Question

The base radii of two right circular cones of the same height are in the ratio 2:3. Find the ratio of their volumes.

Answer 1

Let the base radii of both cones be r₁ and r₂.
Now, according to the question, we have:

$\frac{r_1}{r_2}=\frac{2}{3}$ …(i)
We know:
Volume of the cone = $\frac{1}{3}\mathrm{\pi }{r}^{2}h$

∴ $\frac{V_1}{V_2}=\frac{{\displaystyle \frac{1}{3}}\mathrm{\pi }{r_{\mathit{1}}}^{2}h}{{\displaystyle \frac{1}{3}}\mathrm{\pi }{r_2}^{2}h}$

= ${\left(\frac{r_1}{r_2}\right)}^2$

= ${\left(\frac{2}{3}\right)}^2$ [Using (i)]

= $\frac{4}{9}$

Hence, the ratio of the volumes of two right circular cones is 4:9.