You may also like

problem icon

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.

problem icon

Tetra Inequalities

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

problem icon

The Dodecahedron Explained

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

Pythagoras for a Tetrahedron

Age 16 to 18 Challenge Level:

coordinate diagram
A natural generalisation of Pythagoras' theorem is to consider a right-angled tetrahedron with four faces, three in mutually perpendicular planes and one in the sloping plane. Then ask "what corresponds to the squares of the lengths of the sides?"

The answer must be "the squares of the areas of the faces". If these areas are $P$, $Q$ , $R$ and $S$ respectively then prove that:

$P^2+ Q^2+ R^2= S^2$
Equivalently: (area $OBC$)$^2 + $(area $OCA$)$^2 + $(area $OAB$)$^2 = $(area $ABC$)$^2$.





The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.
Copyright © 1997 - 2019. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.