How this instrument works
A regular polygon with n sides and one side length s is fixed entirely by those two numbers, because equal sides and equal angles force everything else. Draw a segment from the center to every vertex and the shape splits into n congruent isosceles triangles, each with base s and height equal to the apothem, a = s ⁄ (2·tan(π ⁄ n)) — the distance from the center to a side's midpoint. Summing the n triangle areas, ½·s·a apiece, gives A = ns² ⁄ (4·tan(π ⁄ n)) directly, with no shortcut hiding inside it beyond that one repeated triangle.
That same apothem ties this shape to a circle in a way worth noticing: A = ½·P·a, exactly the form of a circle's A = ½·C·r with the apothem standing in for the radius. As n climbs, tan(π ⁄ n) shrinks toward π ⁄ n, the polygon's corners soften, and the two formulas converge — a triangle and a hundred-sided shape both obey the identical A = ½·P·a relationship, they just look less and less like a circle's version of it as n gets small. Separately, every regular polygon's exterior angles sum to exactly 360° no matter how many sides it has (a triangle's three 120° turns and a dodecagon's twelve 30° turns both close a full circle), even though the interior angle itself, (n − 2)·180° ⁄ n, changes with every n.
The formula only makes sense for whole n of 3 or more — a shape with two sides cannot close, and tan(π ⁄ n) itself goes undefined exactly at n = 2, where the geometry stops making sense right along with the algebra. Fix the side length and let n climb, instead of the usual case of fixing n and lengthening the side, and area grows roughly with the square of n while perimeter grows only in step with n itself — a hundred-sided shape with 4-unit sides covers dramatically more ground than a triangle with the same edge, out of proportion to how much longer its boundary is. This sheet exists because pinning down one specific n, as the standalone triangle, pentagon, and hexagon pages already do, means a fresh formula for every shape; letting n itself be an input collapses all of them, and everything beyond, into the one relationship above, with area and perimeter returned together instead of one page at a time.
- Enter how many sides your polygon has into Number of sides, n — any whole number 3 or greater works, from a triangle up to shapes with hundreds of sides.
- Enter the length of one side into Side length; because a regular polygon's sides are all equal, that single figure is all the shape needs.
- Read Area for the enclosed surface and Perimeter for the total boundary length — both recalculate together the moment either input changes.
- Divide Area by half of Perimeter to recover the apothem, useful when a real object's center-to-edge distance is what actually needs checking.
Worked example — hexagon check, n = 6
Set Number of sides, n to 6 and Side length to 4. Perimeter returns 6 × 4 = 24.0 immediately, and Area returns 41.569219381653056 — the exact value of 24√3, since a regular hexagon's own closed-form area, (3√3 ⁄ 2)s², and this general n-gon formula, ns² ⁄ (4·tan(π ⁄ n)), must agree once n is fixed at 6, and they do, digit for digit.
Change Number of sides, n to 3 with Side length left at 4, and the same formula still holds: Area lands on 6.928203230275511, which is 4√3, the area of an equilateral triangle with side 4, while Perimeter drops to 3 × 4 = 12.0. One sheet, one formula, covering the triangle and the hexagon and everything with more sides in between, without switching pages or reaching for a different equation each time n changes.
Questions
What is the general formula for a regular polygon's area?
A = ns² ⁄ (4·tan(π ⁄ n)), where n is the number of sides and s is one side length. It comes from splitting the polygon into n congruent isosceles triangles that meet at the center, each with base s and height equal to the apothem, s ⁄ (2·tan(π ⁄ n)); summing all n triangle areas gives the formula directly, valid for any whole n from 3 upward.
How is this different from separate area-only and perimeter-only polygon calculators?
Those sheets each return one figure for one fixed shape. This one takes any Number of sides, n, and returns Area and Perimeter together in a single pass, useful whenever a job needs both figures at once — ordering edging strip and estimating material for the same tile or plot without running the numbers twice.
How do I find the apothem from the Area and Perimeter results?
Divide Area by half of Perimeter: a = 2A ⁄ P. This mirrors the circle identity A = ½Cr exactly, because a regular polygon's apothem plays the same role as a circle's radius — as n grows large, the polygon's boundary and its apothem approach a circle's circumference and radius, and the two formulas become the same statement.
Why must Number of sides, n be at least 3?
Fewer than three sides cannot close a simple polygon — two segments only fold back on each other. The formula reflects this directly: tan(π ⁄ n) is undefined at n = 2, since π ⁄ 2 sends tangent to infinity, so the area expression breaks down exactly where the geometry does.
Which regular polygons can be built with just a compass and straightedge?
Only those where n is a power of 2 times a product of distinct Fermat primes (3, 5, 17, 257, 65537) — the Gauss–Wantzel theorem, proved in 1837. That covers the triangle, square, pentagon, hexagon, and pentadecagon, but rules out a regular heptagon or nonagon by straightedge and compass alone, no matter how carefully they're drawn.
Does a regular polygon with more sides always have a larger area?
Only when side length stays fixed — this calculator shows why. At s = 4, a triangle (n = 3) encloses 6.928203230275511 square units, a hexagon (n = 6) encloses 41.569219381653056, and a dodecagon (n = 12) encloses about 179.14: area grows roughly with n² for fixed s, far faster than perimeter's plain n·s growth, because higher-n shapes fan more identical triangles out from the same center.