Answer to Question #147432 in Geometry for solid mensuration

Question

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

1. What is the altitude 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 $a$

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

$$h=20\ cm$$

The altitude of the cone is 20 cm.