Area of a Tetrahedron's Base

tetra.html

Exploration

A tetrahedron is a three-dimensional geometric object bounded by four triangular faces. Given a tetrahedron with vertices O(0,0,0), A(a,0,0), B(0,b,0) and C(0,0,c) show that the areas of the triangular faces are related by the equation

where is the area of the triangle whose vertices are x, y and z. Note the similarity to the Pythagorean Theorem for right triangles.

Approach

Compute the area of ΔABC using Heron's formula and compare it to the areas of the other triangles.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Compute the semi-perimeter, s, of ΔABC.

Clear[AB,AC,BC];
s=(AB+AC+BC)/2

Compute the areas of ΔABC using Heron's formula. Replace the lengths of each side by expressions in a, b and c, the coordinates on the axes.

Clear[a,b,c];
A1=Expand[s(s-AB)(s-AC)(s-BC)] //. {
    AB^2->a^2+b^2, AB^4->(a^2+b^2)^2,
    AC^2->a^2+c^2, AC^4->(a^2+c^2)^2,
    BC^2->b^2+c^2, BC^4->(b^2+c^2)^2}

Replace certain expressions with the areas of the triangles involved.

A2=Expand[A1] //. {
   a^2*b^2->(2 Area[AOB])^2,
   a^2*c^2->(2 Area[AOC])^2,
   b^2*c^2->(2 Area[BOC])^2}