Regular Polygon Calculator
6
Area259.808
Perimeter60.000
Interior Angle120.00°
Exterior Angle60.00°
Apothem8.660
A regular polygon has all sides equal and all interior angles equal, like an equilateral triangle, a square, or a hexagon. From just the number of sides and the side length, this calculator finds the area, perimeter, interior angle, exterior angle, and apothem. It works for any regular polygon with three or more sides.
Formula
Area = (perimeter × apothem) / 2; apothem = s / (2·tan(π/n))
- n
- Number of sides (n ≥ 3)
- s
- Length of each side
- apothem
- Perpendicular distance from the centre to the midpoint of a side
How it works
- Enter the number of sides (a whole number of at least 3) and the length of one side.
- The perimeter is sides × side length. The interior angle is (sides - 2) × 180 / sides and the exterior angle is 360 / sides. The apothem (centre-to-edge distance) is side / (2·tan(π/sides)), and the area is half the perimeter times the apothem.
Worked example
A regular hexagon (6 sides) has a side length of 10.
- Perimeter = 6 × 10 = 60; interior angle = (6-2)×180/6 = 120°; exterior angle = 360/6 = 60°.
- Apothem = 10 / (2·tan(π/6)) = 10 / (2 × 0.5774) ≈ 8.66.
- Area = (60 × 8.66) / 2 ≈ 259.808.
Area ≈ 259.808, perimeter = 60, interior angle = 120°, exterior angle = 60°, apothem ≈ 8.66.
Frequently asked questions
- What is the apothem?
- The apothem is the perpendicular distance from the centre of the polygon to the midpoint of any side. It acts like the radius of the inscribed circle and is the key length used to compute the area.
- Why do the interior and exterior angles relate to the number of sides?
- The exterior angles of any polygon always sum to 360°, so each one is 360 divided by the number of sides. The interior angle is its supplement, found from (sides - 2) × 180 divided by the number of sides.
- Does this work for any polygon?
- It works only for regular polygons, where every side and angle is equal, and the number of sides must be a whole number of at least 3. Irregular polygons need their vertices specified individually.