Regular Polygon Calculator

Calculate the area, perimeter, interior angle, and apothem of any regular polygon (3-12 sides) from its side length or circumradius. Free online polygon calculator.

Regular Polygon Properties

Area

93.5307

square units

Perimeter

36

units

Sides (n)6
Side Length6
Interior Angle120°
Apothem5.1962
Circumradius6

Number of Sides

6
312

Known Measurement

Formulas Used

  • Area = (n × s²) / (4 × tan(π/n)) = (6 × 6²) / (4 × tan(0.5236)) = 93.5307
  • Perimeter = n × s = 36
  • Interior angle = (n − 2) × 180° / n = (6 − 2) × 180° / 6 = 120°
  • Apothem = s / (2 × tan(π/n)) = 5.1962

Quick Answer

To calculate regular polygon properties with n sides of length s: Area = (n x s^2) / (4 x tan(pi/n)). Perimeter = n x s. Interior angle = (n - 2) x 180 / n degrees. Apothem = s / (2 x tan(pi/n)). For a regular hexagon with side 6: Area = 93.53 square units, Perimeter = 36 units, Interior angle = 120 degrees, Apothem = 5.20 units.

Key Facts

  • Formula: Area = (n x s^2) / (4 x tan(pi/n))
  • Formula: Interior angle = (n - 2) x 180 / n degrees
  • Formula: Apothem = s / (2 x tan(pi/n))
  • A regular hexagon with side 6 has area 93.53 and perimeter 36
  • As n grows, a regular polygon approaches a circle of the same circumradius
  • The apothem is the distance from center to the midpoint of any side
  • Area from circumradius R: Area = (n/2) x R^2 x sin(2 x pi/n)
  • The sum of interior angles of any polygon is (n - 2) x 180 degrees

Frequently Asked Questions

Use Area = (n x s^2) / (4 x tan(pi/n)), where n is the number of sides and s is the side length. For a hexagon (n = 6) with side 6: Area = (6 x 36) / (4 x tan(30 degrees)) = 216 / 2.309 = 93.53 square units. Equivalently, Area = (1/2) x perimeter x apothem.

The apothem is the perpendicular distance from the center to the midpoint of any side. It equals s / (2 x tan(pi/n)). For a hexagon with side 6, the apothem is 6 / (2 x 0.5774) = 5.20 units. Area = (1/2) x apothem x perimeter.

Interior angle = (n - 2) x 180 / n degrees. Triangles: 60 degrees. Squares: 90. Pentagons: 108. Hexagons: 120. The sum of all interior angles is always (n - 2) x 180 degrees, and each exterior angle is 360 / n.

The circumradius R is the distance from center to any vertex. Side length s = 2 x R x sin(pi/n), then use the standard formulas. For area directly: Area = (n/2) x R^2 x sin(2 x pi/n).

Regular polygons tile a plane only when their interior angle divides 360 evenly. The hexagon angle is 120 degrees (360 / 120 = 3), so three fit around a point. The pentagon angle is 108 degrees, which leaves a 36-degree gap.

A polygon where all sides are equal and all interior angles are equal. Examples: equilateral triangle (3), square (4), regular pentagon (5), regular hexagon (6), up to the regular dodecagon (12). This calculator covers n = 3 through 12.