CalConverters

Regular Polygon Calculator: Side, Area, Angles

Find the side, apothem, circumradius, area, and interior/exterior angles of any regular polygon (3 to 1000 sides) from just one known measure.

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The five regular-polygon formulas

A regular polygon has n equal sides and n equal interior angles. Given the side length aa, every other measure follows from formulas built around π/n\pi/n:

P=naA=14na2cot⁡πn=nr2tan⁡πnP = na \qquad A = \frac{1}{4}na^2\cot\frac{\pi}{n} = nr^2\tan\frac{\pi}{n}

r=a2cot⁡πn=Rcos⁡πnR=a2csc⁡πnr = \frac{a}{2}\cot\frac{\pi}{n} = R\cos\frac{\pi}{n} \qquad R = \frac{a}{2}\csc\frac{\pi}{n}

x=(n−2)×180°ny=360°nx = \frac{(n-2)\times 180°}{n} \qquad y = \frac{360°}{n}

Here rr is the apothem (center to the midpoint of a side) and RR is the circumradius (center to a vertex) — rr is always shorter than RR for the same polygon.

Tip: π/n\pi/n inside cot/csc/tan must be in RADIANS, not degrees — a common slip when computing these formulas by hand.

Solving from any known measure

This calculator also works backward: give it n plus any ONE of side/apothem/circumradius/perimeter/area, and it solves for the side first, then derives everything else.

KnownSolve for side aThen
Side aa— (already known)P=naP = na, rr, RR, AA
Apothem rra=2rtan⁡(π/n)a = 2r\tan(\pi/n)PP, RR, AA
Circumradius RRa=2Rsin⁡(π/n)a = 2R\sin(\pi/n)PP, rr, AA
Perimeter PPa=P/na = P/nrr, RR, AA
Area AAa=4Atan⁡(π/n)/na = \sqrt{4A\tan(\pi/n)/n}PP, rr, RR

The interior and exterior angles never need to be "solved" — they depend only on n, not on the size of the polygon.

Worked example: regular pentagon (n=5), side a=5

P=5×5=25P = 5 \times 5 = 25 r=52cot⁡36°≈3.4410R=52csc⁡36°≈4.2533r = \frac{5}{2}\cot 36° \approx 3.4410 \qquad R = \frac{5}{2}\csc 36° \approx 4.2533 A=14(5)(25)cot⁡36°≈43.0119A = \frac{1}{4}(5)(25)\cot 36° \approx 43.0119 x=3×180°5=108°y=360°5=72°x = \frac{3 \times 180°}{5} = 108° \qquad y = \frac{360°}{5} = 72°

Worked example: regular hexagon (n=6), side a=6

A=14(6)(36)cot⁡30°≈93.5307x=120°y=60°A = \frac{1}{4}(6)(36)\cot 30° \approx 93.5307 \qquad x = 120° \qquad y = 60°

A regular hexagon's interior angle (120°) is a common one to memorize — it's why hexagonal tiles fit together perfectly with no gaps, unlike pentagons.

Angles by number of sides

nNameInterior angle xExterior angle y
3Triangle60°120°
4Square90°90°
5Pentagon108°72°
6Hexagon120°60°
8Octagon135°45°
10Decagon144°36°
12Dodecagon150°30°
Tip: The two angles at any vertex always sum to 180° — interior + exterior = a straight line. If your interior and exterior angles don't add to 180°, re-check the arithmetic.

Common mistakes

  • Confusing apothem and circumradius. The apothem (r) reaches a side's midpoint; the circumradius (R) reaches a vertex — R is always the longer of the two.
  • Using degrees instead of radians in cot(π/n). The π/n term is a radian angle even though it looks like a fraction of 180°.
  • Mixing up interior and exterior angles. Interior angles grow toward 180° as n increases; exterior angles shrink toward 0°.

Where this shows up

  • Tiling and paving. Only equilateral triangles, squares, and regular hexagons tile a flat surface with no gaps — their interior angles (60°, 90°, 120°) divide evenly into 360°.
  • Hardware and signage. Hex bolts and nuts are regular hexagons; a STOP sign is a regular octagon.
  • Honeycomb structure. Bees build hexagonal cells because a regular hexagon encloses the most area for the least perimeter among shapes that tile perfectly.

An equilateral triangle (n=3) and a square (n=4) are simply the smallest two cases of the same regular-polygon formulas used here — everything from a pentagon to a 1000-gon follows the identical π/n\pi/n pattern.

Frequently asked questions

What is a regular polygon?
A polygon that is both equilateral (all sides equal) and equiangular (all interior angles equal) — a triangle (n=3) with equal sides is a regular polygon, and so is a square (n=4).
What are the formulas for a regular polygon?
With n sides and side length a: perimeter P = na, area A = (1/4)na²cot(π/n), apothem r = (a/2)cot(π/n), and circumradius R = (a/2)csc(π/n). The interior angle is (n−2)×180°/n and the exterior angle is 360°/n.
What is the apothem of a regular polygon?
The apothem is the distance from the center to the midpoint of any side — it's always shorter than the circumradius, which reaches all the way to a vertex. Confusing the two is the most common mistake with these shapes.
Why is the interior angle (n−2)×180°/n?
The sum of all interior angles of any n-sided polygon is (n−2)×180° (you can split any polygon into n−2 triangles from one vertex), and a regular polygon splits that total evenly across its n equal angles.
Why do interior and exterior angles always add up to 180°?
At each vertex, the interior angle and the exterior angle sit on a straight line — turning through the exterior angle is what takes you from one side's direction to the next, so together they always form a straight angle, 180°.
How do I find the area if I only know the apothem or circumradius?
Convert to the side first: from the apothem, a = 2r·tan(π/n); from the circumradius, a = 2R·sin(π/n). Once you have the side, area follows from A = (1/4)na²cot(π/n).
Is a square a regular polygon?
Yes — a square is the regular polygon with n = 4, and an equilateral triangle is the regular polygon with n = 3. Both are special cases of the same formulas here.
Why does a large n look like a circle?
As n grows, a regular polygon's vertices get closer together and it hugs its own circumradius more tightly — in the limit, an n-gon with infinitely many infinitesimally short sides IS a circle, which is why a stop sign (n=8) already looks fairly round.

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