Question

Derivation of formula for lateral area of frustum of a right circular cone

Answer 1

Answer:

Lateral Area, AL

The lateral area of the frustum of a right circular cone is equal to one-half the sum of the circumference of the bases multiplied by slant height. See the derivation of the formula for lateral area of frustum of a right circular cone.

Step-by-step explanation:

Frustum of a Right Circular Cone

The lateral area of frustum of a right circular cone is given by the formula

 

A=π(R+r)L

 

where

R = radius of the lower base

r = radius of the upper base

L = length of lateral side

 

Derivation of Formula

Lateral area of right circular cone is the difference of the areas of sectors of a circle of radii L1 and L2 and common central angle θ. See the figure below

 

frustum-of-cone-developement.jpg          Figure for the Derivation of Formula for Lateral Area of Right Circular Cone

 

By ratio and proportion:

L1R=LR−r

L1=RLR−r

 

From the figure:

L2=L1−L

L2=RLR−r−L

L2=RL−(R−r)LR−r

L2=rLR−r

 

The length of arc is the circumference of the base:

s1=2πR

s2=2πr

 

From the figure:

A=12s1L1−12s2L2

A=12(2πR)(RLR−r)−12(2πr)(rLR−r)

A=πR2LR−r−πr2LR−r

A=πR2L−πr2LR−r

A=π(R2−r2)LR−r

A=π(R−r)(R+r)LR−r

A=π(R+r)L