Question

the derivatio of the formula of frustum of cone

Answer 1

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
 

          

 

By ratio and proportion:
L1R=LR−rL1R=LR−r

L1=RLR−rL1=RLR−r
 

From the figure:
L2=L1−LL2=L1−L

L2=RLR−r−LL2=RLR−r−L

L2=RL−(R−r)LR−rL2=RL−(R−r)LR−r

L2=rLR−rL2=rLR−r

 
The length of arc is the circumference of the base:
s1=2πRs1=2πR

s2=2πrs2=2πr
 

From the figure:
A=12s1L1−12s2L2A=12s1L1−12s2L2

A=12(2πR)(RLR−r)−12(2πr)(rLR−r)A=12(2πR)(RLR−r)−12(2πr)(rLR−r)

A=πR2LR−r−πr2LR−rA=πR2LR−r−πr2LR−r

A=πR2L−πr2LR−rA=πR2L−πr2LR−r

A=π(R2−r2)LR−rA=π(R2−r2)LR−r

A=π(R−r)(R+r)LR−rA=π(R−r)(R+r)LR−r

A=π(R+r)LThe lateral area of frustum of a right circular cone is given by the formula