Download Area of the frustum of a cone

Frustum of a Cone
A frustum of a cone is the section of the surface between two cones.
The surface area of a cone can be related to the area of a sector of a circle:
Suppose the central angle of the sector is θ radians and the radius is ρ. The
area of the sector is proportional to θ
A = kθ
A
πρ2
1
k=
=
= ρ2
θ
2π
2
1
A = kθ = ρ2 θ
2
This is related to the surface area of a cone because if we cut a cone along
it’s side and unroll it, we get a sector
A=
1 2
1
2πr
ρ θ = s2 ·
= πrs
2
2
s
µ
Area of Frustrum =
Area of
larger cone
¶
µ
−
Area of
smaller cone
= πr2 (s + ∆s) − πr1 s
= π(r2 − r1 )s + πr2 ∆s
By similar triangles,
r2
r1
=
s + ∆s
s
r2 s = r1 s + r1 ∆s
(r2 − r1 )s = r1 ∆s
Therefore, the area of a frustum is:
Afrustum = πr1 ∆s + πr2 ∆s
µ
¶
r1 + r2
= 2π
∆s
2
= 2πr∆s
¶