Regular Polygon Area
A = (n·l²)/(4·tan(π/n))P = n·lThe area of a regular polygon is the number of sides times the side squared, divided by four times the tangent of π over the number of sides.
It holds for any polygon with equal sides and angles: equilateral triangle, square, pentagon, hexagon and so on. The specific formulas for each shape are special cases of this one.
The perimeter is simply the number of sides times the side length.
It only works for **regular** polygons. If sides or angles differ, split the figure into triangles and add the areas.
Increasing the number of sides at a fixed perimeter, the figure grows ever closer to a circle, and the area tends towards that of the corresponding circle.
A 100-sided polygon is visually indistinguishable from a circle at any ordinary screen or print size.
It was along exactly this path that Archimedes estimated π, computing inscribed and circumscribed polygons with growing side counts to bracket the value from above and below.
- Mechanical parts with polygonal outlines, such as nuts and flanges.
- Towers, gazebos and structures with polygonal floor plans.
- Tiling and mosaic patterns with regular shapes.
- Board layouts, game maps and spatial data grids.
Frequently asked questions
No. The formula assumes all sides and angles are equal. For irregular ones, split the figure into triangles and add each area.
Because the polygon is divided into equal triangles from the centre, and the tangent relates half a side to the apothem — the distance from the centre to the middle of a side.
Mathematically no. In practice, above a few dozen sides the result converges on the circle’s area and the difference stops being meaningful.
The more sides, the larger the area. The limit of that progression is the circle, which encloses the largest possible area for a given perimeter.