Regular Polygon Area Calculator
Calculate the area of any regular polygon by entering the number of sides and side length.
Formula
A = (n × l²) / (4 × tan(π/n)), valid for any n ≥ 3.
Area of a regular polygon
In a regular polygon all N sides share the same length L, and the N interior angles are all equal too. You can find the area with A = (N·L²) / (4·tan(π/N)). The same number drops out of A = (1/2)·N·L·a, where a is the apothem, the distance from the center to the midpoint of a side. Plug in N = 5 and L = 10 m and you get A = (5·100) / (4·tan(36°)) ≈ 172.05 m².
Each interior angle measures (N-2)·180°/N, and they add up to (N-2)·180°. Add more sides and the shape gets rounder; push N high enough and the area closes in on π·r², the area of the circle it is creeping toward. When the polygon is irregular, none of that helps, so reach for the shoelace (Gauss) formula A = (1/2)|Σ(x_i·y_{i+1} − x_{i+1}·y_i)| applied to the vertices in order.
Applications
You will spot regular polygons in architecture (the Pentagon, octagonal towers) and in coin design: the UK 20p and 50p coins are heptagonal, and because their width is the same in every direction, vending machines still read them. They turn up in floor and roof tiling, where triangles, squares and hexagons each cover the plane unaided, and in robotics, where wrapping obstacles in polygons keeps path planning manageable.
FAQ
Which regular polygons tile the plane? Three of them, and only three: the equilateral triangle (N=3), the square (N=4) and the regular hexagon (N=6). They work because their interior angles divide 360° without leaving a gap.
What is the apothem? Measure straight from the center of the polygon to the middle of any side and that distance is the apothem: a = L / (2·tan(π/N)). It happens to equal the radius of the inscribed circle.
How is the perimeter calculated? Just multiply the number of sides by their length: P = N·L. A pentagon with L = 10 ends up at P = 50 m.
Why does the area approach π·r² as N grows? Inscribe the polygon in a circle of radius r and watch what happens as you add sides: it leaves less and less of the disk uncovered, until eventually polygon and circle are the same shape.
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Area of any regular polygon
Pentagon, hexagon, octagon, decagon. Each regular polygon has its own area formula, and memorising them all is impractical. This calculator handles any of them: you enter the number of sides and the side length, and it works out the area.
Behind it sits the general formula, the one that holds for every regular polygon, so it works from the equilateral triangle to figures with dozens of sides. It helps with design, architecture, geometry exercises and projects using regular shapes of any number of sides.
It all runs in the browser. A versatile calculator that spares you memorising a different formula for each polygon.