Circumradius of Regular Polygon Calculator
The circumradius of a regular polygon is the distance from its centre to any corner, equal to the radius of the circle that passes through every vertex. It tells you how big a circle you need to draw a regular polygon by hand, and it pairs naturally with the apothem to describe the shape fully. This tool computes the circumradius from the number of sides and the side length using an exact trigonometric formula, and also reports the apothem for comparison. Enter a whole number of sides and a side length in any unit.
Circumradius formula
Circumradius = s / (2 * sin(pi / n))
Apothem = s / (2 * tan(pi / n))
Circumscribed diameter = 2 * circumradius
Here n is the number of sides and s is the side length, with the angle pi / n in radians. The circumradius reaches a vertex; the apothem reaches a side midpoint.
How the circumradius works
- The circumradius is the radius of the circle through every vertex of the polygon.
- It is always longer than the apothem, which reaches only to a side midpoint.
- The two relate by apothem equals circumradius times cosine of pi over n.
- As the side count grows, the polygon fills its circumscribed circle ever more closely.
- The side count must be a whole number of three or more, so smaller values are invalid.
Circumradius: frequently asked questions
What is the circumradius of a regular polygon?
The circumradius is the distance from the centre of a regular polygon to any of its vertices, the radius of the circumscribed circle that passes through every corner. It is longer than the apothem, which only reaches the midpoint of a side. Every vertex of a regular polygon lies on this circle.
What is the formula for the circumradius?
For a regular polygon with n sides each of length s, the circumradius is R = s / (2 * sin(pi / n)), where the angle pi / n is in radians. This exact trigonometric result comes from the isosceles triangle formed by the centre and two adjacent vertices, whose apex angle is 2 * pi / n.
How does the circumradius relate to the apothem?
The apothem is the circumradius times the cosine of pi / n, since the apothem and circumradius are the two legs of the same right triangle from the centre. The apothem reaches a side midpoint, the circumradius reaches a vertex, so the circumradius is always the larger of the two.
What units does the circumradius use?
The circumradius is in the same length unit as the side length you enter. If the side is in metres, the circumradius is in metres. The reported apothem uses the same unit. Keep your input consistent and the outputs follow directly with no unit conversion.
What is the minimum number of sides?
A polygon needs at least three sides, so the smallest case is an equilateral triangle. This calculator treats any side count below three as invalid, and the side count must be a whole number, since a polygon cannot have a fractional number of corners.
Official sources
- National Institute of Standards and Technology: Digital Library of Mathematical Functions: trigonometric functions.
- National Institute of Standards and Technology: SI Units and measurement.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.