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Circumcircle of a triangle
From Latin: circum- "around"
A circle which passes through all three vertices of a triangle
Also "Circumscribed circle".
Try this Drag the orange dots on each vertex to reshape the triangle. Note that the circumcircle always passes through all three points.
(If there is no image below, see support page.)

The circumcircle always passes through all three vertices of a triangle. Its center is at the point where all the perpendicular bisectors of the triangle's sides meet. See the page on the circumcenter of a triangle for more about this point.

Note that the center of the circle can be inside or outside of the triangle, or at the midpoint of the hypotenuse. Adjust the triangle above and try to obtain these three cases.

Special case - right triangles
In the special case of a right triangle, the hypotenuse is a diameter of the circumcircle, and its center is exactly at the midpoint of the hypotenuse. This is the same situation as Thales Theorem, where the diameter subtends a right angle to any point on a circle's circumference.

If you drag the triangle in the figure above you can create this same situation.
Attributes
Circumradius
  1. The radius of the circumcircle. The radius is given by the formula

    where a,b,c are the lengths of the sides of the triangle.

  2. In the special case of an equilateral triangle, where all three sides (a,b,c) are have the same length, there is a simpler formula:

    where s is the length of any side of the triangle.

  3. In the special case of a right triangle, the radius is half the length of the hypotenuse.
Circumcenter The center of the circumcircle. The point where the perpendicular bisectors of each side intersect. See Circumcenter of a triangle.
Construction of a triangle's circumcircle
It is possible to construct the circumcenter and circumcircle of a triangle with just a compass and straightedge. Construction of the Circumcircle of a Triangle has an animated demonstration of the technique, and a worksheet to try it yourself.
Things to Ponder
If the triangle is a right triangle, what do you notice about the size of the circumscribed circle? (use the figure above to check) answer Its diameter is the same as the triangle's hypotenuse
Challenge question: What is the circumference of the circumcircle of a triangle whose sides are 6,8 and 10 centimeters? answer 31.42 cm.   It is an expanded "3-4-5" right triangle, with the 10cm hypotenuse as the diameter. Therefore 10π.

Related triangle topics

General

Triangle types

Triangle centers

Congruence and Similarity

Triangle quizzes and exercises

(C) 2007 Copyright John Page