Circumcenter
In short: A triangle’s circumcenter is the single point sitting the same distance from all three corners, found where the perpendicular bisectors of the sides cross.
GeometryGeometryG-C.A.3
What is Circumcenter?
If three villages must share one well with equal walks for everyone, the answer is the circumcenter of the triangle they form. Each side has a perpendicular bisector, and every spot on it is equally far from that side’s two ends; two such lines already intersect at the winning point, and the third passes through it as well.
The shared distance is the circumradius, and the circle of that radius touches all three corners. Its center lies inside an acute triangle, on the longest side of a right triangle at the midpoint, and outside an obtuse triangle.
How to spot it
- A problem asks for a location equally far from three towns, three trees or three corners of a lot.
- A circle must pass through all three vertices of a triangle, and its center or radius is requested.
- The words circumscribed circle, circumradius or perpendicular bisectors of the sides appear.
- A right triangle is given and the center of the circle around it is wanted.
The method
- Find the midpoint and the slope of two of the sides.
- Write the perpendicular bisector of each, using the negative reciprocal slope.
- Solve the two equations together; the solution is the center.
- Measure from the center to each corner; the three distances must agree.
Circumcenter: solved example
Problem
Three wells sit at \(P(0, 0)\), \(Q(8, 0)\) and \(R(0, 6)\) on a ranch map. Where should a pump house go so it is the same distance from all three?
Solution
The perpendicular bisector of \(\overline{PQ}\) is the vertical line \(x = 4\). The perpendicular bisector of \(\overline{PR}\) is the horizontal line \(y = 3\). They cross at \((4, 3)\).
Check the distances: to \(P\), \(\sqrt{16 + 9} = 5\); to \(Q\), \(\sqrt{16 + 9} = 5\); to \(R\), \(\sqrt{16 + 9} = 5\). The pump house goes at \((4, 3)\), 5 units from each well.
Try it
Question 1
A right triangle has legs 7 and 24, and hypotenuse 25. What is the radius of the circle through its three corners?
Show the answer
The center is the midpoint of the hypotenuse, so the radius is half of it: \(25 \div 2 = 12.5\).
Question 2
For an obtuse triangle, does the center of the surrounding circle fall inside or outside the triangle?
Show the answer
Outside, beyond the longest side, on the far side from the obtuse corner.
Circumcenter vs. Incenter
The circumcenter comes from perpendicular bisectors of the sides and reaches the corners equally, so its circle wraps around the triangle. The incenter comes from angle bisectors and reaches the sides equally, so its circle nestles inside. One is equidistant from vertices; the other from walls.
Keep going
Lessons
Worksheets and quizzes
- Geometric Constructions – Worksheet
- Geometric Constructions – Quiz
- Theorems About Triangles – Worksheet
- Theorems About Triangles – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry