Hello, reiward!

This one takes a *lot* of preliminary work . . .

A right circular cone is inscribed in a regular tetrahedron with edge 10 cm.

Find the volume of the cone. Consider the base of the tetrahedron (and the cone).

Code:

A
*
/|\
/ | \
/ | \
/ | \
/ | \
/ * * * \
/* | *\
10 * | * 10
* | *
/ | \
/* | *\
/ * O * * \
/ * | * * \
/ | 60° * \
/ * | * \
/ * | * * \
/ * | * 30° * \
B *---------------*-*-*---------------* C
: - - - 5 - - - D - - - 5 - - - :

The base of the tetrahedron is equilateral triangle

. . with: .

is an altitude of the triangle: .

The base of the cone is the inscribed circle with center

. . Its radius is

We find that is a 30°-60° right triangle with

. . Hence: .

The radius of the cone is: .

Consider a vertical cross-section of the tetrahedron,

. . cut through the top vertex and Code:

V *
|\
| \
| \
| \
h | \ 10
| \
| \
| \
| \
O *---------* C
_
10/√3

We have: .

The height of the cone is: .

Therefore, the volume of the cone is:

. . .