Polygon Calculator
Regular polygon area, side, apothem, radius and angles.
About this tool
Enter the number of sides and whichever measurement you know (side, circumradius, apothem, perimeter or area). The tool calculates all the others, the angles and the number of diagonals, and draws the shape.
Regular polygon formulas
A regular polygon has all sides and angles equal. With n sides of length s: the interior angle is (n − 2) × 180° ÷ n and the exterior angle 360° ÷ n; the circumradius (centre to a corner) is R = s ÷ (2 sin(180°/n)); the apothem (centre to the middle of a side) is a = s ÷ (2 tan(180°/n)); and the area is ½ × perimeter × apothem. A regular hexagon with 10 cm sides has an area of 259.8 cm², an apothem of 8.66 cm and 120° interior angles. Its circumradius equals its side, which is why six equilateral triangles fit inside, and why hexagons tile a floor or a honeycomb with no gaps. A square of area 100 has 10-unit sides. The number of diagonals is n(n − 3) ÷ 2. As n grows, the polygon approaches a circle: a 1,000-gon's area is within 0.001% of its circumscribed circle. Honest limits: the tool handles regular polygons only; irregular shapes need their coordinates, as in the shoelace formula. The circle calculator and the triangle solver are related.
Frequently asked questions
What is the interior angle of a hexagon?
120°. In general it is (n − 2) × 180° ÷ n.
How do I find the area of a regular polygon?
Half the perimeter times the apothem, or n × s² ÷ (4 tan(180°/n)).
What is an apothem?
The distance from the centre of a regular polygon to the midpoint of a side.
How many diagonals does an octagon have?
20, from n(n − 3) ÷ 2 = 8 × 5 ÷ 2.
Is my data sent anywhere?
No. Everything is calculated in your browser, and nothing is stored or sent.