Regular polygon calculator — area, angles, apothem and diagonals

How to use the regular polygon calculator

1. Enter the Number of sides, a whole number of 3 or more (3 for a triangle, 6 for a hexagon, 8 for an octagon).
2. Enter the Side length.
3. Read Area and Perimeter.
4. Check the other results: Interior angle, Exterior angle, Apothem (inradius), Circumradius, Sum of interior angles and Diagonals.

What it does and when to use it

Regular polygons appear in floor tiles, hexagonal garden beds, octagonal gazebos and tables, bolt heads, signs and many logos. Builders need the interior angle to cut the joints, the area to buy flooring or paving, and the circumradius or apothem to know how much space the finished shape takes. In class, the tool checks answers on interior angles and polygon areas, and shows how a polygon approaches a circle as the number of sides grows.

If you only need angles, the angle calculator has a polygon mode. The area calculator and the perimeter calculator include regular polygons alongside other shapes, with the formula written out.

How it works

With n sides of length s, and the central angle 180°/n:

• Apothem: s ÷ (2 × tan(180°/n))
• Circumradius: s ÷ (2 × sin(180°/n))
• Area: n × s × apothem ÷ 2 (n triangles, each with base s and height equal to the apothem)
• Perimeter: n × s
• Sum of interior angles: (n − 2) × 180°; each interior angle is that sum ÷ n
• Exterior angle: 360° ÷ n
• Diagonals: n × (n − 3) ÷ 2

Worked examples

An octagonal table top, step by step. 8 sides of 5 units.

1. Half the central angle: 180° ÷ 8 = 22.5°.
2. Apothem: 5 ÷ (2 × tan 22.5°) = 5 ÷ 0.8284271247 = 6.035533906.
3. Area: 8 × 5 × 6.035533906 ÷ 2 = 120.7106781.
4. Diagonals: 8 × 5 ÷ 2 = 20.

The tool shows Apothem (inradius): 6.035533906 and Area: 120.7106781. Each corner is 135°. The circumradius, 6.532814824, tells you the top fits on a round base about 13.07 units across.

A pentagon. 5 sides of 10 give Interior angle: 108°, an exterior angle of 72°, an area of 172.0477401 and 5 diagonals.

An equilateral triangle. 3 sides of 6 give an area of 15.58845727 and 0 diagonals, since every corner already connects to every other.

Limits and tips

• Decimal numbers of sides are refused with a message.
• The side length must be greater than zero. There is no unit menu; area is in your unit squared.
• Results show up to 10 significant figures and are calculated in your browser.