Name: nishi

Who is asking: Student
Level: Secondary

Question: how do i prove (a simply as possible) why the bond angles of a tetrahedral polygon are 109.5 degrees? *i already have two explanations that i don't understand. one is about "theory of dot products" and "vectors" and a hook-like symbol w/ a cosine, and the other has an incomprhensible diagram w/ difficult notation- PLEASE BE SIMPLE! thanks sooo much

Hi Nishi,

Imagine that you create the tetrahedron by joining opposite vertices on the faces of a cube as in the diagram below. To simplify the arithmetic assume that the cube has sides of length 2.

Let O be the point at the center of the cube (equidistant from the eight vertices) as in the diagram below. Construct the triangle OPQ.

Consider the triangle OPQ. Let R be the midpoint of PQ.

Since the sided of the cube are of length 2, a use of Pythagoras' Theorem gives |PQ| = . Also, since O is at the center of the cube, |OR| = 1. From the diagram the tangent of the angle ROP is . Thus angle QOP = 2 x angle ROP = 2 x arctan() = 109.5o

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