Decagon Calculator
Calculate all properties of a regular decagon: area, perimeter, angles, radii, and diagonals
Calculate Decagon Properties
Length of one side of the regular decagon
Decagon Properties
Key Formulas:
• Area = 2.5 × a² × √(5 + 2√5) ≈ 7.694 × a²
• Circumradius R = ½a × (1 + √5) ≈ 1.618 × a
• Inradius r = ½a × √(5 + 2√5) ≈ 1.539 × a
• Interior angle = 144°, Exterior angle = 36°
Example Calculation
Regular Decagon with Side Length 2 units
Given: Side length a = 2 units
Perimeter: P = 10 × 2 = 20 units
Area formula: A = 2.5 × a² × √(5 + 2√5)
Area calculation: A = 2.5 × 4 × √(5 + 2√5) ≈ 7.694 × 4 ≈ 30.78 sq units
Circumradius: R = ½ × 2 × (1 + √5) ≈ 3.236 units
Inradius: r = ½ × 2 × √(5 + 2√5) ≈ 3.078 units
Angle Properties
Interior angle: α = (n-2) × 180° / n = 8 × 180° / 10 = 144°
Exterior angle: β = 360° / n = 360° / 10 = 36°
Sum of interior angles: 10 × 144° = 1440°
Central angle: 360° / 10 = 36°
Decagon Quick Facts
Key Formulas
Perimeter
P = 10 × a
Area
A ≈ 7.694 × a²
Circumradius
R ≈ 1.618 × a
Inradius
r ≈ 1.539 × a
Diagonals
n(n-3)/2 = 35
Calculation Tips
A decagon has 10 equal sides and 10 equal angles
The golden ratio (φ ≈ 1.618) appears in decagon formulas
Interior angles sum to 1440°
Apothem equals the inradius
Understanding Regular Decagons
What is a Decagon?
A decagon is a polygon with ten sides and ten vertices. A regular decagon has all sides equal in length and all interior angles equal. The word "decagon" comes from the Greek words "deka" (ten) and "gonia" (angle).
Properties
- •10 equal sides and 10 equal vertices
- •Interior angle of 144° each
- •35 diagonals in total
- •Exhibits rotational symmetry of order 10
Mathematical Relationships
Key Constants:
- √5 ≈ 2.236
- Golden ratio φ = (1 + √5)/2 ≈ 1.618
- Area coefficient ≈ 7.694
Real-World Applications
- •Architecture and building design
- •Geometric art and pattern design
- •Engineering and mechanical parts