Answer to Question #147373 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.

3. What is the surface area of the cone?
4. What is the volume of the space between the cone and the cube?

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 $a$

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

Slant height $L$ is

$$L=\sqrt{r^2+h^2}=\sqrt{(\dfrac{a}{2})^2+a^2}=\dfrac{\sqrt{5}a}{2}$$

3. Surface area of a cone = Base Area + Curved Surface Area of a cone

$$A=\pi r^2+\pi rL$$

$$A=\pi((\dfrac{a}{2})^2+\dfrac{a}{2}(\dfrac{\sqrt{5}a}{2}))=\dfrac{\pi a^2}{4}(1+\sqrt{5})$$

$$A=\dfrac{\pi (20)^2}{4}(1+\sqrt{5})=100\pi(1+\sqrt{5}) (cm^2)$$

$$\approx1016.64(cm^2)$$

The surface area of the cone is $100\pi(1+\sqrt{5})\ cm^2\approx1016.64\ cm^2.$

4. The volume of the cube is

$$V_{cube}=a^3$$

The volume of the cone is

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

The volume of the space between the cone and the cube is

$$V_{space}=V_{cube}-V_{cone}$$

$$=a^3-\dfrac{\pi a^3}{12}=\dfrac{ a^3}{12}(12-\pi)$$

$$V_{space}=\dfrac{(20)^3}{12}(12-\pi)=\dfrac{2000(12-\pi)}{3}(cm^3)$$

$$\approx5905.605(cm^3)$$

The volume of the space between the cone and the cube is

$\dfrac{2000(12-\pi)}{3}\ cm^3\approx 5905.605 \ cm^3.$