To find $D$, we simply write the equation for 3 spheres with radii being the lengths of the 3 tetrahedron legs going out of the plane. Doing that, point $B=A + a\cdot(cos(\alpha), sin(\alpha),0)=(2.71429,4.19913,0)$ Next we will find point $D$, and having done so, the z-coordinate of $D$ is your height answer. From your example, $a=5$ $b=6$ $c=7$ Use the law of cosines to get angle $\alpha$ at point A in the xy plane. Here is a general solution that is easy enough to understand.
You should check that the solutions actually satisfy the given conditions, because I haven't done that. Here's a python script from import symbolsįrom import nonlinsolve This is a complicated formula also, but it doesn't have all those square roots!Īlternatively, just solve it with coordinate geometry. Some care is necessary in using the formula, because as explained in the article, the formula depends on the pairings of opposite sides of the tetrahedron, and there are actually two tetrahedra, with different volumes, for a given pairing of six lengths. I found another discussion of a tetrahedral volume formula (due to the renaissance painter Pierro della Francesca!) in Kevin Brown's Math Pages. It doesn't look pleasant, but it should do the job. At the bottom of the same Wikipedia page, there is a Heron-type formula for the volume of a tetrahedron, due to David Robbins. We can find the area of a triangle, given the lengths of the sides, by Heron's formula. We know that the volume of a tetrahedron is one-third the area of the base times the height, so if we can find the are of $\triangle ABC$ and the volume of the tetrahedron, we are home free.
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