Area of a Regular Polygon
The area of a regular polygon can be calculated using different known values. The two most common methods are using the side length or the apothem (the perpendicular distance from the center to a side).
Area Formulas
From Side Length
When you know the number of sides and the side length.
From Apothem
When you know the apothem and number of sides.
Using Perimeter & Apothem
The most elegant form of the area formula.
Understanding the Formulas
From Side Length
The formula A = (ns²)/(4tan(π/n)) works by dividing the polygon into n isosceles triangles from the center. Each triangle has a base of s and a height equal to the apothem. The tangent function relates the side length to the apothem through the central angle.
From Apothem
When you know the apothem a, you can first find the side length as s = 2a × tan(π/n), then compute the area. Alternatively, use A = n × a² × tan(π/n) directly, which is derived by substituting the side-apothem relationship into the standard formula.
Example Calculations
- Regular Hexagon (n=6), s=5: A = (6 × 25) / (4 × tan(30°)) = 150 / 2.309 = 64.95 sq units
- Square (n=4), s=10: A = (4 × 100) / (4 × tan(45°)) = 400 / 4 = 100 sq units
- Pentagon (n=5), s=8: A = (5 × 64) / (4 × tan(36°)) = 320 / 2.906 = 110.11 sq units
Tips
- Ensure your calculator is in radians when using π/n.
- The apothem is always shorter than the circumradius.
- As n increases, the area approaches πR² (area of a circle).