Answer to Question #150546 in Geometry for Sm

Question

The surface area of a sphere inscribed in a regular tetrahedron is 144pi cm^2.
1. What is the radius of the sphere?

2. What is the altitude of the tetrahedron?

Answer 1

The surface area of a sphere inscribed in a regular tetrahedron is $144\pi\space cm^2$

1. What is the radius of the sphere?

Surface area of a sphere $S=4\pi r^2$

$r-$ radius of sphere

$r=\sqrt{\frac{S}{4\pi}}=\sqrt{\frac{144\pi}{4\pi}}=\sqrt{36}=6.$

Answer: radius of the sphere is $6\space cm.$

2. What is the altitude of the tetrahedron?

For a regular tetrahedron of edge length a:

Altitude $h=\sqrt{\frac{2}{3}}a$

$a=2r\sqrt{6}=12\sqrt{6}$

$h=\sqrt{\frac{2}{3}}12\sqrt{6}=24.$

Answer:altitude of the tetrahedron is $24\space cm.$