Reach for Polydron

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

Tetra Inequalities

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

Tetra Square

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

Pythagoras for a Tetrahedron

Age 16 to 18Challenge Level

 Consider a right-angled tetrahedron with vertices at $O(0,0,0)$, $A(a, 0, 0)$, $B(0, b, 0)$ and $C(0, 0, c)$.   Let the area of face $AOB$ be $P$, the area of $BOC$ be $Q$ and the area of $COA$ be $R$.  Also let the slanted face $ABC$ have area $S$. ($S$ is not shown on the diagram above!).   Can you prove that $P^2+ Q^2+ R^2= S^2$?

Equivalently: (area $OBC$)$^2 +$(area $OCA$)$^2 +$(area $OAB$)$^2 =$(area $ABC$)$^2$.

Extension

If you enjoyed this question, you might like explore STEP Support Programme Foundation Assignment 5 which asks a question about the perpendicular distance of face $ABC$ from the origin.