Regular Polygon Calculator
Instantly calculate interior angles, exterior angles, area, and perimeter of any regular polygon. Perfect for geometry students, architects, and tile pattern designers.
About this calculator
A regular polygon has n equal sides of length s and n equal angles. The sum of all interior angles is (n − 2) × 180°, so each interior angle equals ((n − 2) × 180°) / n. Each exterior angle is the supplement of the interior angle and equals 360° / n — all exterior angles of any convex polygon sum to exactly 360°. The perimeter is simply P = n × s. The area is derived by dividing the polygon into n isosceles triangles meeting at the center: Area = (n × s²) / (4 × tan(π / n)). As n increases, a regular polygon increasingly approximates a circle.
How to use
Take a regular hexagon (n = 6) with a side length of 5 units. Interior angle: ((6 − 2) × 180) / 6 = (4 × 180) / 6 = 720 / 6 = 120°. Exterior angle: 360 / 6 = 60°. Perimeter: 6 × 5 = 30 units. Area: (6 × 5²) / (4 × tan(π / 6)) = (6 × 25) / (4 × 0.5774) = 150 / 2.3094 ≈ 64.95 square units. Enter numberOfSides = 6, sideLength = 5, and select your desired calculation to verify each result.
Frequently asked questions
How do you find the interior angle of a regular polygon with any number of sides?
The formula is Interior Angle = ((n − 2) × 180°) / n, where n is the number of sides. The logic is that any polygon can be split into (n − 2) triangles, each contributing 180°. Dividing by n distributes the total angle sum equally among all vertices. For a regular pentagon (n = 5) each interior angle is ((5 − 2) × 180) / 5 = 108°. This formula only applies to regular (equilateral and equiangular) polygons; irregular polygons have varying interior angles.
What is the relationship between interior and exterior angles of a regular polygon?
For any convex polygon, each interior angle and its adjacent exterior angle are supplementary, meaning they add up to 180°. For a regular polygon, every exterior angle equals 360° / n. This means the exterior angles always sum to 360° regardless of the number of sides — a fact useful in navigation and robotics when calculating turning angles. For a regular octagon, each exterior angle is 360° / 8 = 45°, and each interior angle is 180° − 45° = 135°.
Why does the area formula for a regular polygon use a tangent function?
The formula Area = (n × s²) / (4 × tan(π / n)) comes from dividing the polygon into n congruent isosceles triangles, each with a base of s and an apex angle of 2π / n radians at the center. The height of each triangle (the apothem) equals s / (2 × tan(π / n)). Multiplying the area of one triangle by n gives the full formula. The tangent function captures the relationship between the side length and the apothem, which grows as the polygon becomes more circle-like with increasing n.