What is a circumscribed circle?

A circumscribed circle (circumcircle) is the smallest circle that passes through all vertices of a polygon. Every triangle has a unique circumcircle, and every regular polygon has one as well.

How do I find the circumradius of a triangle?

Use the formula R = abc / (4K), where a, b, c are the side lengths and K is the triangle's area. The area can be found using Heron's formula.

Does every polygon have a circumscribed circle?

No. All triangles and all regular polygons have circumscribed circles, but irregular polygons with 4+ sides may not. A polygon that does is called cyclic.

Where is the circumcenter of a triangle?

The circumcenter is the intersection of the perpendicular bisectors of the sides. For acute triangles it is inside, for right triangles it is on the hypotenuse midpoint, and for obtuse triangles it is outside.

What is the circumradius of a regular hexagon?

For a regular hexagon with side length s, the circumradius R equals s. This is because sin(π/6) = 0.5, so R = s / (2 × 0.5) = s.

How does the area ratio change with number of sides?

As the number of sides n increases, the regular polygon fills more of its circumscribed circle. The ratio circle area / polygon area approaches 1 as n → ∞.

Circumscribed Circle Calculator

Calculate the circumscribed circle (circumcircle) of a triangle or regular polygon. Find the circumradius, circle area, area ratios, and perimeter comparisons in one view.

Mode

Side a

Side b

Side c

Planning notes, formulas, and examples

About the Circumscribed Circle Calculator

The circumscribed circle — also called a circumcircle — is the unique circle that passes through all vertices of a polygon. For any triangle, a circumcircle always exists and is unique, with its center (the circumcenter) equidistant from all three vertices. For regular polygons, the circumscribed circle is the circle that touches every vertex of the polygon.

This calculator handles two modes: triangles defined by three side lengths and regular polygons defined by their number of sides and side length. For triangles, the circumradius R is calculated using the elegant formula R = abc / (4K), where a, b, c are the side lengths and K is the area found via Heron's formula. For regular polygons, R = s / (2 sin(π/n)), where s is the side length and n is the number of sides.

Understanding circumscribed circles is essential in computational geometry, CAD design, mesh generation for finite element analysis, and many areas of mathematics. The ratio between the circumscribed circle's area and the inscribed polygon reveals important insights about geometric efficiency and how polygons approximate circles as the number of sides increases. Use this calculator to explore circumradii, compare areas and perimeters, and visualize the relationship between shapes and their circumscribed circles.

When This Page Helps

Use this when you need the circumradius or full circumcircle from triangle data for geometry proofs, surveying, mesh generation, or CAD layout. It keeps the triangle measurements and the resulting circle tied together so you can verify that the construction matches the sides and angles you entered.

How to Use the Inputs

Choose between Triangle mode (3 sides) or Regular Polygon mode (n sides + side length).
Enter your measurements or click a preset to load common examples.
For triangles, enter the three side lengths — the calculator verifies they form a valid triangle.
For polygons, enter the number of sides (≥ 3) and the side length.
Review the circumradius, areas, ratios, and visual comparisons below.
Open the reference table to compare circumradii across different polygon types.

Formula used

Triangle: R = abc / (4K), where K = √(s(s−a)(s−b)(s−c)) and s = (a+b+c)/2
Regular Polygon: R = s / (2 sin(π/n))
Circle Area = πR²
Polygon Area = ½ n s² / tan(π/n)

Example Calculation

Result: For mode=5, sidea=10, sideb=15, the tool returns the solved circumscribed circle outputs shown in the result cards.

This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in circumscribed circle formulas and reports derived values, checks, and classifications automatically.

Tips & Best Practices

A right triangle's circumradius equals half its hypotenuse — try the 3-4-5 preset to verify.
For equilateral triangles, R = a / √3.
As the number of polygon sides → ∞, the area ratio approaches 1 (polygon → circle).
The circumcenter of an acute triangle lies inside, of a right triangle on the hypotenuse, and of an obtuse triangle outside.
Regular polygon circumradii decrease relative to perimeter as n grows.

When To Use This Calculator

Calculate the circumscribed circle (circumcircle) of a triangle or regular polygon. Find the circumradius, circle area, area ratios, and perimeter comparisons in one view. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.

How To Check The Result

Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.

Practical Notes

Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

A circumscribed circle (circumcircle) is the smallest circle that passes through all vertices of a polygon. Every triangle has a unique circumcircle, and every regular polygon has one as well.