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Circumscribed Circle Calculator

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A triangle with a circumscribed circle

Given a triangle's sides, the circumscribed circle calculator computes the properties of a circle that pass through each of the vertices of the triangle.

A circumscribed circle is also known as a circumcircle. Every triangle has one.

Key Terms:

  1. Circumcircle - the circle that surrounds any triangle that passes through all of its points.
  2. Circumradius - the radius of the circumcircle.
  3. Circumcenter - the center of the circumcenter, which is also where the perpendicular bisectors of the triangle's sides intersect.

How to Use This Calculator

  1. Enter the three side lengths of the triangle (must be greater than zero)

Assuming all three side lengths are greater than zero, they must also satisfy the Triangle Inequality Theorem:

  • a + b > c
  • a + c > b
  • b + c > a

Understanding the Formula

To calculate the properties of the circumcircle, the half perimeter of the triangle S must be computed.

S=12(a+b+c)S=\frac{1}{2}\left(a+b+c\right)

Next, the area of the triangle A can be found.

A=S(S−a)(S−b)(S−c)A=\sqrt{S\left(S-a\right)\left(S-b\right)\left(S-c\right)}

Note: this formula for area given three sides and the half perimeter is known as Heron's formula.

Properties of the Circumcircle

The radius of the circle:

r=abc4Ar=\frac{abc}{4A}

The diameter of the circle:

d=2rd=2r

The circumference of the circle:

c=2πrc=2\pi r

The area of the circle:

a=πr2a=\pi r^2

Additional Properties of the Triangle

The perimeter of the triangle is

p=a+b+cp=a+b+c

To reiterate, the area of the triangle is

A=S(S−a)(S−b)(S−c)A=\sqrt{S\left(S-a\right)\left(S-b\right)\left(S-c\right)}

The area ratio of the circle to the triangle is

Area Ratio=πr2S(S−a)(S−b)(S−c)\text{Area Ratio}=\frac{\pi r^2}{\sqrt{S\left(S-a\right)\left(S-b\right)\left(S-c\right)}}