Pentagon Calculator
Calculate area, perimeter, diagonal, and other properties of regular pentagons
Calculate Pentagon Properties
Choose which measurement you know to calculate all other properties
Enter a positive value to calculate all pentagon properties
Pentagon Properties
Regular Pentagon Diagram
Example Calculation
Regular Pentagon with Side Length 5 units
Given: Side length (a) = 5 units
Area: 5² × √(25 + 10√5) / 4 ≈ 43.01 sq units
Perimeter: 5 × 5 = 25 units
Diagonal: 5 × (1 + √5) / 2 ≈ 8.09 units
Height: 5 × √(5 + 2√5) / 2 ≈ 7.69 units
Interior angle: 108°
Number of diagonals: 5
Pentagon Properties
Sides
A pentagon has exactly 5 sides
Diagonals
5 diagonals that form a pentagram
Interior Angle
Each interior angle = 108°
Sum of Angles
Total interior angles = 540°
Golden Ratio Connection
Golden Ratio φ
φ = (1 + √5) / 2 ≈ 1.618
Diagonal Ratio
Diagonal / Side = φ (golden ratio)
Pentagram
Diagonals form a perfect pentagram
Architecture
Pentagon building, flower petals
Understanding Regular Pentagons
What is a Regular Pentagon?
A regular pentagon is a 5-sided polygon where all sides have equal length and all interior angles are equal (108°). The name comes from Greek: "Penta" meaning five and "gon" meaning angle.
Key Properties
- •Sides: 5 equal sides
- •Interior angles: Each 108°, sum = 540°
- •Diagonals: 5 equal diagonals forming pentagram
- •Symmetry: 5-fold rotational and reflectional
Pentagon Formulas
Area: A = a² × √(25 + 10√5) / 4 ≈ 1.720a²
Perimeter: P = 5a
Diagonal: d = a × (1 + √5) / 2 ≈ 1.618a
Height: h = a × √(5 + 2√5) / 2 ≈ 1.539a
Inradius: r = a / (2 × tan(36°)) ≈ 0.688a
Note: Where 'a' represents the side length of the regular pentagon.
Golden Ratio Connection
Golden Ratio φ
φ = (1 + √5) / 2 ≈ 1.618. The ratio of diagonal to side equals φ.
Pentagram
The five diagonals intersect to form a smaller pentagon inside, creating a pentagram.
Nature & Art
Found in flower petals, starfish, and classical architecture due to aesthetic appeal.