Every measure of a regular polygon
A regular polygon has all sides and all angles equal — an equilateral triangle, a square, a pentagon, a hexagon, and so on. From just the number of sides and one side length, this calculator returns the area, perimeter, interior and exterior angles, the apothem, and the circumradius.
The formulas
With n sides of length s: perimeter = n times s. The interior angle is (n – 2) times 180, divided by n; the exterior angle is 360 divided by n. The apothem — the distance from center to the middle of a side — is s divided by (2 tan(180/n)), and the area is one half times perimeter times apothem.
Apothem and circumradius
The apothem is the inradius, the radius of the circle that fits inside touching each side. The circumradius, s divided by (2 sin(180/n)), reaches the corners. As n grows, both close in on each other and the polygon approaches a circle.
Related geometry tools
For the limiting circle itself, use the circle calculator; for triangle area by other methods, the triangle area calculator.
Worked example
A regular hexagon (n = 6) with side 5 has a perimeter of 30, interior angles of 120 degrees, an apothem of about 4.33, and an area of about 64.95.
FAQ
Why must n be at least 3?
Fewer than three sides cannot enclose a region, so a polygon needs a minimum of three.
Does this work for irregular polygons?
No — the formulas assume all sides and angles are equal. Irregular shapes must be split into triangles and summed.