Exploring Analyic Geometry with Mathematica®

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Area of a Tetrahedron's Base



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 "tetra_2.gif" is the area of the triangle whose vertices are x, y and z. Note the similarity to the Pythagorean Theorem for right triangles.


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


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Compute the semi-perimeter, s, of ΔABC.



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.

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}


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing