Decagon Area Calculator
Calculate the area of a regular decagon using side length, perimeter, or radii
Calculate Decagon Area
Length of one side of the regular polygon
Area Calculation Results
Area Formula: A = (n × a × r) / 2
• n = number of sides (10)
• a = side length (0.000 units)
• r = apothem (0.000 units)
• For decagon: A ≈ 7.694 × a²
Example Calculations
Decagon with 5 cm side length
Given: Side length a = 5 cm
Step 1: Calculate apothem r = a / (2 × tan(18°))
Step 2: r = 5 / (2 × 0.324920) = 7.694 cm
Step 3: Apply area formula A = (10 × a × r) / 2
Step 4: A = (10 × 5 × 7.694) / 2 = 192.36 cm²
Alternative: A = 7.694 × a² = 7.694 × 25 = 192.36 cm²
Decagon with 3 cm side length
Given: Side length a = 3 cm
Using simplified formula: A = 7.694208843 × a²
Calculation: A = 7.694208843 × (3)² = 7.694 × 9
Result: A = 69.25 cm²
Area Formulas
General Polygon
A = (n × a × r) / 2
Where r = a / (2 × tan(π/n))
Decagon Simplified
A = 7.694 × a²
Or A = 2.5 × a² / tan(18°)
From Apothem
A = n × a × r / 2
From Perimeter
A = P × r / 2
Decagon Constants
Calculation Methods
Triangulation: Divide into 10 triangles
Apothem method: A = (perimeter × apothem) / 2
Direct formula: A = 7.694 × side²
General polygon: A = n × s² / (4 × tan(π/n))
How to Find the Area of a Decagon
Area Formula Derivation
To find the area of a regular decagon, we use the general polygon area formula: A = ½ × n × a × r, where n is the number of sides, a is the side length, and r is the apothem.
Step-by-Step Process
- 1.Identify the side length (a) of the decagon
- 2.Calculate the apothem: r = a / (2 × tan(18°))
- 3.Apply the area formula: A = (10 × a × r) / 2
- 4.Simplify to: A = 7.694 × a²
Mathematical Background
Decagon Properties:
- • 10 equal sides and vertices
- • Central angle = 360° / 10 = 36°
- • Interior angle = 144°
- • Can be divided into 10 congruent triangles
Alternative Methods
- •From perimeter: A = P × apothem / 2
- •From circumradius: Calculate side first, then area
- •Triangulation: Sum of 10 triangle areas