Pentagon Calculator
Calculate area, perimeter, diagonal, apothem, and radii from any known value
Pentagon Calculator
Golden Ratio (φ): 1.618034
The diagonal-to-side ratio in a regular pentagon equals φ.
How to Use the Pentagon Calculator
- Select Input Type: Choose whether to calculate from side, area, perimeter, diagonal, or apothem
- Enter Value: Input the known measurement
- View Results: All properties are calculated instantly including area, perimeter, diagonals, and radii
- Copy Values: Click the copy button next to any result
Understanding Regular Pentagons
A regular pentagon is a five-sided polygon with all sides equal in length and all interior angles equal (108° each). It has deep connections to the golden ratio and appears throughout nature, art, and architecture.
Key Properties:
- 5 equal sides and 5 equal angles
- Interior angle: 108° each
- Sum of angles: 540°
- 5 diagonals that form a pentagram (star)
- 5-fold rotational symmetry
- 5 lines of symmetry
Golden Ratio Connection:
The pentagon is intimately connected to the golden ratio (φ ≈ 1.618). The ratio of a diagonal to a side equals φ, and the diagonals divide each other in the golden ratio.
Pentagon Formulas
Basic Formulas
Radius & Apothem
Practical Applications
Architecture & Design
- The Pentagon building (US Department of Defense)
- Pentagonal floor tiles and patterns
- Window and skylight designs
- Decorative elements and logos
Nature & Science
- Starfish and sand dollars
- Many flowers (hibiscus, morning glory)
- Fruit cross-sections (apples, okra)
- Molecular structures
Sports & Games
- Soccer ball panels (black pentagons)
- Home plate in baseball
- Gaming dice (dodecahedron faces)
- Board game elements
Symbolism
- Pentagram (five-pointed star)
- Religious and mystical symbols
- National flags and emblems
- Corporate logos
Frequently Asked Questions
How do I find the area of a regular pentagon?
Use A = (s² × √(25+10√5)) / 4 ≈ 1.7205 × s², where s is the side length. Alternatively, A = (1/2) × Perimeter × Apothem.
What is the apothem of a pentagon?
The apothem is the perpendicular distance from the center to the midpoint of any side. For side s: apothem ≈ 0.6882 × s.
What is the relationship between a pentagon and the golden ratio?
The diagonal-to-side ratio equals the golden ratio (φ ≈ 1.618). The diagonals also divide each other in this ratio.
What are the interior angles of a regular pentagon?
Each interior angle is 108°. The sum of all interior angles is 540° (formula: (n-2) × 180° where n=5).