Circumscribed Circle Calculator

What Is a Circumscribed Circle?

A circumscribed circle (commonly known as a circumcircle) is a circle that passes through every vertex of a given polygon. When a polygon possesses a circumcircle, the polygon is said to be inscribed in the circle, and the polygon itself is termed a cyclic polygon.

The center of the circumcircle is the circumcenter, and its radius is denoted as the circumradius ($R$). Every triangle and regular polygon always has a unique circumcircle. For quadrilaterals, a circumcircle exists if and only if the opposite angles sum to 180 degrees (cyclic quadrilateral).

Circumradius Formulas for Common Geometric Shapes

The circumradius formula varies depending on the geometry of the inscribed polygon:

1. General Triangle (SSS)

For any triangle with side lengths $a$, $b$, and $c$, the circumradius $R$ is related to the triangle's area:

$$R = \frac{a \cdot b \cdot c}{4 \cdot \text{Area}}$$

Where the area is computed using Heron's formula with semiperimeter $s = (a + b + c) / 2$:

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

2. Equilateral Triangle

For an equilateral triangle where all sides equal $a$:

$$R = \frac{a}{\sqrt{3}} = \frac{a\sqrt{3}}{3}$$

3. Right-Angled Triangle

By Thales's theorem, the hypotenuse $c$ of a right triangle is the diameter of its circumcircle. Therefore:

$$R = \frac{c}{2} = \frac{\sqrt{a^2 + b^2}}{2}$$

4. Rectangle and Square

The circumcircle of a rectangle passes through its four corners, with the diagonal $d$ acting as the diameter:

$$R = \frac{\sqrt{l^2 + w^2}}{2}$$

For a square of side $a$, this simplifies to:

$$R = \frac{a\sqrt{2}}{2} = \frac{a}{\sqrt{2}}$$

5. Regular Polygon with $n$ Sides

For any regular $n$-sided polygon with side length $a$:

$$R = \frac{a}{2 \cdot \sin(\pi / n)}$$

Properties of the Circumscribed Circle

• Area of Circumcircle: $A_{\text{circle}} = \pi R^2$
• Circumference: $C = 2\pi R = \pi D$
• Diameter: $D = 2R$
• Coverage Ratio: The ratio of the polygon area to the circumcircle area demonstrates the packing efficiency of the shape.

Explore Related Geometry Tools

Discover more mathematical tools from our collection:

• Circumcenter of a Triangle Calculator: Find circumcenter coordinates and perpendicular bisector lines.
• Circle Calculator: Convert between radius, diameter, circumference, and circle area.
• Area of a Circle Calculator: Calculate circle area from various parameters.
• Equilateral Triangle Calculator: Solve side, height, perimeter, and area.
• Right Triangle Calculator: Solve legs, hypotenuse, and trigonometric ratios.