Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will discuss what a circumradius of a polygon is. We will then go on to find the length of the circumradius of a triangle and of a regular polygon using some very useful formulas.

Suppose an architect is designing a building so that it is dome shaped, and the top of it has a glass ceiling in the shape of a pentagon with equal side lengths (also called a regular pentagon), and a circular beam around it. As she is designing the glass ceiling, she realizes that she will need to have supportive beams that run from each of the vertices of the pentagonal window to the center of the circular supportive beam as shown in the image.

This pentagonal glass ceiling along with the circular beam that goes around it is actually a quite fascinating concept in mathematics! Any circle drawn around a polygon, or two-dimensional shape with straight sides, in such a way that it passes through each of the vertices of the polygon is called a circumcircle. In this scenario, the pentagonal glass ceiling is the polygon, and the circular beam is the circumcircle.

Another important characteristic that a polygon with a circumcircle possesses is a circumradius. A circumradius of a polygon is the radius of the polygon's circumcircle. It can also be thought of as a line segment that goes from any vertex of the polygon to the center of the circumcircle.

Looking back at our pentagonal glass ceiling, can you identify which parts of it would represent a circumradius of the glass ceiling? If you are thinking that each of the supportive beams that run from the vertices of the pentagon to the center of the circumcircle would be a circumradius of the pentagonal glass ceiling, you are correct! Let's take a look at how to find the length of a circumradius of a polygon.

## Formula for the Circumradius of a Triangle

Not all polygons have circumcircles, so not all polygons have a circumradius. However, all triangles and all regular polygons do have these characteristics, so let's consider the formulas for finding the length of the circumradius of a triangle and for a regular polygon.

We'll start with a triangle. If a triangle has side lengths a, b, and c, then we can find the length of its circumradius using the following formula:

R = (abc) / √((a + b + c)(b + c - a)(c + a - b)(a + b - c))

This may look like a complicated formula, but when we plug in values for a, b, and c we'll find that it really isn't too bad. For example, suppose we have a triangle with side lengths 6 inches, 8 inches, and 10 inches.

To find the length of this triangles circumradius, we simply let a = 6, b = 8, and c = 10, and we plug these values into our formula and simplify.

We end up with the R = 5, so the length of the circumradius of the triangle with side lengths 6 inches, 8 inches, and 10 inches is 5 inches. That was pretty easy! Told ya it wasn't so bad! Let's consider regular polygons.

## Formula for the Circumradius of a Regular Polygon

A regular polygon is a polygon that has sides of equal length, like the pentagonal glass ceiling in our opening example. Suppose that now that the architect has her design made, she hires a builder to help make her design a reality. She tells the builder that the sides of the pentagonal ceiling will have a length of 7 feet each.

As the builder is trying to gather materials, he realizes that he needs to know the lengths of the supportive beams that run from the vertices of the pentagon to the center of the circular beam. In other words, he needs to know the length of the circumradius of the pentagonal glass ceiling. Thankfully, we have another nice formula for the length of a circumradius of a regular polygon.

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