Regular Polygon Calculator
Compute the area, perimeter, and interior angles of any regular polygon from the number of sides and one known measurement such as side length, circumradius, or apothem. Great for architecture, tiling, and geometry.
About this calculator
A regular polygon has n equal sides and n equal interior angles. Each interior angle measures (n − 2) × 180° / n degrees. If the side length s is known, the perimeter is simply P = n × s, and the area is A = (n × s²) / (4 × tan(π/n)). When the circumradius R (centre to vertex) is known, the side length is derived as s = 2R × sin(π/n). When the apothem a (centre to midpoint of a side) is known, the side is s = 2a × tan(π/n). All three starting measurements lead to the same set of formulas once s is established, making any single measurement sufficient to fully describe the polygon.
How to use
Find the area of a regular hexagon (6 sides) with a side length of 4 units. Enter sides = 6, select 'Side' as the known measurement, and enter value = 4. The interior angle is (6 − 2) × 180 / 6 = 120°. The area formula gives A = (6 × 4²) / (4 × tan(π/6)) = (6 × 16) / (4 × 0.5774) = 96 / 2.3094 ≈ 41.57 square units. The perimeter is P = 6 × 4 = 24 units. Select 'Area' or 'Perimeter' in the Calculate field to see each result.
Frequently asked questions
How do you find the area of a regular polygon when you only know the circumradius?
First convert the circumradius R to the side length using s = 2R × sin(π/n), where n is the number of sides. Then apply the area formula A = (n × s²) / (4 × tan(π/n)). For a regular pentagon (n = 5) with R = 6 units, s = 2 × 6 × sin(36°) ≈ 7.05 units, and A ≈ (5 × 49.7) / (4 × 0.7265) ≈ 117.97 square units. This calculator performs the conversion automatically when you select 'Circumradius' as your known measurement.
What is the interior angle of a regular polygon and how does it change with more sides?
The interior angle of a regular polygon with n sides is (n − 2) × 180° / n. A triangle has 60°, a square has 90°, a hexagon has 120°, and a dodecagon (12 sides) has 150°. As n increases toward infinity, the interior angle approaches 180° and the polygon approaches a circle. This progression is why regular hexagons tile a flat surface perfectly — their 120° angles combine in groups of three to form 360°.
Why do regular polygons with more sides increasingly resemble a circle?
As the number of sides n grows, each side becomes shorter relative to the polygon's size, and the vertices pack closer together along what would be a circular arc. The area formula A = (n × s²) / (4 × tan(π/n)) converges to πr² as n → ∞, and the perimeter converges to 2πr. Engineers exploit this by approximating circular cross-sections with high-sided polygons in computer graphics and structural design. A 16-sided polygon already looks visually circular to most observers.