Sphere in a Pentagonal Pyramid - Answer
We know that the surface of the sphere will be tangent to the center of the base and also to
each triangular face at a point that is on the vertical-center-line and a little below the top of each
triangle:
We can visualize a right triangle which has a small leg of the height of the pyramid, another leg
as the distance from the pentagon’s center to the center of one of its sides, and a hypotenuse
with the length of the distance from the vertex of a triangle to the center of its opposite side
(height of the triangle). We know that the sphere will be tangent to this hypotenuse.
We know that the distance from the center of the pentagon to the center of a side is its
apothem. The equation for the apothem of this pentagon is:
𝑎𝑝𝑜𝑡ℎ𝑒𝑚 =
© 2011, Leon Hostetler, www.leonhostetler.com
1
2 tan
𝜋 ≈ 0.68819
5
Sphere in a Pentagonal Pyramid - Answer
We can find the height of each of the pyramid’s triangular faces by splitting the triangle into two
right triangles and using the Pythagorean theorem. We find that the height of the triangle is:
3
� ≈ 0.86603
4
With the above two pieces of information can find the height of the pyramid by using the
Pythagorean theorem again. We find that the height is:
2
3
1
� −�
𝜋� ≈ 0.52573
4
2 tan
5
If we think of the above triangle as the right half of an isosceles triangle with a perimeter of
3.10844 and an area of 0.36181(found using Heron’s formula), we can find the radius of the
incircle using the formula:
So our answer is 0.2328 meter
𝑟=
2𝑎
2(0.36181)
≈
≈ 0.2328
3.10844
𝑝
© 2011, Leon Hostetler, www.leonhostetler.com