would help us estimate the area of a circle. Consider a circle of - PowerPoint PPT Presentation

D AY 143 – A RGUMENTS ABOUT AREA OF A CIRCLE

I NTRODUCTION Let us also have a look at the area of a circle, it is given by 𝜌𝑠 2 where 𝑠 is the radius and 𝜌 a constant number that is irrational. The constant 𝜌 is normally approximated using 3.14159265…. Despite the formula, 𝜌𝑠 2 , there are a number of arguments that are used to estimate the area of a circle by first estimating what 𝜌 should be. In this lesson, we are going to give an informal argument for the formulas for area of a circle.

V OCABULARY Area of a circle Amount of two dimensional space occupied by a circle Radius The line segments from the center of the circle to the circumference

Estimation of area of a circle We would like to come up with an argument that would help us estimate the area of a circle. Consider a circle of radius 𝑠 and center O. Pick a point, T, on the circle and connect it to the center. Using a compass of radius OT, mark several points on the arc of the circle such the distance from one mark to the other is 𝑠 = 𝑃𝑈. Connect the points together to get a hexagon.

Drawing the diameters from the vertices of the hexagon, we get six equilateral triangles. S M 𝑠 𝑠 B 𝑠 T O W Z Since the interior angle of an equilateral triangle is 1 2 𝑠 2 sin 60 1 60° , the area of one triangle is 2 𝑏𝑐 sin 𝜄 = Since there are six such triangles, the area of hexagon which is an approximation of that of the circle would be 1 2 𝑠 2 × 0.866 × 6 = 2.598𝑠 2

Comparing this with 𝜌𝑠 2 , we get that 𝜌 is approximated using as 2.598 units. If we increase the number of sides so that we have a 12 sided figure, the angle at QON would be 30° hence the area of the small triangle would be 1 1 2 𝑠 2 × 0.5 2 𝑏𝑐 sin 30° = R Q N M 𝑠 𝑠 S P O U T

T he area of the whole figure, dodecagon would be = 1 2 𝑠 2 × 0.5 × 12 = 3𝑠 Comparing this with 𝜌𝑠 2 , we get that 𝜌 is approximated using as 3 units.

Example Approximate 𝜌 for by estimating the area of a circle of radius 𝑠 using an inscribed regular polygon having 20 sides. Solution 360 The central angle for this figure would be 20 = 18° The area of one triangle enclosed by the radii and the chord which is one side of the polygon is 1 2 𝑏𝑐 sin 𝜄 = 1 2 × 𝑠 2 sin 18 = 0.1545𝑠 2 . Area of the polygon is 20 × 0.1545𝑠 2 = 3.09𝑠 2 Comparing 3.09𝑠 2 with 𝜌𝑠 2 , we get that 𝜌 = 3.09

HOMEWORK Approximate the area of a circle of radius 𝑠 using an inscribed regular polygon having 18 sides.

A NSWERS TO HOMEWORK Area = 3.078r 2

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