MHB Radius of Sphere Inscribed in Square Pyramid: Michelle's Q&A

[MarkFL]

Here is the question:

How would you find the radius of a sphere inscribed in a square pyramid?

So we got a problem in Geo today and we don't know how to solve it.
So what we got is:
-A pyramid with a square base
-a sphere inscribed in the pyramid
-base edge of 10
-slant height of 10

Please explain how to find the radius and of the sphere! Thanks!

I have posted a link there to this thread so the OP can view my work.

 

[MarkFL]

Hello Michelle,

The sphere will be tangent to the sides of the pyramid along the slant heights. Here is a diagram of a cross section of the pyramid and the sphere through the center of the sphere and long one of the two lines of bilateral symmetry for the square base (i.e. containing two opposing slant heights of the sides):

View attachment 2168

We know the cross-section of the pyramid is an equilateral triangle having side lengths 10, and the cross-section of the sphere is a circle having the radius of the sphere. From the diagram, we see that we may state:

$$\tan\left(30^{\circ} \right)=\frac{r}{5}$$

Hence:

$$r=5\tan\left(30^{\circ} \right)=\frac{5}{\sqrt{3}}=\frac{5}{3}\sqrt{3}$$