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# 3d geometry

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A regular tetrahedron is a triangular pyramid in which each face is an equilateral triangle. If the height of a regular tetrahedron is 20 inches then what is the volume of the tetrahedron?

May 29, 2021

Let the side length be $s$. Draw the height and draw the bottom point connected to two vertices of the base. Because the height hits the bottom at the centroid, the length of each of those legs are half the height of the triangles, which is $\frac{s\sqrt{3}}{4}$. This forms a right angle with the height, and the hypotenuse is $s$. Using the Pythagorean theorem: $\sqrt{\left(\frac{s\sqrt3}{4}\right)^2+20^2}=s$. The end is near.