Formula and Method for the Perimeter of a Regular Polygon
A regular polygon is a closed shape with n straight, equal-length sides and n equal interior angles — think of an equilateral triangle (n = 3), a square (n = 4), a regular pentagon (n = 5), or a regular hexagon (n = 6). Because every side has the same length, the perimeter is simply the number of sides times the length of one side: P = n × s. This calculator also derives the interior angle, exterior angle, and area from the same two inputs.
How the calculation works
Enter the number of sides (n, an integer of 3 or more) and the side length (s), then choose the unit it is measured in. The calculator multiplies n by s to get the perimeter. It finds each interior angle from the polygon angle-sum theorem, (n − 2) × 180° / n, and each exterior angle as 360° / n (the two are always supplementary, adding to 180°). For area, the calculator splits the polygon into n congruent isosceles triangles that meet at the center and sums their areas, which simplifies to A = (n × s²) / (4 × tan(π/n)).
Common mistakes
- Confusing perimeter with area: a regular hexagon with 10 ft sides has a perimeter of 6 × 10 = 60 ft, not an area of 60 ft² — those are two different quantities with different units.
- Using a non-integer or too-small n: a polygon must have a whole number of sides, and the minimum is 3 (a triangle); n = 1 or n = 2 is not a polygon.
- Mixing units: keep the side length in one consistent unit — convert inches to feet, or centimeters to meters, before entering the value.
Real-world applications
- Fencing, edging, and trim projects use the perimeter to determine how much linear material is needed around a polygonal plot or structure.
- Framing, tiling, and paving layouts use the interior and exterior angles to cut miters and joints accurately for hexagonal or octagonal patterns.
- Architecture and landscape design use the area formula to estimate material coverage for gazebos, planters, and paved courtyards built as regular polygons.
- Manufacturing and CAD work rely on the same angle formulas to lay out bolt patterns, nuts, and other polygonal components.