr ℓ
Finding the Surface Area of a Right Cone
Lateral Surface Slant Height
ℓ
Base
Slide 180 / 311
LA = πr ℓ = π(6)(8) LA = 48π units2 LA = 150.80 units2 SA = πr2 + πr ℓ = π(6)2 + π(6)(8) = 36π + 48π SA = 84π units2 SA = 263.89 units2 Example: Find the lateral area and surface area of the right cone. = 8 r = 6
ℓ
click
Cones
click click click click click click click click
SLIDE 31 Slide 181 / 311
Example: Find the lateral area and surface area of the right cone. h = 8 C = 12π units C = 2πr 12π = 2πr 2π 2π 6 units = r 62 + 82 = ℓ2 36 + 64 = ℓ2 100 = ℓ2 10 units = ℓ
Cones
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Slide 182 / 311
Example: Find the lateral area and surface area of the right cone. h = 8 C = 12π units SA = πr2 + πr ℓ = π(6)2 + π(6)(10) = 36π + 60π SA = 96π units2 SA = 301.59 units2 LA = πr ℓ = π(6)(10) LA = 60π units2 LA = 188.50 units2
Cones
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72 Find the lateral area of the right cone, in square units. r = 4
ℓ = 9 Slide 184 / 311
r = 4
ℓ = 9
73 Find the surface area of the right cone, in square units.
Slide 185 / 311
74 Find the lateral area of the right cone, in square units. h = 9 Base Area = 16π units2
Slide 186 / 311
75 Find the surface area of the right cone, in square units. h = 9 Base Area = 16π units2
SLIDE 32
Slide 187 / 311
76 Find the length of the radius of the right cone if the lateral area is 50π units2?
ℓ = 10 Slide 188 / 311 ℓ = 10
77 Find the height of the right cone if the lateral area is 50π units2?
Slide 189 / 311
78 Find the slant height of the right cone if the surface area is 45π units2 and the diameter of the base is 6 units?
Slide 190 / 311
79 Find the height of the right cone if the surface area is 45π units2 and the diameter of the base is 6 units?
Slide 191 / 311
80 The Department of Transportation keeps 4 piles of road salt for snowy days. Each conical shaped pile is 20 feet high and 30 feet across at the base. During the summer the piles are covered with tarps to prevent erosion. How much tarp is needed to cover the conical shaped piles so that no part of them are exposed?
Slide 192 / 311
Volume of a Prism
Return to Table of Contents
SLIDE 33 Slide 193 / 311
The volume of a solid is the amount of cubic units that a solid can hold. Where area used square units, volume will use cubic units.
Prisms Slide 194 / 311
Base height Base ℓ w H V = BH Specific Prisms Rectangular Prism: V = ℓwH Cube: V = s3
Finding the Volume of a Prism
Prisms Slide 195 / 311
Does a prism need to be a right prism for the volume formula to work? Think of a ream of paper Stacked nicely it has 500 sheets. If the stack is fanned, it still has 500 sheets. So the volume doesn't change if the prism, stack of paper, is right or oblique. The formula V = BH works for all prisms.
Prisms Slide 196 / 311
Example: Find the volume of the rectangular prism with a length of 2, a width of 6, and a height of 5. V = ℓ w H V = 2(6)(5) V = 60 units3
Prisms Slide 197 / 311
Example: The volume of a box is 48 ft3. If the height is 4 ft and width is 6 ft, what is the length? V = ℓ w H 48 = ℓ(6)(4) 48 = 24 ℓ 24 24 2 ft = ℓ
Prisms Slide 198 / 311
Example: Find the volume of the prism shown below. 10 6 11 Since it has a base that is a right triangle, we need to find the base
- f the triangle using Pythagorean Theorem.
62 + b2 = 102 36 + b2 = 100 b2 = 64 b = 8 units
Prisms
Next, calculate the area of your base, B B = (1/2)(8)(6) = 24 units2 Finally, calculate your Volume. V = BH V = 24(11) V = 264 units3
SLIDE 34 Slide 199 / 311
Example: The volume of a cube is 64 m3, what is area of one face? V = s3 64 = s3 4 m = s Area of one face A = 4(4) A = 16 m
2
Prisms Slide 200 / 311
4 in 7 in 30° 4 in. x in. Because the base is a regular polygon, we need to calculate the side length. To begin, figure out the central angle & top angle in the triangle. = 60° = central angle = 30° = top angle of the triangle. 360 6 60 2
Click Click Click
Example: Find the volume of the prism with a height 7 in. and hexagon base with an apothem of 4 in.
Prisms Slide 201 / 311
4 in 7 in Example: Find the volume of the prism with a height 7 in. and hexagon base with an apothem of 4 in. Then, calculate the side length of your base. s = 2(2.31) = 4.62 in Next, find the value of x using trigonometry, or special right triangles (if it applies). tan 30 = 4tan30 = x x = 4√3 = 2.31 in. 3 x 4
Prisms
30° 4 in. x in.
Click Click Click Click
Slide 202 / 311
4 in 7 in Example: Find the volume of the prism with a height 7 in. and hexagon base with an apothem of 4 in. Next, use your value of s to find the Perimeter of your base P = 6(4.62) = 27.72 in
Prisms
Click
Then, calculate the area of your base, B B = (1/2)aP = (1/2)(2.31)(27.72) = 32.02 in2 Finally, calculate your Volume. V = Bh V = 32.02(7) V = 224.14 in3
Click Click Click Click Click Click
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81 What is the volume of a rectangular prism with edges of 4, 5, and 7?
Slide 204 / 311
82 What is the volume of a cube with edges of 5 units?
SLIDE 35
Slide 205 / 311
83 If the volume of a rectangular prism is 64 u3 and has height 8 and width 4, what is the length?
Slide 206 / 311
84 If a cube has volume 27 u3, what is the cubes surface area?
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85 Find the volume of the prism. 15 12 20
Slide 208 / 311
86 Find the volume of the prism. 7 2 6 6 6
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87 Find the volume of the prism. 8 11 The base is a regular hexagon.
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88 A high school has a pool that is 25 yards in length, 60 feet in width, and contains the depth dimensions shown in the figure below. If one cubic yard is about 201.974 gallons, how much water is required to fill the pool? Shallow end Deep end 3 ft 9 ft 2 yds 4 yds 19 yds
SLIDE 36 Slide 211 / 311
Volume of a Cylinder
Return to Table of Contents
Slide 212 / 311
base base height r r
Finding the Volume of a Cylinder
V = Bh V = πr2h
Slide 213 / 311
Example: Find the volume of the cylinder with a radius of 4 and a height of 11. V = π(4)2 (11) V = 176π units3 V = 552.92 units3
Cylinders Slide 214 / 311
Example: The surface area of a cylinder is 96π units2, and its radius is 4 units. What is the volume? V = π(4)2 (8) V = 128π units3 V = 402.12 units3 SA = 2πr2 + 2πrh 96π = 2π(4)2 + 2π(4)h 96π = 32π + 8πh
64π = 8πh 8π 8π h = 8 units
Cylinders Slide 215 / 311
89 Find the volume of the cylinder with radius 6 and height 8.
Slide 216 / 311
90 Find the volume of the cylinder with a circumference of 18π units and a height of 6.
SLIDE 37
Slide 217 / 311
r = 8 h
91 Find the volume of the cylinder with a surface area of 653.12 u2 & a radius of 8 units. Use 3.14 as your value of π.
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92 The volume of a cylinder is 108π u3, and the height is 12 units. What is the surface area?
Slide 219 / 311
93 The height of a cylinder doubles, what happens to the volume? A Doubles B Quadruples C Depends on the cylinder D Cannot be determined
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94 The radius of a cylinder doubles, what happens to the volume? A Doubles B Quadruples C Depends on the cylinder D Cannot be determined
Slide 221 / 311
24" 4" 3" 95 A 3" hole is drilled through a solid cylinder with a diameter of 4" forming a tube. What is the volume of the tube?
Slide 222 / 311
Volume of a Pyramid
Return to Table of Contents
SLIDE 38 Slide 223 / 311 Finding the Volume of a Pyramid
V = 1/3 Bh
Square Base (B) Slant Height ( ) Pyramid's Height (h)
ℓ
Slide 224 / 311
Example: Find the volume of the pyramid. 5 4 6 V = 1/3 Bh B = 5(4) = 20 h = 6 units V = 1/3 (20)(6) V = 40 units3
Volume of Pyramids Slide 225 / 311
Example: Find the volume of the pyramid. 8 8 5 8 8 5 4 h
click for extra diagram
Volume of Pyramids Slide 226 / 311
96 Find the volume of the pyramid. 7 6 5
Slide 227 / 311
97 Find the volume of the pyramid. 6 6 8
Slide 228 / 311
98 Find the volume of the pyramid. 12 12 10
SLIDE 39 Slide 229 / 311
Example: Find the volume of the pyramid. a a = 4 lateral edge = 8 Regular Pentagonal Pyramid First, find the side length of the regular pentagon using the apothem & trigonometric ratios. = 72° = central = 36° = top angle of the . 72 2 360 5 tan36 = x = 4tan36 = 2.91 Therefore, s = 2(2.91) = 5.82 x 4
Volume of Pyramids
Click Click Click Click Click
Slide 230 / 311
Example: Find the volume of the pyramid. Next, find the slant height of the pyramid using the lateral edge, the value of x from the previous slide & Pyth. Theorem. 8 2.91
ℓ
2.912 + ℓ 2 = 82 8.4681 + ℓ 2 = 64 ℓ 2 = 55.5319 ℓ = 7.45 Then, use the slant height & apothem w/
find the height. 7 . 4 5 4 h
C l i c k
42 + h2 = 7.452 16 + h2 = 55.5319 h2 = 39.5319 h = 6.29
Volume of Pyramids
Click Click Click Click Click Click Click Click
a a = 4 lateral edge = 8 Regular Pentagonal Pyramid
Click
Slide 231 / 311
Example: Find the volume of the pyramid. Last, find the Area of your Base & Volume. B = 1/2 aP B = 1/2 (4)(29.1) B = 58.2 units2 V = 1/3 Bh V = 1/3 (58.2)(6.29) V = 122.03 units3
Volume of Pyramids
Click Click Click Click Click Click
a a = 4 lateral edge = 8 Regular Pentagonal Pyramid
Slide 232 / 311
99 Find the volume of the right pyramid.
a
a = 5 h = 12 Regular Octagonal Pyramid
Slide 233 / 311
100 Find the volume of the right pyramid. 8 11 The base is a regular hexagon.
Slide 234 / 311
A truncated pyramid is a pyramid with its top cutoff parallel to its base. Find the volume of the truncated pyramid shown. 2 2 6 6 9 3 Vtruncated = Vbig - Vsmall Bbig = 6(6) = 36 hbig = 3 + 9 = 12 Vbig = 1/3 (36)(12) Vbig = 144 units3 Bsmall = 2(2) = 4 hsmall = 3 Vsmall = 1/3 (4)(3) Vsmall = 4 units3 Vtruncated = 144 - 4 Vtruncated = 140 units3
Volume of Pyramids
SLIDE 40
Slide 235 / 311
101 Find the volume of the truncated pyramid.
2 2 8 8 12 3
Slide 236 / 311
102 The table shows the approximate measurements of the Red Pyramid in Egypt and the Great Pyramid of Cholula in Mexico. Approximately, what is the difference between the volume of the Red Pyramid and the volume of the Great Pyramid of Cholula? A 6,132,867 cubic meters B 4,455,000 cubic meters C 2,777,133 cubic meters D 1,677,867 cubic meters Length Width Height Red Pyramid 220 m 220m 104 m Great Pyramid of Cholula 450 m 450 m 66 m
Answer
Slide 237 / 311
103 Salt water comes in cylindrical containers that measure 10 feet high and have a diameter of 8 feet. Determine the height of the aquarium that should be used in the design. Show that your design will be able to store at least 3 cylindrical containers of water. When you finish, enter your value for h1 into your SMART Responder. The Geometryville Aquarium is building a new tank space for coral reef fish shown in the figure below. The laws say that the dimensions of the tank must have a maximum length of 14 feet, a maximum width of 10 feet and a maximum height of 16 feet. w h1 h2
ℓ Slide 238 / 311
Volume of a Cone
Return to Table of Contents
Slide 239 / 311
r height S l a n t H e i g h t
ℓ Finding the Volume of a Cone
V = 1/3 Bh V = 1/3πr2 h
Slide 240 / 311
Example: Find the volume of the cone. 9 7 V = 1/3 πr2 h V = 1/3 π(7)2 (9) V = 147π units3 V = 461.81 units3
Volume of a Cone
SLIDE 41 Slide 241 / 311
Example: Find the volume of the cone. 12 4 V = 1/3 πr2 h V = 1/3 π(4)2 (8.94) V = 47.68π units3 V = 149.79 units3 r = 4, so d = 8 With the right triangle, use Pythagorean Theorem to find the height of the pyramid. h2 + 82 = 122 h2 + 64 = 144 h2 = 80, h = √80 = 8.94
Volume of a Cone Slide 242 / 311
Example: Find the volume of the cone, with lateral area of 15π units2 and a slant height 5 units. LA = πr ℓ 15π = πr(5) 15π = 5πr 5π 5π 3 units = r 1) You know the Lateral area & slant height, so use the Lateral Area formula to calculate the radius.
Volume of a Cone
Click Click Click Click Click
Slide 243 / 311
h2 + 32 = 52 h2 + 9 = 25 h2 = 16 h = 4 Note: 3-4-5 Pyth. Triple 2) Next, use the slant height & radius to calculate the height of the cone using Pythagorean Theorem.
Volume of a Cone
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Slide 244 / 311
V = 1/3 πr2 h V = 1/3 π(3)2 (4) V = 12π units3 V = 37.70 units3 3) Last, calculate the volume of the cone.
Volume of a Cone
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104 What is the volume of the cone? 8 d = 10
Slide 246 / 311
105 What is the volume of the cone? r = 4 = 9 ℓ
SLIDE 42 Slide 247 / 311
106 What is the volume of the cone? 10 40°
Slide 248 / 311
107 What is the volume of the truncated cone? r = 8 r = 4 6 6
Slide 249 / 311
Surface Area & Volume of Spheres
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Slide 250 / 311
Recall the Definition of a Circle The locus of points in a plane that are the same distance from a point called the center of the circle.
X
Y
Every point on the above circle is the same distance from the origin in the x, y plane.
Y X
Spheres Slide 251 / 311
The locus of points in space that are the same distance from a point.
Y X Z
Every point on the sphere above on the left side, is the same distance from the origin in space, the x, y, z plane.
X Y
Y X
Spheres Slide 252 / 311
Y X Z
The Great Circle of a sphere is found at the intersection
- f a plane and a sphere when the plane contains the center
- f the sphere.
Spheres
SLIDE 43 Slide 253 / 311
Y X Z
Great Circles
Each of these planes intersects the sphere, and the plane contains the center of the sphere
Slide 254 / 311
International Date Line
Great Circles
The Earth has 2 Great Circles: Can you name them?
Click to reveal picture
The Equator The Prime Meridian w/ the International Date Line
click click
Slide 255 / 311 Great Circle
The Great Circle separates the Sphere into two equal halves at the center of the sphere.
Slide 256 / 311 Each half is called a Hemisphere Slide 257 / 311 Cross Sections
A Cross Section is found by the intersection of a plane and a solid.
Cross - Section
(Click the top hemisphere to see the cross section.)
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.
small circles great circle The farther the cross section of the sphere is taken from its center the smaller the circle.
Cross Sections
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8 2 8 r
Example: Find the radius of the cross section of the sphere that has a radius of 8 if the cross section is 2 from the center. 22 + r2 = 82 4 + r2 = 64 r2 = 60 r = √60 = 2√15 = 7.75
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4 Example: A cross section of a sphere is 4 units from the center of the sphere and has an area of 16π units2. What is area of the great circle? Leave your answer in terms of π. 16π = πr2 r = 4 units in the cross section 42 + 42 = r2 32 = r2 r =√32 = 4√2 = 2.83 = radius of sphere A = π(√32)2 A = 32π units2
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108 What is the area of the cross section of a sphere that is 6 units from the center of the sphere if the sphere has radius 8 units?
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109 What is the area of the great circle if a cross section that is 3 from the center has a circumference of 10π?
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110 The circumference of the great circle of a sphere is 12π units and a cross section has a circumference of 8π units. How far is the cross section from the center?
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r S.A. = 4πr2
Finding the Surface Area of the Sphere
Why is there no formula for lateral area? A sphere doesn't have any bases, so the lateral area is the same as the surface area.
Click to reveal
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r V = πr3 4 3
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Example: Find the surface area & volume of a sphere with radius of 6 ft. SA = 4π(6)2 SA = 144π units2 SA = 452.39 units2 V = π(6)3 V = 288π units3 V = 904.78 units3 4 3
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Example: Find the surface area & volume of a sphere that a great circle with area 24π units2? SA = 4π(4.9)2 SA = 96.04π units2 SA = 301.72 units2 V = π(4.9)3 V = 156.87π units3 V = 492.81 units3 4 3 24π = πr2 π π r2 = 24 r = 4.90 units
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Example: A cross section of a sphere has area 36π units2 and is 10 units from the center, what is the surface area & volume
Radius of Cross Section 36π = πr2 π π r2 = 36 r = 6 units Radius of Sphere 102 + 62 = R2 136 = R2 R = √136 = 11.66 units
Finding the Volume of the Sphere
click click click click click click click
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SA = 4π(√136)2 SA = 544π units2 SA = 1,709.03 units2 V = π(√136)3 V = 2,114.69π units3 V = 6,643.50 units3 4 3 Example: A cross section of a sphere has area 36π units2 and is 10 units from the center, what is the surface area & volume
Finding the Volume of the Sphere
click click click click click Click
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111 Find the surface area of a sphere with radius 10.
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112 Find the volume of a sphere with radius 10.
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113 What is the surface area of a sphere if a cross section 7 units from the center has an area of 50.26 units2?
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114 What is the volume of a sphere if a cross section 7 units from the center has an area of 50.26 units2?
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115 The volume of a sphere is 24π units3. What is the area
- f a great circle of the sphere?
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116 A recipe calls for half of an orange. Shelly use an
- range that has a diameter of 3 inches. She wraps
the remaining half of orange in plastic wrap. What is the amount of area that Shelly has to cover?
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Cavalieri's Principle
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SLIDE 47 Slide 277 / 311 Cavalieri's Principle
If two solids are the same height, and the area of their cross sections are equal, then the two solids will have the same volume.
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14 14 14 Which solid has the greatest volume? 224π 703.72 None: All of the solids have the same volume.
Click
Cavalieri's Principle
2π 8 4π 4 4 224π 703.72 224π 703.72
Click Click Click
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Example: A sphere is submerged in a cylinder. Both solids have a radius of 4. What is the volume of the cylinder not occupied by the sphere? volume of cylinder - volume of sphere
Cavalieri's Principle
π(4)2 (8) - 4/3 π(4)3 128π - 256/3 π
128/3 π units3 Click Click Click Click
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The result shows that the left over volume is equal to what other solid? cone According to Cavalieri, what can be said about the cross section? The cross section of the great circle of the sphere is equal to the circle cross section of the cylinder.
Click
Cavalieri's Principle
Click
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Example: What is the radius of a sphere made from the cylinder of modeling clay shown? If you are using clay to model both solids, what measurement is the same? Volume 15 5
Cavalieri's Principle
Click
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Therefore, calculate the volume of the cylinder first.
Cavalieri's Principle
V = π(5)2 (15) V = 375π units3 Then create an equation to represent the problem and solve for r. 375π = 4/3 πr3 375 = 4/3 r3 281.25 = r3 r = 6.55 units
Click Click Click Click Click Click
15 5
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117 These 2 solids have the same volume, find the value of x. 11 r = 6 11 x 9
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118 These 2 solids have the same volume, find the value of x.
12 x 12 10 8
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Prism C B = 20 in2 x Prism D B = 20 in2 y Two prisms each with a base area of 20 square inches are shown. Which statements about prisms C and D are true. Select all that
- apply. (Statements are on the next slide.)
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119 Two prisms each with a base area of 20 square inches are shown. Which statements about prisms C and D are true. Select all that apply. A If x > y, the area of a vertical cross section of prism C is greater than the area of a vertical cross section of prism D. B If x > y, the area of a vertical cross section of prism C is equal to the area of a vertical cross section of prism D. C If x > y, the area of a vertical cross section of prism C is less than the area of a vertical cross section of prism D. D If x = y, the volume of prism C is greater than the volume of prism D, because prism C is a right prism. E If x = y, the volume of prism C is equal to the volume of prism D because the prisms have the same base area. F If x = y, the volume of prism C is less than the volume of prism D because prism D is an oblique prism.
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Similar Solids
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Corresponding sides of similar figures are similar. The prisms shown are similar. Find the values of x and y. 4 x 2 6 9 y 4 6 = 4 6 =
Similar Solids
x 9 36 = 6x 6 6 6 = x 4y = 12 4 4 y = 3 2 y
Click Click Click Click Click Click Click Click
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4 x 2 6 9 y The ratio of similarity, k, is the common value that is multiplied to preimage to get to the image.
- Hint: it's the ratio of image : preimage
If the smaller prism is the preimage, then the value of k is If the larger prism is the preimage, then the value
click for the hint
Similar Solids
3/2 2/3
Click Click
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120 The pyramid on the left is the preimage and is similar to the image on the right. Find the value
8 8 16 h 2 x y 3
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121 The pyramid on the left is the preimage and is similar to the image on the right. Find the value
8 8 16 h 2 x y 3
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122 The pyramid on the left is the preimage and is similar to the image on the right. Find the value
8 8 16 h 2 x y 3
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4 6 2 6 9 3 Consider the example of the prisms from earlier. The ratio of similarity from the smaller solid to the larger is 2:3. Calculate the surface area of both solids. How do they compare? SAsmall = 2(6)(2) + 16(4) = 88 units2 SAbig = 2(3)(9) + 24(6) = 198 units2 SA Similarity ratio = 88:198 = 4:9 = 22:32 How do their volumes compare? Vsmall = 2(4)(6) = 48 units3 Vbig = 6(3)(9) = 162 units3 V Similarity ratio = 48:162 = 8:27 = 23:33
Similar Solids
Click Click Click Click Click Click Click Click Click Click Click Click
Slide 294 / 311 Comparing Similar Figures
length in image length in preimage
= k
area in image area in preimage
= k2
volume in image volume in preimage
= k3
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How many times bigger is the surface area of the sphere to the right? How many times bigger is the volume of the sphere to the right? r = 3 r = 9 Example: How many times bigger is the radius of the sphere to the right? 3 times bigger 9 times bigger 27 times bigger
Comparing Similar Figures
Click Click Click
SAsmall = 4π(3)2 = 36π units2 SAbig = 4π(9)2 = 324π units2 Vsmall = 4/3 π(3)3 = 36π units3 Vbig = 4/3 π(9)3 = 972π units3
Click Click Click Click Click Click Click Click
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123 The scale factor of 2 similar pyramids is 4. If the surface area of the larger one is 64 units2, what is surface area of the smaller one?
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124 The scale factor of 2 similar right square pyramids is
- 3. If the area of the base of the larger one is 36 u2
and its height is 12, what is the volume of the smaller one?
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125 An architect builds a scale model of a home using a scale of 2 in to 5 ft. Given the view of the roof of the model, how much roofing material is needed for the house? 12 in 6 in 8 in 5 in 4 in 3 in
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PARCC Sample Questions
The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing this unit, you should be able to answer these questions. Good Luck! Return to Table
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Question 6/11
Daniel buys a block of clay for an art project. The block is shaped like a cube with edge lengths of 10 inches. Daniel decides to cut the block of clay into two pieces. He places a wire across the diagonal of one face of the cube, as shown in the figure. Then he pulls the wire straight back to create two congruent chunks of clay. PARCC Released Question - PBA - Calculator Section Topic: Intro to 3-D Solids
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126 Part A - Question #1: Daniel wants to keep one chunk of clay for later use. To keep that chunk from drying out, he wants to place a piece of plastic sheeting
- n the surface he exposed when he cut through the cube.
Determine the newly exposed two-dimensional cross section. A Triangle B Parallelogram C Rectangle D Rhombus E Square
Question 6/11
PARCC Released Question - PBA - Calculator Section Topic: Intro to 3-D Solids
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127 Part A - Question #2: Daniel wants to keep one chunk of clay for later use. To keep that chunk from drying out, he wants to place a piece of plastic sheeting
- n the surface he exposed when he cut through the cube. Find the
area of this newly exposed two-dimensional cross section. Round your answer to the nearest whole square inch.
Question 6/11
PARCC Released Question - PBA - Calculator Section Topic: Intro to 3-D Solids
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128 Part B: Daniel wants to reshape the other chunk of clay to make a set of clay spheres. He wants each sphere to have a diameter of 4
- inches. Find the maximum number of spheres that Daniel can
make from the chunk of clay. Show your work.
Question 6/11
PARCC Released Question - PBA - Calculator Section Topic: Cavaleri's Principle
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Question 10/11
The Farmer Supply is building a storage building for fertilizer that has a cylindrical base and a cone-shaped top. The county laws say that the storage building must have a maximum width of 8 feet and a maximum height of 14 feet. Topics: Volume of a Prism, Volume of a Cylinder, and Volume of a Cone PARCC Released Question - PBA - Calculator Section
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129 Dump Trucks deliver fertilizer in loads that are 4 feet tall, 6 feet
wide & 12 feet long. Farmer Supply wants to be able to store 2 dump-truck loads of fertilizer. Determine the height of the cylinder, h1, and a height of the cone, h2, that Farmer Supply should use in the design. Show that your design will be able to store at least two dump-truck loads of
- fertilizer. When you finish, enter your value for h1 into your
Responder.
Question 10/11
Topics: Volume of a Prism, Volume of a Cylinder, and Volume of a Cone PARCC Released Question - PBA - Calculator Section - SMART Response Format
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130 A rectangle will be rotated 360º about a line which contains the point of intersection of its diagonals and is parallel to a side. What three-dimensional shape will be created as a result of the rotation? A a cube B a rectangular prism C a cylinder D a sphere
Question 4/7
PARCC Released Question - EOY - Non-Calculator Section Topic: Intro to 3-D Solids
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131 The table shows the approximate measurements of the Great
Pyramid of Giza in Egypt and the Pyramid of Kukulcan in Mexico. Approximately, what is the difference between the volume of the Great Pyramid of Giza and the volume of the Pyramid of Kukulcan? A 1,945,000 cubic meters B 2,562,000 cubic meters C 5,835,000 cubic meters D 7,686,000 cubic meters PARCC Released Question - EOY - Calculator Section Topic: Volume of a Pyramid
Question 8/25
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Question 11/25
Two cylinders each with a height of 50 inches are shown. Topic: Cavaleri's Principle PARCC Released Question - EOY - Calculator Section
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132 Which statements about cylinders P and S are true? Select all that apply. A If x = y, the volume of cylinder P is greater than the volume
- f cylinder S, because cylinder P is a right cylinder.
B If x = y, the volume of cylinder P is equal to the volume of cylinder S, because the cylindres are the same height. C If x = y, the volume of cylinder P is less than the volume of cylinder S, because cylinder S is slanted. D If x < y, the area of a horizontal cross section of cylinder P is greater than the area of a horizontal cross section of cylinder S. E If x < y, the area of a horizontal cross section of cylinder P is equal to the area of a horizontal cross section of cylinder S. F If x < y, the area of a horizontal cross section of cylinder P is less than the area of a hoizontal cross section of cylinder S.
Question 11/25
Topic: Cavaleri's Principle PARCC Released Question - EOY - Calculator Section
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133 Part A The outer surface of the pipe is coated with protective material. How many square feet is the outer surface of the pipe? Give your answer to the nearest integer. A steel pipe in the shape of a right circular cylinder is used for drainage under a road. The length of the pipe is 12 feet and its diameter is 36
- inches. The pipe is open at both ends.
Question 13/25
Topic: Surface Area of a Cylinder PARCC Released Question - EOY - Calculator Section
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134 Part B A wire screen in the shape of a square is attached at one end of the pipe to allow water to flow through but to keep people from wandering into the pipe. The length of the diagonals of the screen are equal to the diameter of the pipe. The figure represents the placement of the screen at the end of the pipe. A 72 B 102 C 125 D 324 E 648 F 1,018 and the area of the screen is ________ square inches.
Question 13/25
Topic: Surface Area of a Cylinder PARCC Released Question - EOY - Calculator Section The perimeter of the screen is approximately ________ inches, Select from each set of answers to correctly complete the sentence.
