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:
. . .