How do I find the base edge of a pentagonal pyramid given the height and slant height? I just need formulas.

Question

If you need them, the height is 4 and the slant height is 5. 

Answer 1

Use the Pythagorean Theorem to find the the length of the apothem:

 

(height)² + (apothem)² = (slant height)²

 

If you draw a line segment from the center of the pentagon to each vertex, it forms 5 congruent triangles.  The angle near the pentagon's center of each triangle is 360/5 = 72°.  The two base angles are 54° each.  The apothem goes from the center to the base edge at a right angle.  Now use a little trig (tangent) to find the length of half of the edge:

 

tan(54°) = apothem/(half the edge length)

half the edge length = (apothem)/(tan(54°))

edge length = (2×apothem)/(tan(54°))