Answer to Question #147434 in Math for solid mensuration

Question

A right circular cone is inscribed in a cube having a diagonal which
measures 20 square root of 3 cm.

2. What is the volume of the cone?

Answer 1

The diagonal $d$ of the cube with the side $a$ is

$$d^2=a^2+a^2+a^2=3a^2$$

$$d=\sqrt{3}a$$

Given $d=20\sqrt{3}\ cm.$

Then

$$a=\dfrac{d}{\sqrt{3}}=\dfrac{20\sqrt{3}}{\sqrt{3}}=20(cm)$$

The radius $r$ of the circle inscribed in the square with the side $a$ is

$$r=\dfrac{a}{2}$$

Then we have the right circular cone inscribed in a cube with the side $a$

$$radius=r=\dfrac{a}{2}, \ height=h=a$$

The volume of the cone is

$$V_{cone}=\dfrac{1}{3}\pi r^2h=\dfrac{1}{3}\pi (\dfrac{a}{2})^2 (a)=\dfrac{\pi a^3}{12}$$

$$V_{cone}=\dfrac{\pi(20)^3}{12}=\dfrac{100\pi}{3}(cm^3)$$

$$\approx104.720(cm^3)$$

The volume of the cone is

$\dfrac{100\pi}{3}\ cm^3\approx 104.720 \ cm^3.$