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

DAY 143 – ARGUMENTS ABOUT

AREA OF A CIRCLE

INTRODUCTION

Let us also have a look at the area of a circle, it is given by 𝜌𝑠2 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, 𝜌𝑠2, 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.

VOCABULARY

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

• n 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. Since the interior angle of an equilateral triangle is 60°, the area of one triangle is

1 2 𝑏𝑐 sin 𝜄 = 1 2 𝑠2 sin 60

Since there are six such triangles, the area of hexagon which is an approximation of that of the circle would be 1 2 𝑠2 × 0.866 × 6 = 2.598𝑠2

O T M S B W Z 𝑠 𝑠 𝑠

Comparing this with 𝜌𝑠2, 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 2 𝑏𝑐 sin 30° = 1 2 𝑠2 × 0.5 O P

Q R S T U 𝑠 𝑠 N M

The area of the whole figure, dodecagon would be = 1 2 𝑠2 × 0.5 × 12 = 3𝑠 Comparing this with 𝜌𝑠2, we get that 𝜌 is approximated using as 3 units.

Example Approximate 𝜌 for by estimating the area of a circle

• f radius 𝑠 using an inscribed regular polygon

having 20 sides. Solution The central angle for this figure would be

360 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 × 𝑠2 sin 18 = 0.1545𝑠2. Area of the polygon is 20 × 0.1545𝑠2 = 3.09𝑠2 Comparing 3.09𝑠2 with 𝜌𝑠2, we get that 𝜌 = 3.09

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

ANSWERS TO HOMEWORK

Area = 3.078r2

THE END