Free Circumscribed Circle Calculator

The Circumscribed Circle and Its Radius

A circumscribed circle, or circumcircle, is the circle that passes through all three vertices of a triangle. Its radius is the circumradius, and its center is the circumcenter—the meeting point of the triangle's perpendicular bisectors. This circle exists for every triangle, but the same cannot be said for other polygons. Among quadrilaterals, for instance, only rectangles (including squares) possess a circumcircle; a non-square rhombus does not.

The Circumscribed Circle Calculator (a free online geometry tool) streamlines the process of finding the circumradius of a triangle, also known as the triangle's circumcircle radius. By inputting just the side lengths $a$, $b$, and $c$, you instantly receive not only the circumradius but also the diameter, circumference, and area of the circumcircle. This calculator eliminates manual computation, making it a reliable aid for students, engineers, and geometry enthusiasts.

Formula for the Circumradius

For any triangle with sides $a$, $b$, $c$, the circumradius $R$ is given by:

$$R = \dfrac{abc}{4A}$$

where $A$ is the area of the triangle. The area can be computed using Heron's formula:

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

with $s = \dfrac{a+b+c}{2}$ (half the perimeter). Once $R$ is known, several derived quantities are easily obtained:

• Diameter: $d = 2R$
• Circumference: $C = 2\pi R$
• Area of the circumcircle: $A_{\text{circle}} = \pi R^2$
• Ratio of circumcircle area to triangle area: $\dfrac{\pi R^{2}}{A}$

Special Triangles Simplified

For an equilateral triangle (side $a$), the circumradius reduces to $R = \dfrac{a}{\sqrt{3}}$, and the circumcenter coincides with the orthocenter. For a right triangle (hypotenuse $c$), the circumradius is simply $R = \dfrac{c}{2}$; the circumcenter is the midpoint of the hypotenuse. These shortcuts make calculations even faster.

Using the Circumradius Calculator

Operating the circumcircle calculator is straightforward. Enter the triangle's side lengths into the fields labeled $a$, $b$, and $c$. The tool automatically computes the circumradius, diameter, circumference, and circular area. Additional information—such as the triangle's area and the ratio between the two areas—appears in the "Additional results" section. This eliminates the risk of arithmetic errors and provides a complete geometric picture.

Constructing a Circumscribed Circle

To draw a circumcircle for a given triangle, locate its circumcenter by constructing the perpendicular bisectors of any two sides. Their intersection is the circumcenter. With a compass set to the circumradius (the distance from the circumcenter to any vertex), draw the circle that passes through all three vertices.