Question

If the radius of the base of a cone is halved, keeping the height same, what is the ratio of the volume of the reduced cone to that of the original cone?

Answer 1

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as

Volume of cone =

Now, let another cone have the same height, that is ‘h’, but the base radius of this cone is half that of the previous one we have talked about, that is ‘’

Now,

The volume of this new cone =

=

Now the ratio between the old cone and the new one would be,

= $\frac{{\displaystyle \frac{1}{3}}{\mathrm{\pi r}}^2\mathrm{h}}{{\displaystyle \frac{\pi r^2\mathrm{h}}{12}}}$=

=

=

Hence the ratio between the volumes of the modified cone and the original cone is