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How to Find the Circumradius of a Triangle

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  • 0:04 Circumradius of a Triangle
  • 1:59 Constructing the Circumradius
  • 3:17 Finding Circumradius Length
  • 4:35 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

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

This lesson explains how to find the circumradius of a triangle We'll look at how to construct the circumradius of a triangle using a straightedge and a compass, and we'll look at a formula we can use to find its length.

Circumradius of a Triangle

Imagine there exists a lake called Clear Circle Lake. It's named so because it is a perfect circle and it is so clear that you can see straight to the bottom of the lake and all the sea creatures within. Now, suppose that there are three towns on the lake called the 'triangle' towns. This is because each town is connected by a bridge over Clear Circle Lake, and those bridges create a triangle.

Because the best viewing area of the lake is in the center of the lake, the mayor of triangle town A decides that he wants to build a bridge from the town to the exact center of the lake for a viewing area for the town residents.

Now comes the interesting part. Every aspect of this scenario has a mathematical name. Consider the following definitions:

  • Vertices of a triangle: this is where the points at which the sides of a triangle meet

  • Circumcircle of a triangle: this is a circle that passes through all vertices of a triangle

  • Circumcenter of a triangle: this is the center of the circumcircle of a triangle, and finally

  • Circumradius of a triangle: this is the radius of the circumcircle of a triangle, or the line segment connecting any vertex of a triangle to the circumcenter of the triangle


circumrad2


Think about Clear Circle Lake and the triangle towns again. As we said, the bridges between the triangle towns form a triangle, so the triangle towns are the vertices of that triangle. Clear Circle Lake is the circumcircle of the triangle formed by the bridges, and the best viewing area at the center of the lake is the circumcenter. Lastly, the bridge that the mayor of town A wants to build would be the circumradius of the triangle.

Huh! Who knew that a lake and some towns could be so mathematical?

As the mayor of town A is creating the blueprints for the bridge, he realizes that to be able to construct the bridge on the blueprint he will also need to figure out the length of the bridge, so he that he can buy the right amount of building materials. In other words, he needs to know how to find the circumradius of a triangle and its length so that he can construct the bridge. Let's take a look at how to do both of these things!

Constructing the Circumradius

As it turns out, constructing a circumradius of a triangle can be done using just a straightedge and a compass. It revolves around the fact that the circumcenter of a triangle is also the point at which the perpendicular bisectors of the sides of the triangle intersect. The perpendicular bisector of a line segment is a line that is perpendicular to the line segment and cuts the line segment exactly in half.

This construction assumes that one is already familiar with constructing perpendicular bisectors using a straightedge and a compass. Also, it's important to note that the circumcenter of a triangle can fall inside or outside of the triangle.

To create a circumradius of a triangle, we use the following steps:

1. Start with triangle ABC.


circumrad3


2. Construct the perpendicular bisector of side AB, and construct the perpendicular bisector of side BC. Call the intersection point of the two perpendicular bisectors M. This is the circumcenter of the triangle, so it's the center of the circumcircle of the triangle.


circumrad4


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