Circumcircle: Definition, Properties & Formula

Instructor: Laura Pennington

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

Circumcircles show up in the world around us often. This lesson will define circumcircles, and we will look at various properties of circumcircles and some formulas we can use to better analyze these fascinating shapes.

Circumcircle

Oh no! The town of Faye has just had a very bad spill of toxic waste by the local power plant. The spill happened in such a way that there is a square area where the risk to the public is at its most, and the entire risk area is enclosed in a circle that passes through each of the vertices of the square as shown in the image.


circumcirc1


When a polygon is enclosed in a circle that passes through all of its vertices, we call that circle the circumcircle of the polygon. For example, since the circular entire risk area passes through each of the vertices of the square high-risk area, we would say that the circle is the circumcircle of the square.

Circumcircles have various characteristics and properties that make them very interesting, and provide formulas for analyzing their different aspects.

Properties and Formulas

Like any circle, a circumcircle has a center point and a radius. We call the center point the circumcenter of the polygon that the circumcircle belongs to. The radius is a line segment from the circumcenter to any point on the circumcircle, and is called the circumradius of the polygon that the circumcircle belongs to.

The area and perimeter of a circumcircle are the same as they would be for any other circle. If a circle has radius r, then the formulas for the area and perimeter of that circle, are as follows:

  • Area of a circle = πr2
  • Perimeter of a circle = 2πr

We see that both of these formulas depend on the radius of a circle, so in the case of a circumcircle, the area and perimeter of a circumcircle both depend on the circumradius of the polygon that the circumcircle belongs to. Therefore, to find the area or the perimeter of a circumcircle of a polygon, we follow these steps:

  1. Find the length of the circumradius of the polygon.
  2. Plug the value you found in step 1 in for r in the appropriate formula.

Obviously, the circumradius of a polygon is going to depend on the type of polygon, so it will be different in each case. Thus, it is a good idea to practice working with the circumcircle of different types of polygons, so we can be comfortable with this concept. Let's look at an example.

Example

Looking back at the toxic spill in the town of Faye, the city council determines that the lengths of the sides of the square high-risk area are 100 meters each. They are working hard to contain the risk area, so they decide that they want to put a large covering over the entire risk area, and then they want to build a solid fence around the entire risk area in order to keep the toxic waste spill from spreading, and to keep people out of the risk area. To do this, they need to know the current area of the entire risk area, and they need to know the perimeter of the entire risk area in order to know the amount of materials they will need.

In other words, we need to find the area and the perimeter of the circumcircle of the square high-risk area. To find these, we just need to first find the circumradius of the square and then we can plug that into our area and perimeter formulas.


circumcirc2


Notice that in a square, the circumradius is one-half the length of a diagonal of the square. It is also the case that the length of the diagonal of a square with side length s is as follows:

  • Length of a diagonal = √(2s^2)

Therefore, the circumradius of a square with side length s is

  • Length of circumradius = (1/2)√(2s^2)

We can simplify this.


circumcirc3


Perfect, so we have that the length of the circumradius of a square with side length s can be found using the following formula:

  • Circumradius = (√(2) / 2) ⋅ s

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