How to Find the Circumcenter of a Triangle

The circumcircle and circumcenter of a triangle

Theory

Circumcircle, Circumcenter and Perpendicular Bisectors

The perpendicular bisectors of the sides in a triangle have one common point of intersection. This point is the center of the circle that passes through all three corners of the triangle. This circle is called the circumscribed circle or the circumcircle, and the center of the circle is called the circumcenter. The radius $R$ of the circumscribed circle is given by

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2 R

When you are asked to find the circumscribed circle of a triangle, you need to construct the perpendicular bisector of two of the sides. The intersection of the sides is where you find the circumcenter. You put the needle point of the compass on the circumcenter and make a circle that intersects all three corners of the triangle.

Notice that the circumcenter can actually be outside of the triangle. This happens when one of the angles is greater than $90$ °.

Example 1

A triangle has the sides $A B = 5$ , $A C = 6$ and $B C = 3$ . Construct the triangle and the corresponding circumcircle.

Before you construct the circumcircle, you begin by constructing the triangle with the given measurements. Begin with the line $A B = 5$ . Set the compass’s radius to 3, and make a faint circle with the center $B$ . Then, set the compass’s radius to 6, and make a faint circle with the center $A$ . The corner $C$ appears as the intersection between the two circles. You end up with the following construction:

Example of construction of circumcenter of triangle 1

Then, you construct the perpendicular bisector for two of the sides. The perpendicular bisectors intersect in the circumcircle, which you call $O$ . Then you construct a circle with a center on the circumcenter $S$ , that also touches all the corners of $△ A B C$ .

Example of construction of circumcenter of triangle 2

You see that the circumcenter

O

is outside of the triangle—that is because

∠ A B C

is greater than

90

°.

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