Question

If A₁, and A₂, denote areas of circular bases of the frustum of a cone and h is its height, then the volume of the frustum is _________.

Answer 1

Let r₁ and r₂ be the radius of the top and bottom faces of the frustum of cone.

∴ $A_1=\mathrm{\pi }{r}_1^2\Rightarrow r_1=\sqrt{\frac{A_1}{\mathrm{\pi }}}$ .....(1)
and
$A_2=\mathrm{\pi }{r}_2^2\Rightarrow r_2=\sqrt{\frac{A_2}{\mathrm{\pi }}}$ .....(2)

Volume of frustum of cone

$$\begin{gathered} =\frac{1}{3}\mathrm{\pi }h\left(r_1^2+r_2^2+r_1{r}_2\right) \\ =\frac{1}{3}\mathrm{\pi }h\left(\frac{A_1}{\mathrm{\pi }}+\frac{A_2}{\mathrm{\pi }}+\sqrt{\frac{A_1}{\mathrm{\pi }}}\sqrt{\frac{A_2}{\mathrm{\pi }}}\right)\left[\mathrm{Using}\left(1\right)\mathrm{and}\left(2\right)\right] \\ =\frac{1}{3}h\left(A_1+A_2+\sqrt{A_1{A}_2}\right) \end{gathered}$$

If A₁, and A₂, denote areas of circular bases of the frustum of a cone and h is its height, then the volume of the frustum is $\underline{\frac{1}{3}h\left(A_1+A_2+\sqrt{A_1{A}_2}\right)}$.