Download Tetrahedra - Cornell Math

Math 2130 Workshop: Reasoning about Regular Tetrahedra I
In this workshop, we will be working out the angle between two faces of a tetrahedron
and its volume without the use of vector algebra.
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1) Suppose the triangle ABC above is the base of a regular tetrahedron with side length
1. Use trigonometry to work out the height and thus the area of ABC.
2) The center of the triangle (E) is where two angle bisectors intersect. Use trigonometry to work out the length of ED and AE.
3) Let F denote the topmost vertex of our regular tetrahedron. We know that F will
be located directly above E and the length of AF will be 1. Thus AEF will form a right
triangle standing out of the page. Use the length of AE you worked out in the previous
problem to solve for EF , the height of the tetrahedron.
4) Recall that the volume of a tetrahedron is 31 the area of the base times the height.
Use your previous answers to find the volume of our tetrahedron.
5) DEF will also form a right triangle. Find the angle ∠F DE, and leave it in terms
of inverse trig.