How this instrument works
A circumscribed circle is the single circle that passes through every corner of a regular polygon at once - not an approximation, an exact fit, since all the corners of a regular polygon sit the same distance from its center by construction. That shared distance is the circumradius R, and it is fixed by just two numbers: the side length s and the side count n. Split the polygon into n identical isosceles triangles meeting at the center, each with two sides of length R and an apex angle of 360°⁄n. Bisect that apex angle and a right triangle falls out with hypotenuse R, opposite side s⁄2, and angle 180°⁄n - so sin(180°⁄n) = (s⁄2) ⁄ R, which rearranges directly into R = s ⁄ (2 sin(180°⁄n)).
The ratio R ⁄ s is not constant across polygons; it climbs steadily as n grows. A triangle's circumradius is only about 0.577 times its side, a square's about 0.707 times, and a hexagon's lands exactly on 1 - a hexagon is the one regular polygon whose circumradius equals its own side length, because six equilateral triangles of that side length tile perfectly around a center at that same radius. Push past six sides and the ratio keeps rising, since a polygon with more corners packed at a fixed side length has to be a bigger shape overall.
Push n toward infinity with s held fixed and the polygon's outline approaches a circle, and the formula quietly turns into the one for that circle: sin(180°⁄n) is nearly 180°⁄n itself once the angle is small and read in radians, so R closes in on s·n ⁄ (2π) - the full perimeter divided by 2π, exactly how a circle's radius is recovered from its own circumference. The lower edge matters too: below three sides there is no polygon left to circumscribe, which is why this sheet refuses n under 3.
- Type how many corners the regular polygon has into Number of sides, n - any whole number from 3 upward.
- Enter the length of one edge into Side length, in whichever unit you want the answer expressed in.
- Read Circumradius for the radius of the circle running through every corner, in that same unit.
- Try the defaults, sides 6 and Side length 1, and confirm Circumradius comes back at 1 - the hexagon's exact case.
- Raise Number of sides, n while Side length stays fixed and watch Circumradius climb, closing in on a circle's own radius.
Worked example — a regular hexagon with side 1
Set Number of sides, n to 6 and Side length to 1, the stop-sign shape. The sheet works R = 1 ⁄ (2·sin(180°⁄6)) = 1 ⁄ (2·sin(30°)) = 1 ⁄ (2 × 0.5) = 1 ⁄ 1, and Circumradius reports 1.0000000000000002 - the trailing digit is ordinary floating-point residue from computing sin(π⁄6) on a computer, not an error in the geometry, which gives exactly 1.
That exact 1 is not a coincidence of these particular numbers. A regular hexagon divides perfectly into six equilateral triangles meeting at its center, and an equilateral triangle has all three sides equal - including the two that happen to be radii of the circumscribed circle. So the hexagon's own edge, the third side of each triangle, is forced to equal that same radius, making the hexagon the unique regular polygon where R = s holds for any size at all.
Questions
What is the circumradius of a regular polygon?
It is the radius of the single circle that passes through every vertex of the polygon at once, given by R = s ⁄ (2 sin(180°⁄n)) for n sides of length s. Every regular polygon has exactly one such circle, found by bisecting the isosceles triangle formed by the center and any two adjacent corners.
Where does the formula R = s ⁄ (2 sin(180°⁄n)) come from?
It comes from bisecting one of the n identical isosceles triangles meeting at the polygon's center. Each has two sides equal to R and an apex angle of 360°⁄n; splitting it down the middle gives a right triangle with hypotenuse R, opposite side s⁄2, and angle 180°⁄n, so sin(180°⁄n) = (s⁄2) ⁄ R - rearranged, that is exactly this sheet's formula.
Why is a regular hexagon the polygon whose circumradius equals its side length?
Because R ⁄ s = 1 ⁄ (2 sin(180°⁄n)) rises steadily as n grows, and n = 6 is exactly where it crosses 1. A triangle's ratio is about 0.577, a square's about 0.707, a pentagon's about 0.851 - all under the side length - while a heptagon's is about 1.152, already past it. The hexagon sits precisely on that boundary because six equilateral triangles of side s tile around a center that is also a distance s away.
How is the circumradius different from the apothem?
The circumradius reaches from center to a vertex; the apothem reaches from center to a side's midpoint, a shorter distance for any polygon past a line segment. The apothem equals R cos(180°⁄n), so the two agree only in the limiting case of a circle. Mixing them up is a common error when someone asks this sheet for a polygon's radius and expects the apothem back instead.
What happens to the circumradius as the number of sides grows very large?
R approaches perimeter ⁄ (2π), the same relationship an ordinary circle has between its own circumference and radius. For large n, sin(180°⁄n) sits very close to 180°⁄n measured in radians, π⁄n, so R ≈ s ⁄ (2π⁄n) = ns ⁄ (2π) - the polygon's full perimeter divided by 2π. A hundred-sided polygon already looks convincingly round by this measure.
Why does this sheet refuse fewer than three sides?
Three is the fewest sides a closed polygon can have, so Number of sides, n cannot drop below it. At n = 3 the formula still holds exactly: an equilateral triangle of side 1 returns a circumradius of 1 ⁄ √3 ≈ 0.577, the smallest ratio any regular polygon produces, since a triangle's corners sit as close to its own edges as a regular shape allows.