How to Calculate Decagon Area
The area of a regular decagon (a 10-sided polygon with all equal sides and angles) can be calculated using several different methods depending on which measurement you have available. Each method ultimately produces the same result, but the approach differs based on whether you know the side length, the apothem, or the circumradius.
Three Methods for Decagon Area
Method 1: From Side Length
Use the direct formula involving the side length and the square root expression.
Method 2: From Apothem
Use the general polygon area formula with the apothem and perimeter.
Method 3: From Circumradius
Use the circumradius to first find the side, then calculate the area.
Alternative: Triangulation
Divide into 10 isosceles triangles from the center.
Method 1: Area from Side Length (Detailed)
When you know the side length s of a regular decagon, the area can be computed directly using the formula: A = (5/2) x s2 x sqrt(5 + 2sqrt(5)). This formula is derived by dividing the decagon into 10 congruent isosceles triangles, each with a base equal to the side length and a height equal to the apothem.
Method 2: Area from Apothem (Detailed)
The apothem is the perpendicular distance from the center of the decagon to the midpoint of any side. If you know the apothem a, you can find the side length using s = 2a x tan(pi/10), then compute the perimeter P = 10s, and finally the area as A = (1/2) x P x a. This is the general formula for the area of any regular polygon.
Method 3: Area from Circumradius (Detailed)
The circumradius R is the distance from the center to any vertex of the decagon. Given R, the area formula simplifies to A = 5R2 x sin(36 degrees). This comes from the fact that the decagon can be split into 10 isosceles triangles, each with two sides equal to R and an included angle of 36 degrees (360/10).
Derivation of the Area Formula
The area formula for a regular decagon can be derived by dividing the shape into 10 congruent isosceles triangles, each radiating from the center. Each triangle has:
- A base equal to the side length s
- A height equal to the apothem a = s / (2 tan(18 degrees))
- An area of (1/2) x s x a
Multiplying by 10 triangles and simplifying using trigonometric identities yields the compact formula A = (5/2) x s2 x sqrt(5 + 2sqrt(5)).
Numerical Approximation
The factor sqrt(5 + 2sqrt(5)) evaluates to approximately 3.07768. Therefore, the area can be approximated as A ≈ 7.6942 x s2. This means the area of a regular decagon is about 7.69 times the square of its side length, making it one of the most area-efficient regular polygons.
Comparison with Other Polygons
- Equilateral triangle: A = 0.433 s2
- Square: A = s2
- Regular pentagon: A = 1.720 s2
- Regular hexagon: A = 2.598 s2
- Regular octagon: A = 4.828 s2
- Regular decagon: A = 7.694 s2
- Circle: A = pi x r2 (the limit as sides approach infinity)