Question

For the two cones to have the same volume, their ______ should be the same.

Answer 1

Volume of a cone $=\frac{1}{3}\pi r^{2}h$
Radius of cone $1=“r_1”$
Radius of cone $2=“r_2”$

Height of cone $1=“h_1”$
Height of cone $2=“h_2”$

Volume of cone $1=\frac{1}{3}\pi r_1^2{h}_1$ …..(1)
Volume of cone $2=\frac{1}{3}\pi r_2^2{h}_2$ ……(2)

Given: The volume is the same
So equating (1) and (2):

Volume of cone 1 = Volume of cone 2
$\frac{1}{3}\pi r_1^2{h}_1=\frac{1}{3}\pi r_2^2{h}_2$
$r_1^2{h}_1=r_2^2{h}_2$

From the given options, if only one of the two quantities is the same and the other is different, then the volumes cannot be the same.
Option A: The height is the same, but because the radius is different, the volume will be different.
Option B: The radius is the same, but because the height is different, the volume will be different.
Option D: This is similar to option B as the diameter is the same as twice the radius.

Therefore, L.H.S. will be equal to R.H.S. only if $r_1=r_2\text{and}{h}_1=h_2$.
Hence, for the two cones to have the same volume, their height and radius should be the same.

$\to$ Option C is correct.