Regular Polygon Calculator
Solve any regular polygon — pentagon, hexagon, octagon, or a 17-gon — from two numbers: how many sides and how long each side is. Enter the side in any unit to get area, perimeter, apothem, circumradius, and the interior angle.
Example: with Number of sides (n) 6 · Side length (a) 10 → Area: 259.808 square units.
- Perimeter (n × a)60 units
- Apothem8.66 units
- Circumradius10 units
Computed by the calculator below using its default values. Change any input to see your own numbers.
A = n·a² / (4·tan(π/n)) — n identical triangles around the center. As n grows, the polygon approaches the circle through its corners.
One formula for every regular polygon
Slice any regular n-gon from its center to each corner and you get n identical isosceles triangles. Each has base a and height equal to the apothem, a/(2·tan(π/n)), so the total area is A = n·a²/(4·tan(π/n)). Set n = 6 and it reduces to the hexagon's (3√3/2)a²; set n = 4 and it collapses to a². Every named polygon formula is this one with a number plugged in.
Two radii describe the polygon: the apothem (center to mid-side, the inscribed circle's radius) and the circumradius (center to corner, a/(2·sin(π/n))). As n climbs, the two radii converge and the polygon hugs its circle — a 60-gon's area is within 0.2% of the circle through its corners, which is essentially how Archimedes cornered π.
How it’s calculated
Area A = n·a²/(4·tan(π/n)); perimeter P = n·a; apothem = a/(2·tan(π/n)); circumradius R = a/(2·sin(π/n)); interior angle = (n−2)·180°/n; exterior angle = 360°/n. n is rounded to the nearest whole number and must be at least 3.
Regular polygons only — all sides and angles equal. Irregular polygons need triangulation or the shoelace formula instead.
Regular polygons with side a = 10
| n | Name | Area | Apothem | Interior angle |
|---|---|---|---|---|
| 3 | Triangle | 43.30 | 2.89 | 60° |
| 4 | Square | 100 | 5 | 90° |
| 5 | Pentagon | 172.05 | 6.88 | 108° |
| 6 | Hexagon | 259.81 | 8.66 | 120° |
| 8 | Octagon | 482.84 | 12.07 | 135° |
| 10 | Decagon | 769.42 | 15.39 | 144° |
| 12 | Dodecagon | 1,119.62 | 18.66 | 150° |
Computed with A = n·a²/(4·tan(π/n)) and apothem a/(2·tan(π/n)); rounded to 2 decimals.
Common mistakes
- Feeding the formula a circumradius or apothem as if it were the side length — the three are different for every n.
- Using degrees inside tan(π/n); the formula's angle is in radians (π/n radians = 180°/n).
- Assuming area scales linearly with n at fixed side length — a 12-gon with the same side has far more than twice a hexagon's area (1,119.6 vs 259.8).
- Applying regular-polygon formulas to a shape with unequal sides; one stretched side invalidates all of them.
Frequently asked questions
What is the area formula for a regular polygon?
A = n·a²/(4·tan(π/n)), where n is the number of sides and a the side length. Equivalently ½ × perimeter × apothem — both come from cutting the polygon into n triangles.
What is the apothem?
The perpendicular distance from the center to the midpoint of a side: a/(2·tan(π/n)). It is the radius of the largest circle that fits inside, and the "height" used in the ½ × P × apothem area form.
How do I get the interior angle?
(n − 2) × 180° ÷ n. A pentagon's is 108°, a hexagon's 120°, an octagon's 135°. The exterior angles always sum to 360°, so each is 360°/n.
Why does a polygon approach a circle as n grows?
With the circumradius held fixed, the corners all sit on one circle and the flat sides shrink toward it. By n = 60 the area gap to the circle is about 0.2% — the idea behind Archimedes' polygon bounds on π.
Does this work for a star polygon?
No — stars like the pentagram are non-convex and trace their perimeter differently. Solve the underlying regular polygon, then handle the star's points as separate triangles.