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Question

If the slant height of a triangular pryamid is 12 and the lateral edge is 15, then what is the height and base edge length?

Answer 1

Assuming this is a regular triangular pyramid (problem does not specify):

The first step is to notice, a lateral face split down the slant height, is a right triangle with ratios (3:4:5), this makes half the base of lateral face 9. So our base equilateral triangle has sides of length 18.

The next step is to notice, the height of this pyramid is one leg of another right triangle. The other leg of the right triangle is the distance from the center of the equilateral triangle to the vertex.The lateral edge 15 is a hypotenuse. To find the height of the pyramid, we first need to calculate the distance from the center of the equilateral triangle to a vertex. Since the triangle has angles (60, 60, 60), cutting three triangles from the center produces triangles with angles (30, 30, 120). We know the length opposite to the angle 120 is 18. We cut this triangle in half to produce (30, 60, 90) triangles. Thus, the base is half of 18 (since we cut out triangle in half), which is 9. 9 is opposite to angle 60, so the side opposite to the angle 30 is 3*sqrt(3), and the hypotenuse is double that: 6sqrt(3).

Our last step is simply to use the pythagorean theorem. 15² = (6√3)² + x²

Solving this gives us: √117.