Circumcircle of a Triangle - Math Open Reference (original) (raw)

A circle which passes through all three verticesof a triangle
Also "Circumscribed circle".

Try this Drag the orange dots on each vertexto reshape the triangle. Note that the circumcircle always passes through all three points.

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

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

The radius of the circumcircle is also called the triangle's circumradius.

For right triangles

In the case of a right triangle, the hypotenuseis a diameterof the circumcircle, and its center is exactly at the midpoint of the hypotenuse. This is the same situation as Thales Theorem, where the diameter subtendsa 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.

For equilateral triangles

In the case of an equilateral triangle, where all three sides (a,b,c) are have the same length, the radius of the circumcircle is given by the formula: where s is the length of a side of the triangle.

If you know all three sides

If you know the length (a,b,c) of the three sides of a triangle, the radius of its circumcircle is given by the formula:

If you know one side and its opposite angle

The diameter of the circumcircle is given by the formula: where a is the length of one side, and A is the angle opposite that side.

This gives the diameter, so the radus is half of that

This is derived from the Law of Sines.

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.

Other triangle topics

General

• Triangle definition
• Hypotenuse
• Triangle interior angles
• Triangle exterior angles
• Triangle exterior angle theorem
• Pythagorean Theorem
• Proof of the Pythagorean Theorem
• Pythagorean triples
• Triangle circumcircle
• Triangle incircle
• Triangle medians
• Triangle altitudes
• Midsegment of a triangle
• Triangle inequality
• Side / angle relationship

Perimeter / Area

• Perimeter of a triangle
• Area of a triangle
• Heron's formula
• Area of an equilateral triangle
• Area by the "side angle side" method
• Area of a triangle with fixed perimeter

Triangle types

• Right triangle
• Isosceles triangle
• Scalene triangle
• Equilateral triangle
• Equiangular triangle
• Obtuse triangle
• Acute triangle
• 3-4-5 triangle
• 30-60-90 triangle
• 45-45-90 triangle

Triangle centers

• Incenter of a triangle
• Circumcenter of a triangle
• Centroid of a triangle
• Orthocenter of a triangle
• Euler line

Congruence and Similarity

• Congruent triangles

Solving triangles

• Solving the Triangle
• Law of sines
• Law of cosines

Triangle quizzes and exercises

• Triangle type quiz
• Ball Box problem
• How Many Triangles?
• Satellite Orbits

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