How to Find the Circumcenter of a Triangle

The circumcenter is the centre of the one circle that passes through all three corners of a triangle. Unlike the centroid it cannot be averaged out of the coordinates, because it is defined by equal distances rather than by balance.

Why perpendicular bisectors

The perpendicular bisector of a segment is the set of every point equidistant from its two ends. That is not a fact about triangles, it is what a perpendicular bisector is.

So take the bisector of one side and every point on it is equally far from those two corners. Take the bisector of a second side and every point on that is equally far from its two corners. Where they cross, a single point is equidistant from all three, and a circle centred there passes through every one of them.

The third bisector is not needed. It has to pass through the same point, because that point already sits equidistant from the two corners the third side connects. Drawing it is a check, not a step.

A worked example

Take a right triangle with corners at (0,0), (6,0) and (0,8). The side from (0,0) to (6,0) lies along the x axis, its midpoint is (3,0), and the line through that at a right angle is simply x = 3.

The side from (0,0) to (0,8) lies along the y axis, its midpoint is (0,4), and the perpendicular there is y = 4. The two cross at (3,4).

Check the distances. From (3,4) to (0,0) is the square root of 9 plus 16, which is 5. To (6,0) is the square root of 9 plus 16 again, also 5. To (0,8) is the square root of 9 plus 16 once more, still 5. All three match, so the circle centred at (3,4) with radius 5 passes through every corner.

Where it lands depends on the triangle

On an acute triangle the circumcenter sits inside. On a right triangle it lands exactly on the midpoint of the hypotenuse, which is why the hypotenuse is a diameter of the circumcircle. On an obtuse triangle it falls outside the shape altogether.

That last case surprises people, but nothing is wrong. The circle still passes through all three corners, and its centre simply happens to lie beyond one of the sides. If your answer lands outside, check whether the triangle is obtuse before assuming an error.

Mistakes that catch people out

• Using angle bisectors instead. Those meet at the incenter, which is a different point.
• Using medians instead. Those meet at the centroid, which is a third different point.
• Assuming it must be inside. On an obtuse triangle the circumcenter falls outside, and that is correct.
• Finding the midpoint and stopping. The midpoint is where the bisector starts. You still need the perpendicular line through it.
• Taking the perpendicular slope wrongly. It is the negative reciprocal. A side of slope 2 needs a bisector of slope minus a half.

Practice questions

Common questions

How do I find the circumcenter of a triangle?

Find the perpendicular bisector of two sides and see where they cross. That point is the same distance from all three corners.

What is the circumcenter?

The centre of the one circle passing through all three corners. It is equidistant from every vertex, which is what defines it.

Do I need all three bisectors?

No. Two is enough, since the third always passes through the same point. Drawing it is a useful check rather than a requirement.

Can the circumcenter fall outside the triangle?

Yes. On an obtuse triangle it lands outside, and on a right triangle it sits exactly on the hypotenuse midpoint.

Is the circumcenter the same as the centroid?

No. The centroid is the balance point and always sits inside. The two coincide only on an equilateral triangle.

What is the difference between the circumcenter and the circumradius?

The circumcenter is a point, given as coordinates. The circumradius is a length, the distance from that point out to any corner.

Related lessons and calculators

• How to Find the Radius of the Circumscribed Circle for the radius from the side lengths, with no coordinates needed.
• How to Find the Centroid of a Triangle From Its Vertices for the balance point, which is a different centre.
• How to Find the Radius of the Inscribed Circle in a Triangle for the incircle, found from angle bisectors instead.
• Distance and Pythagoras Calculator for checking the three distances come out equal.
• Area of a Triangle: Every Formula and When to Use Each for the overview of every area method in one place.
• National Council of Teachers of Mathematics for curriculum standards and how this topic is sequenced in school.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.