Wednesday, December 18, 2013

Volume of a Tetrahedon

In this article I give a rough outline of a derivation of the volume formula for a tetrahedron, given its four vertices.

Any set of three points in 3D are co-planar. If they are not co-linear, then they uniquely define a plane. They also uniquely define a triangle. With polygons of more than three sides/vertices, you need to specify which edges are drawn and which are not.  In a program, you might imply the edges according to the ordering of the points, but the order then is a means of indicating which edges to draw. You could not simply draw all of the possible edges and get a polygon, generally. But with a triangle, you take every pairing of points and draw an edge between them.

If you add a fourth point which is not in the same plane, you now have a tetrahedron (tretra = four)—a four sided polyhedron which is uniquely defined by four points. As in the case of the triangle, you draw an edge between every pairing of points. The number of combinations of 2 points is    
and so there are 6 edges.
A tetrahedron is a special case of a slanted pyramid. The same volume formula applies, namely,
                                                          
In order to use the formula, we must pick a face to serve as the base.  We choose the face with vertices P1, P2, and P3, in the above illustration. (It doesn't matter which ones as long as you're consistent.)  We need to calculate the area of the base and find the height of the pyramid. The height is the distance from P4 to the plane defined by the base points (the measure is taken perpendicular to the plane).

First we find the area. A well known formula for finding the area of a triangle given the length of two sides, say a and b, and the included angle, say θ,  is      
(With some minor trig you can figure this out, but if you just want to see it proved, see Area of Triangles Without Right Angles [1].) The cross-product comes in handy here since
where v and u are vectors and θ is the included angle. Thus, the area is
To find the distance from P4 to the base plane, we first find the plane. The equation of a plane can be expressed in terms of a normal vector and any point on the plane. The normal vector for our plane is
and the equation of the plane is
(This is a 3D analogue to the point-slope form of the equation of a line, albeit, it also resembles the general form.)

The height from the plane to P4 is the component of the vector (P4 – P1) which is in the direction of the normal vector, namely ([2], p. 679),

This leads to the volume formula
Sources:
  1. Math is Fun, Area of Triangles Without Right Angleshttp://www.mathsisfun.com/algebra/trig-area-triangle-without-right-angle.html
  2. Stewart, J., Calculus: Early Transcendentals, 3rd Ed., Brooks/Cole Publishing Company, 1995
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