Problem:
What is the volume of tetrahedron with edge lengths , , and ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
By the converse of the Pythgorean Theorem, each face containing apex is a right triangle with right angle at . Hence segments , and are mutually perpendicular. The area of the right triangle is . If is considered to be the base of the tetrahedron, the height is . The volume of the tetrahedron is therefore .
Note: There is a complicated formula, analogous to Heron's formula for the area of a triangle, giving the volume of a tetrahedron in terms of the edge lengths.
The problems on this page are the property of the MAA's American Mathematics Competitions