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Polygon Circumradius Calculator

Find the circumradius of a regular polygon.

How it is calculated

For a regular polygon with n equal sides of length s, the circumradius (centre to a corner) is R = s / (2 × sin(π/n)) and the inradius, or apothem (centre to the middle of a side), is r = s / (2 × tan(π/n)). The angle π/n is in radians. Both come from the right-angled triangle formed by the centre, the middle of one side and one corner, where the angle at the centre is half the angle subtended by the side.

The number of sides is rounded to a whole number, so 5.6 is treated as 6. It must be at least 3, and the side must be greater than zero, otherwise there is no result. Results are shown to four decimals in the units you used for the side. The polygon must be regular: unequal sides or angles have no single circumradius.

Worked example

A regular pentagon, n = 5, with side 10. The angle π/5 is 36°. sin 36° is about 0.5878, so R = 10 / (2 × 0.5878) = 10 / 1.1756 = 8.5065 units. tan 36° is about 0.7265, so r = 10 / (2 × 0.7265) = 10 / 1.4531 = 6.8819 units. A circle of radius 8.5065 passes through all five corners and a circle of radius 6.8819 touches the middle of every side.

Try this example →

Common mistakes

  • Entering the polygon's width or diameter instead of one side length. The input is a single edge.
  • Mixing up circumradius and inradius. The circumradius reaches the corners and is always the larger one; the inradius reaches the sides, which matters for fitting a shape inside a round hole.
  • Typing a number of sides below 3. Two sides do not enclose an area, so there is no result.
  • Using the formula for a shape that only looks regular. Unequal sides or angles mean the tool's answer does not describe it.
  • Entering a fractional side count, such as 5.5, and expecting something between a pentagon and a hexagon. It is rounded to a whole number first.

About this calculator

For a regular polygon with n sides of length s, this calculator finds the circumradius R = s / (2 sin(π/n)) — the distance from the center to each vertex — and the inradius (apothem) r = s / (2 tan(π/n)), the distance from the center to the middle of each side. Both come from splitting the polygon into n identical isosceles triangles with a central angle of 2π/n.

Architects and carpenters use the circumradius to lay out gazebos, pavilions, and cupolas on a circular foundation, machinists use it to space bolt holes evenly on a bolt-circle pattern, and manufacturers of nuts and hex stock use twice the inradius (the across-flats distance, i.e. the wrench size) to check part dimensions against twice the circumradius (the corner-to-corner distance).

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