Verifying a Formula (a) Given a circular sector with radius and central angle (see

Chapter 7, Problem 54

(choose chapter or problem)

Verifying a Formula

(a) Given a circular sector with radius L and central angle \(\theta\) (see figure), show that the area of the sector is given by

\(S=\frac{1}{2} L^{2} \theta\).

(b) By joining the straight-line edges of the sector in part (a), a right circular cone is formed (see figure) and the lateral surface area of the cone is the same as the area of the sector. Show that the area is \(S=\pi r L\), where r is the radius of the base of the cone. (Hint: The arc length of the sector equals the circumference of the base of the cone.)

(c) Use the result of part (b) to verify that the formula for the lateral surface area of the frustum of a cone with slant height L and radii \(r_{1}\) and \(r_{2}\) (see figure) is \(S=\pi\left(r_{1}+r_{2}\right) L\). (Note: This formula was used to develop the integral for finding the surface area of a surface of revolution.)

Text Transcription:

S=1/2 L^2 theta

S=pi rL

r_1

R_2

S=pi(r_1+r_2)L