⬟ Pentagon Calculator
Calculate properties of a regular pentagon
How to Use This Calculator
Enter Side Length
Input the length of one side of the regular pentagon.
Click Calculate
Press the "Calculate Pentagon Properties" button to compute all pentagon measurements.
Review Results
View the calculated area, perimeter, apothem, and circumradius.
Formula
Area = (1/4)√(5(5 + 2√5)) × s²
Perimeter = 5s
Apothem = s / (2tan(π/5))
Circumradius = s / (2sin(π/5))
Where:
- s = Side length
- √5 ≈ 2.236
- Apothem = Distance from center to midpoint of a side
- Circumradius = Distance from center to vertex
Example: Calculate pentagon properties for side length = 10
Area ≈ (1/4)√(5(5 + 2√5)) × 100 ≈ 172.05 square units
Perimeter = 5 × 10 = 50 units
About Pentagon Calculator
The Pentagon Calculator helps you find the area, perimeter, apothem, and other properties of a regular pentagon. A regular pentagon has five equal sides and five equal angles (each 108°).
When to Use This Calculator
- Geometry: Calculate pentagon properties for geometric problems
- Architecture: Design pentagonal buildings or structures
- Education: Learn and practice pentagon mathematics
- Design: Create pentagonal patterns, logos, or graphics
- Art: Calculate dimensions for pentagonal art and designs
- Engineering: Design pentagonal components
Why Use Our Calculator?
- ✅ Complete Properties: Calculates area, perimeter, apothem, and radius
- ✅ Instant Results: Get all pentagon measurements immediately
- ✅ Step-by-Step Display: See calculation formulas with your values
- ✅ Works with Any Units: Meters, feet, inches, or any unit
- ✅ 100% Accurate: Precise geometric calculations
- ✅ Completely Free: No registration required
Understanding Regular Pentagons
A regular pentagon is a five-sided polygon with all sides equal and all angles equal (108°). Key properties:
- Five Sides: All sides have equal length
- Five Angles: Each interior angle = 108° (sum = 540°)
- Golden Ratio: The diagonal-to-side ratio equals the golden ratio φ ≈ 1.618
- Symmetry: Has 5-fold rotational symmetry
- Star Shape: Connecting alternate vertices creates a pentagram (star)
- Nature: Appears in many flowers (e.g., morning glories, hibiscus)
Real-World Applications
Architecture: The Pentagon building in Washington D.C. is one of the world's largest office buildings, designed as a regular pentagon.
Nature: Many flowers have pentagonal symmetry, including roses, hibiscus, and morning glories. This is common in plant structures.
Design: Pentagons are used in logo design, badges (especially military), and decorative patterns.
Mathematics: Pentagons are closely related to the golden ratio, making them important in mathematical and artistic contexts.
Frequently Asked Questions
What is a regular pentagon?
A regular pentagon is a five-sided polygon with all sides equal in length and all interior angles equal to 108°. It has 5-fold rotational symmetry.
How do you calculate pentagon area?
For a regular pentagon, area = (1/4)√(5(5 + 2√5)) × s², where s is the side length. This simplifies to approximately 1.7205 × s².
What is the golden ratio connection?
In a regular pentagon, the ratio of diagonal length to side length equals the golden ratio φ ≈ 1.618. This makes pentagons special in mathematics and art.
What is the interior angle of a regular pentagon?
Each interior angle of a regular pentagon is 108°. The formula is: (n-2) × 180° / n, where n=5, giving (5-2) × 180° / 5 = 108°.
Can pentagons tile a plane?
Regular pentagons alone cannot tile a plane. However, irregular pentagons can tile the plane, and pentagonal tiling patterns exist.
What is a pentagram?
A pentagram is a five-pointed star created by connecting alternate vertices of a pentagon. It's a symbol used in many cultures and appears in religious and artistic contexts.
Why do many flowers have pentagonal symmetry?
Five-fold symmetry is common in flowers due to genetic factors and evolutionary advantages. It allows efficient packing of reproductive organs and is visually appealing to pollinators.