Regular Polygon Calculator
Calculate area, perimeter, and other properties of a regular polygon
Calculation Results
Enter regular polygon measurements to calculate all properties.
Polygon Name:
Number of Sides (n):
Side Length (s): units
Perimeter (P): units
Apothem (a): units
Radius (R): units
Area (A): square units
Central Angle: °
Interior Angle: °
Regular Polygon Diagram
Regular Polygon Formulas
Key Properties
- Number of Sides (n): Count of equal-length sides
- Side Length (s): Length of each side
- Perimeter (P): Sum of all sides (n × s)
- Apothem (a): Perpendicular distance from center to a side
- Radius (R): Distance from center to any vertex
- Area (A): Space contained within the polygon
- Central Angle: Angle between two radii to adjacent vertices
- Interior Angle: Angle between two adjacent sides inside the polygon
Calculation Formulas
- Perimeter (P): P = n × s
- Area (A): A = ½ × n × s × a or A = ½ × P × a
- Apothem (a): a = s / (2 × tan(π/n))
- Radius (R): R = s / (2 × sin(π/n))
- Central Angle: 360° / n
- Interior Angle: (n-2) × 180° / n
Where: n = number of sides, s = side length, a = apothem, R = radius
Example Calculation
For a regular hexagon (6 sides) with:
Side length = 10 unitsCalculations:
- Perimeter = 6 × 10 = 60 units
- Apothem ≈ 10 / (2 × tan(π/6)) ≈ 8.66 units
- Radius ≈ 10 / (2 × sin(π/6)) = 10 units
- Area = ½ × 6 × 10 × 8.66 ≈ 259.81 units²
- Central Angle = 360° / 6 = 60°
- Interior Angle = (6-2) × 180° / 6 = 120°
About Regular Polygons
A regular polygon is a two-dimensional shape with equal-length sides and equal angles between sides. Regular polygons are both equilateral (all sides equal) and equiangular (all angles equal). Common examples include equilateral triangles, squares, regular pentagons, and hexagons.
Real-World Applications
- Architecture: Pentagon building, hexagonal tiles
- Nature: Honeycomb cells, snowflakes
- Design: Logos, patterns, and decorative elements
- Engineering: Bolt heads, nuts, and mechanical parts
Common Regular Polygons
- Triangle (n=3): Equilateral triangle
- Quadrilateral (n=4): Square
- Pentagon (n=5): Regular pentagon
- Hexagon (n=6): Regular hexagon
- Heptagon (n=7): Regular heptagon
- Octagon (n=8): Regular octagon
How to Use the Regular Polygon Calculator
The Regular Polygon Calculator helps compute various properties based on input values. Here's how to use it:
1. Select Calculation Method
Choose what measurements you know:
- Side Length & Number of Sides - Most common method
- Radius & Number of Sides - When you know the circumradius
- Apothem & Number of Sides - When you know the apothem
- Area & Number of Sides - To find other dimensions
- Perimeter & Number of Sides - To find side length
2. Enter Your Values
Input positive numbers in the appropriate fields:
Example 1: Side length = 10, Number of sides = 6 (hexagon)
Example 2: Radius = 8, Number of sides = 5 (pentagon)
3. Get Results
The calculator will compute all properties:
- Polygon name (if common)
- Side length
- Perimeter
- Apothem
- Radius
- Area
- Central and interior angles