Regular Polygon Calculator

Background.

The Quanta regular polygon calculator handles every regular polygon from a triangle to a 1000-gon. Tell it how many sides the shape has and give it any single measurement — a side, the apothem, the circumradius, the across-flats distance, the across-corners distance, the area or the perimeter — and it returns all eleven standard properties: area, perimeter, side length, apothem, circumradius, both across-diameters, the interior and exterior angles, the sum of the interior angles, and the number of diagonals. With the default hexagon of side 10, the area comes back as 259.8076211353, the apothem as 8.6602540378, the circumradius as exactly 10, and the interior angle as exactly 120°.

A regular polygon is one where all the sides are the same length and all the angles are the same size. That is a strong enough condition that, once you fix the number of sides, the shape has only one degree of freedom: a single measurement pins down everything else. This page is built around that fact, which is why the second dropdown asks which measurement you happen to have rather than assuming it is the side length. Working backwards from an area or from a measured diameter is often the real problem — and it is the part most polygon formulas sheets leave you to rearrange yourself.

The geometry behind all of it is one triangle. Join the centre of a regular n-gon to every corner and the shape falls into n identical isosceles triangles. Cut one of those in half down its own axis of symmetry and you have a right triangle whose short side is half a polygon side, whose long side is the apothem, whose hypotenuse is the circumradius, and whose angle at the centre is π ÷ n. Every formula on this page is that right triangle, written out. The apothem is ½ × side × cot(π ÷ n); the circumradius is ½ × side × csc(π ÷ n); and the area is n copies of ½ × side × apothem, which tidies up to ¼ × n × side² × cot(π ÷ n) — the three equations Wolfram MathWorld numbers (3), (5) and (7) on its Regular Polygon page.

The angles need no trigonometry at all. Triangulating an n-gon from one corner produces n − 2 triangles, and Euclid proved in Elements Book I, Proposition 32 that the interior angles of a triangle sum to two right angles. So the polygon's interior angles sum to (n − 2) × 180°, each interior angle in a regular polygon is that divided by n, and each exterior angle is the 180° remainder — which works out as 360° ÷ n, and is why the exterior angles of any convex polygon always add to a full turn no matter how many sides it has.

Two practical notes belong beside the answer. First, across flats is a real measurement only when the number of sides is even, because an odd polygon has no pair of opposite parallel sides; for odd n the figure reported is the diameter of the inscribed circle, which is what twice the apothem always means. Second, these formulas describe convex regular polygons. Star polygons such as the pentagram are also called regular under a broader definition and have a different area, and an irregular shape has no single apothem or circumradius at all — for that you need the vertex coordinates and the shoelace formula.

This page also covers the hexagon case in full, including the fastener question: enter across flats 19 with 6 sides and it returns a side length of 10.9696551146, because a hexagon's across-flats distance is its side times √3. That is the same reduction MathWorld makes when it notes that a regular hexagon's inradius, circumradius and area 'can be computed directly from the formulas for a general regular polygon'.

What is regular polygon calculator?

A regular polygon is a plane figure with n straight sides, all of equal length, meeting at n equal angles. The familiar members are the equilateral triangle (n = 3), the square (n = 4), the regular pentagon (n = 5), the regular hexagon (n = 6) and the regular octagon (n = 8), and the family continues indefinitely, converging on a circle as n grows. Two circles are associated with every regular polygon: the inscribed circle, which touches the midpoint of every side and whose radius is called the apothem or inradius, and the circumscribed circle, which passes through every vertex and whose radius is the circumradius. Wolfram MathWorld's Regular Polygon entry gives the standard relations for a polygon of n sides of length a: the inradius r = ½a·cot(π/n) as its Equation (3), the circumradius R = ½a·csc(π/n) as its Equation (5), and the area A = ¼na²·cot(π/n) as its Equation (7). The perimeter is simply na. The angle results are independent of the trigonometry and follow from triangulation: cutting an n-gon into n − 2 triangles from a single vertex, and applying Euclid's Elements Book I, Proposition 32 — 'the sum of the three interior angles of the triangle equals two right angles' — gives a total of (n − 2) × 180° for the interior angles of any simple convex n-gon, regular or not. Regularity then makes each one equal, at (n − 2) × 180° ÷ n, with an exterior angle of 360° ÷ n. The diagonal count, n(n − 3) ÷ 2, comes from counting rather than geometry: each of the n vertices joins to n − 3 non-adjacent vertices, and halving corrects for counting each diagonal from both ends.

How to use this calculator.

1. 01Enter the number of sides. Whole numbers from 3 to 1000 — 3 is an equilateral triangle, 4 a square, 5 a pentagon, 6 a hexagon, 8 an octagon, 12 a dodecagon.
2. 02Choose which measurement you already have from the second dropdown. Side length is the usual one, but apothem, circumradius, across flats, across corners, area and perimeter all work equally well and are often what you actually measured.
3. 03Enter that value in any unit. Lengths come back in the same unit; the area comes back in that unit squared; angles are always in degrees.
4. 04Read the area at the top, then the perimeter and the side length. If you want to sanity-check the area, confirm it equals half the perimeter times the apothem — that identity holds for every regular polygon.
5. 05Use the apothem when you need the distance from the centre to the middle of a side — for laying out a paving pattern, a bolt circle, or the inside radius of a socket. Use the circumradius when you need the distance from the centre to a corner, which is the radius of the circle you would draw the polygon inside.
6. 06For fasteners and machined parts, use Across flats. A hex nut or bolt head is specified by that measurement, and entering it with 6 sides gives you the side length and the across-corners distance you need for clearance.
7. 07Note that across flats is a real distance only for an even number of sides. With an odd number there are no opposite parallel sides, and the figure shown is the inscribed circle's diameter — still twice the apothem, but not a caliper measurement.
8. 08For an irregular shape, this page is the wrong tool: it has no single apothem or circumradius. Use the irregular polygon area calculator with your vertex coordinates instead.

The formula.

ONE TRIANGLE DOES ALL THE WORK. Join the centre of a regular n-gon to each of its n corners and the shape splits into n identical isosceles triangles, each with a polygon side as its base and the two circumradii as its equal sides. The angle at the centre in each is 2π ÷ n, since n of them fill a full turn. Now cut one of those triangles in half along its own axis of symmetry. What is left is a right triangle whose side opposite the centre is a ÷ 2 (half a polygon side), whose side adjacent to the centre is the apothem r, whose hypotenuse is the circumradius R, and whose angle at the centre is π ÷ n. Basic trigonometry on that one figure gives everything: tan(π ÷ n) = (a ÷ 2) ÷ r, so r = ½a·cot(π ÷ n); and sin(π ÷ n) = (a ÷ 2) ÷ R, so R = ½a·csc(π ÷ n). These are MathWorld's Equations (3) and (5).

AREA. Each of the n isosceles triangles has base a and height r, so its area is ½ar, and the polygon's area is n × ½ar = ½ × (na) × r = ½ × perimeter × apothem. Substituting r = ½a·cot(π ÷ n) gives A = ¼na²·cot(π ÷ n), MathWorld's Equation (7). The half-perimeter-times-apothem form is the one worth remembering, because it is the exact two-dimensional analogue of a circle's A = ½ × circumference × radius, and it shows why the polygon's area approaches πR² as n grows: the apothem creeps up towards the circumradius and the perimeter towards 2πR.

ANGLES, WITHOUT ANY TRIGONOMETRY. Pick one vertex of an n-gon and draw every diagonal from it. The polygon falls into n − 2 triangles. Euclid's Elements Book I, Proposition 32 states that 'the sum of the three interior angles of the triangle equals two right angles', so the polygon's interior angles total (n − 2) × 180°. This holds for any simple convex polygon, regular or not. Regularity adds that all n angles are equal, so each is (n − 2) × 180° ÷ n. The exterior angle is the supplement, 180° − (n − 2) × 180° ÷ n, which simplifies to 360° ÷ n — and since there are n of them, the exterior angles always total exactly 360°, however many sides the polygon has.

DIAGONALS. Take any vertex. It can be joined to n − 1 other vertices, but two of those joins are sides rather than diagonals, so it contributes n − 3 diagonals. Doing that at all n vertices counts every diagonal twice, once from each end, so the total is n(n − 3) ÷ 2. Equivalently, it is the number of vertex pairs C(n, 2) minus the n pairs that are sides. A triangle gets 3 × 0 ÷ 2 = 0, a square 4 × 1 ÷ 2 = 2, a hexagon 6 × 3 ÷ 2 = 9. This result is derived here rather than cited, because it is a two-line counting argument rather than an empirical or standards-governed value.

WORKING THE DEFAULT ALL THE WAY THROUGH. n = 6, side = 10. Then π ÷ 6 is 30°, cot 30° = √3 = 1.7320508076 and csc 30° = 2. The apothem is ½ × 10 × 1.7320508076 = 8.660254037844386, displayed as 8.6602540378. The circumradius is ½ × 10 × 2 = 10 exactly — equal to the side, which is the hexagon's signature property and the reason it is six equilateral triangles. The perimeter is 6 × 10 = 60. The area is ½ × 60 × 8.660254037844386 = 259.80762113533159, displayed as 259.8076211353. Across flats is 2 × 8.660254037844386 = 17.3205080757, which is the side times √3. Across corners is 2 × 10 = 20. The interior angle is 4 × 180 ÷ 6 = 120° exactly, the exterior is 360 ÷ 6 = 60°, the interior angles sum to 720°, and there are 6 × 3 ÷ 2 = 9 diagonals.

A CROSS-CHECK THAT USES NO TRIGONOMETRY AT ALL. MathWorld's separate Regular Hexagon page gives closed forms for n = 6: area (3√3 ÷ 2)a², inradius (√3 ÷ 2)a, circumradius a. At a = 10 those are 259.8076211353, 8.6602540378 and 10 — identical to the values above, reached without a single trigonometric function. The square (area exactly a²) and the equilateral triangle (area (√3 ÷ 4)a²) provide two further trig-free checks at other values of n. All three are in the test suite rather than only in this paragraph.

ROUNDING STAGE. Every calculation, including the trigonometry, runs at 40 significant digits, with π computed as 4 × arctan(1) rather than typed in. Whichever measurement you supply is converted to a side length first, and that side length is not rounded before the area, apothem, radii and diameters are derived from it — rounding it first would put visible error into the area, which is quadratic in the side. The angles are computed by exact fraction arithmetic instead of through the trigonometric path, which is why a hexagon returns exactly 120° rather than 119.999999999, and the diagonal count is exact integer arithmetic. Rounding happens once, at the return boundary, to 10 decimal places.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — 'Regular Polygon' (live revision retrieved 2026-07-29). Equation (3) gives the inradius r = ½a·cot(π/n), Equation (5) the circumradius R = ½a·csc(π/n), and Equation (7) the area A = ¼na²·cot(π/n) for a regular polygon of n sides of length a. All three confirmed on retrieval. Open access.
2. [2]Wolfram Research, MathWorld — 'Regular Hexagon' (live revision retrieved 2026-07-29). Gives the n = 6 case in closed form with no trigonometry — area (3√3/2)a², inradius (√3/2)a, circumradius a — which is the independent cross-check used on this page's general formulas, and states that a regular hexagon's properties 'can be computed directly from the formulas for a general regular polygon'. Open access.
3. [3]Euclid, Elements, Book I, Proposition 32 — 'In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.' Applied to the n − 2 triangles a polygon divides into, this gives the (n − 2) × 180° interior-angle sum. David E. Joyce web edition of the Heath translation, Clark University; statement confirmed verbatim on retrieval 2026-07-29. Open access.
4. [4]Wolfram Research, MathWorld — 'Polygon Diagonal' (retrieved 2026-07-29). Recorded for transparency: this entry does NOT give the total diagonal count. It gives the number of regions the diagonals cut the interior into, C(n,4) + C(n−1,2). Rather than attribute n(n − 3)/2 to a source that does not state it, this page derives that count in full from the counting argument. Open access.