Perimeter of a Regular Polygon
The perimeter of a regular polygon is the total distance around the outside of the shape. Since all sides of a regular polygon are equal in length, the perimeter is simply the number of sides multiplied by the side length.
Formulas
Perimeter from Side
When you know the side length.
Side from Perimeter
When you know the perimeter.
Perimeter from Circumradius
When you know the circumradius R.
How It Works
Finding the Perimeter
The perimeter formula P = n × s is the simplest polygon formula. If a regular hexagon has sides of 5 units, its perimeter is 6 × 5 = 30 units. This works for any regular polygon regardless of the number of sides.
Finding the Side Length
If you know the total perimeter and the number of sides, you can find each side's length by dividing: s = P / n. For example, if a pentagon has a perimeter of 45 units, each side is 45 / 5 = 9 units.
Perimeter Examples
- Equilateral Triangle: P = 3s. A triangle with side 7 has P = 21 units.
- Square: P = 4s. A square with side 10 has P = 40 units.
- Regular Pentagon: P = 5s. A pentagon with side 6 has P = 30 units.
- Regular Hexagon: P = 6s. A hexagon with side 4 has P = 24 units.
- Regular Octagon: P = 8s. An octagon with side 3 has P = 24 units.
Relationship Between Perimeter and Area
For a regular polygon, knowing the perimeter and apothem gives the area directly: A = (1/2) × P × a. This elegant relationship shows that the area equals half the perimeter times the apothem (the distance from the center perpendicular to a side).
Isoperimetric Inequality
Among all regular polygons with the same perimeter, the one with more sides encloses a greater area. In the limit, a circle (infinite sides) encloses the maximum possible area for a given perimeter. This is known as the isoperimetric inequality.