Regular Polygon Calculator (any number of sides)

A regular n-sided polygon has n equal sides and n equal angles. Triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, decagon, and on up to shapes with so many sides they are indistinguishable from a circle.

Perimeter: P = n × s

Interior angle (per vertex): angle = (n − 2) × 180° / n

Apothem (inradius, centre to mid-side): r = s / (2 × tan(π/n))

Circumradius (center to vertex): R = s / (2 × sin(π/n))

Area: A = (n × s² / 4) × cot(π/n) = (1/2) × P × r

The area formula has a clean interpretation: a regular polygon’s area equals half its perimeter times its apothem. This generalizes the triangle formula (½ × base × height) to any regular polygon.

Reference values for common polygons (s = 1 unit):

As n increases, the polygon approaches a circle. For very large n with side length 1, the circumscribed circle has radius ≈ s × n / (2π).

Where this matters in real life:

• Gazebo and pavilion floors. Most are octagonal, hexagonal, or pentagonal. The same formulas size them all.
• Architectural cupolas and rooftop lanterns. Usually square, six- or eight-sided. A seven- or nine-sided one is a deliberate oddity, and the same formulas size it.
• Coins. Britain’s 50p and 20p are seven-sided, though each side bulges slightly so the coin measures the same width in every direction and still works in a vending machine. Australia’s 50-cent piece has twelve straight sides.
• Geodesic dome panels. A geodesic dome’s panels are triangles, but the projected ground footprint is often a regular polygon (10, 12, 15 sides).
• Road signs. Stop signs are octagons and yield signs triangles, and in the US the school-zone and school-crossing signs are pentagons, a shape kept for that one meaning.

Worked example: building a heptagonal (7-sided) gazebo

You want a unique 7-sided gazebo with 4-ft sides.
Perimeter = 28 ft of edge.
Area = 3.634 × 16 = 58.1 sq ft of floor space.
Apothem = 1.038 × 4 = 4.15 ft (the distance from center to the middle of each wall).
Interior angle = 128.57° (each corner is slightly more “open” than the 120° of a hexagon).

Worked example: designing a polygonal flower bed

You want a 9-sided (nonagonal) decorative bed with 3-ft sides.
P = 27 ft of bed border.
A = 6.182 × 9 = 55.6 sq ft.
Apothem = 1.374 × 3 = 4.12 ft, so the centre is a 4.12 ft reach from the middle of any side, about the limit for weeding without stepping into the bed.

The limit. As n grows the polygon closes in on a circle from both sides at once, and the numbers are startling. At n = 1000 the polygon fills 99.9993% of the circle drawn through its corners, and it overshoots the circle drawn inside it by only 0.0003%. Squeezed between those two, π is pinned to five decimal places.

That squeeze is exactly Archimedes’ method. He worked up to 96 sides by hand and reported π as lying between 223/71 and 22/7, which is 3.1408 to 3.1429. His bracket was correct and it held as the standard value in Europe for centuries.