Regular Polygon Calculator
Calculate the area, perimeter, interior and exterior angles, apothem and circumradius of any regular polygon.
How to use this calculator
- 1Enter the number of sides — 3 for a triangle, 6 for a hexagon, and so on.
- 2Choose which length you know: the side, the apothem (centre to edge), or the circumradius (centre to corner).
- 3Everything else is derived, including both angles and the diagonal count.
How the calculation works
Area = (n × s²) ÷ (4 × tan(π/n)) apothem = s ÷ (2·tan(π/n)) circumradius = s ÷ (2·sin(π/n)) interior angle = (n − 2)×180° ÷ n- n
- Number of sides
- s
- Side length
- apothem
- Perpendicular distance from the centre to the midpoint of a side
- circumradius
- Distance from the centre to any vertex
The area formula comes from splitting the polygon into n identical isosceles triangles meeting at the centre. Each has base s and height equal to the apothem, giving area = n × (s × apothem) ÷ 2 = (perimeter × apothem) ÷ 2.
Because all three lengths are tied together through the half-angle π/n, knowing the number of sides plus any one of side, apothem or circumradius fixes the entire shape — which is why this calculator accepts whichever you happen to have.
The interior angle formula follows from the fact that any n-sided polygon can be cut into (n − 2) triangles, each contributing 180°.
Worked example
A regular hexagon with 10 cm sides
- 1.Half-angle: π ÷ 6 = 30°, and tan(30°) = 0.577350.
- 2.Apothem: 10 ÷ (2 × 0.577350) = 8.660254 cm.
- 3.Perimeter: 6 × 10 = 60 cm.
- 4.Area: (perimeter × apothem) ÷ 2 = (60 × 8.660254) ÷ 2 = 259.807621 cm².
- 5.Interior angle: (6 − 2) × 180 ÷ 6 = 120°, and the exterior angle is 360 ÷ 6 = 60°.
Result: 259.8076 cm², apothem 8.6603 cm, interior angle 120°
What makes a polygon "regular"
A regular polygon satisfies two conditions at once: every side is the same length, and every interior angle is the same size. Both are required. A rhombus has equal sides but unequal angles; a rectangle has equal angles but unequal sides; neither is regular. Only when both hold does the shape gain the symmetry that makes a single side length enough to determine everything else about it.
That symmetry is why one measurement plus the side count is sufficient. A regular polygon can be decomposed into n identical isosceles triangles radiating from the centre, so the entire shape is fixed once you know how many there are and how big one of them is.
Three radii, one shape
Regular polygons have two distinct "radius" measurements that are easy to confuse.
- Apothem (inradius) — the perpendicular distance from the centre to the midpoint of a side — the radius of the largest circle that fits inside. This is the one used in the area formula.
- Circumradius — the distance from the centre to a vertex — the radius of the circle passing through all the corners. Always larger than the apothem.
- The ratio between them — apothem ÷ circumradius = cos(π/n), which approaches 1 as sides are added. For a triangle the apothem is only half the circumradius; for a dodecagon it is about 96.6% of it.
Why hexagons are everywhere
The regular hexagon has a property no other regular polygon shares: its circumradius is exactly equal to its side length. It is six equilateral triangles fitted together, which is also why its interior angle is exactly 120° — and 120 divides evenly into 360, meaning three hexagons meet perfectly at every vertex with no gaps and no overlap.
Only three regular polygons tile the plane this way: the triangle, the square and the hexagon, because only 60°, 90° and 120° divide 360° evenly. Of those three, the hexagon encloses the most area for a given perimeter, which is the standard explanation for honeycomb structure — it is the tiling that walls off the most space for the least wax.
Approaching the circle
As the side count rises, a regular polygon converges on a circle. This is not merely a visual resemblance — it is how π was first computed to useful precision. Archimedes bounded the circle between inscribed and circumscribed regular polygons, doubling the side count repeatedly until he had 96-sided polygons, which pinned π between 3¹⁰⁄₇₁ and 3¹⁄₇. The "fills X% of its circumcircle" figure in the results is the modern form of exactly that squeeze: it climbs toward 100% as sides are added, and how fast it climbs is what made the method practical.
What this assumes, and where it stops
Assumptions
- The polygon is regular — all sides equal and all interior angles equal. Irregular polygons need their area computed another way, such as the shoelace formula from vertex coordinates.
- The shape is planar and convex, which every regular polygon is.
Limitations
- Does not handle irregular or star (self-intersecting) polygons.
- Angles are reported in degrees; the internal trigonometry works in radians.
Common questions
What is the apothem, and why does the area formula use it?
The apothem is the perpendicular distance from the centre to the midpoint of a side. It is used because splitting the polygon into n triangles from the centre makes the apothem the height of each one — so area = (perimeter × apothem) ÷ 2, which is the same structure as ½ × base × height for a single triangle.
Why is a hexagon's circumradius the same as its side length?
Because a regular hexagon is exactly six equilateral triangles meeting at the centre. Each triangle has all three sides equal, and two of those sides are circumradii while the third is a side of the hexagon — so they must all be the same length. No other regular polygon has this property.
Which regular polygons tile a flat surface?
Only the equilateral triangle, the square and the regular hexagon. A tiling needs the interior angle to divide 360° evenly so shapes meet without gaps: 60°, 90° and 120° do; a pentagon's 108° does not, which is why regular pentagons cannot tile a plane on their own.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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