Derive the formula for the volume of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.

Let ABC is cone, which is converted into a frustum by cutting the top. Let r₁ and r₂ be the radius of two ends and h be its height.
In ΔABG and ΔADF, DF∥BG ∴ ΔABG ∼ ΔADF
DF/BG = AF/AG = AD/AB
⇒ r²/r₁ = h₁ – h/h₁ ⇒ r²/r₁ = 1 – h/h₁ = 1 – 1/l₁
⇒ r₂/r₁ = 1 – h/h₁ ⇒ h/h₁ = 1 – r₂/r₁ = (r₁ – r₂/r₁)
⇒ h₁/h = (r₁/r₁ – r₂) ⇒ h₁ = r₁h/r₁ – r₂
Volume of frustum = Volume of cone ABC- Volume of cone ADE
= (1/3)πr₁²h₁ – (1/3)πr₂²(h₁ – h) = 1/3π[r₁²h₁ – r₂²(h₁ – h)]
= (1/3)π[r₁²(r₁h/r₁ – r₂) – r₂²(r₁h/r₁ – r₂) – h)] = (1/3)π[r₁³h/r₁ – r₂) – r₂²(r₁h – r₁h + r₂h/r₁ – r₂)]
= (1/3)π [(r₁³h/r₁ – r₂) – (r₂³h/r₁ -r₂)] = (1/3)πh[r₁³ – r₂³/r₁ – r₂] = (1/3)πh [(r₁ – r₂)(r₁² + r₁r₂ + r₂²/r₁ – r₂] = (1/3)πh(r₁² + r₁r₂ + r₂².