Regular Polygon Area Calculator
Calculate Polygon Area
Calculate area of any regular polygon from side length, apothem, radius, or number of sides. Visualize polygon shape and properties.
Polygon Area Result
259.81 square units
Polygon Visualization
Formula Used:
Where: A = area, n = number of sides, s = side length
Interior Angle
Each interior angle = 180 × (n-2) / n
Sum of interior angles = 180 × (n-2)
Step-by-Step Calculation:
Symmetry Properties
Rotational symmetry: 6-fold
Reflection symmetry: 6 lines
Dihedral group: Dih6
Shape Name: Regular Hexagon
Area Approximation: ≈ 260 square units
Applications: Architecture, engineering, computer graphics, nature patterns
A regular hexagon with side length 10 units has an area of 259.81 square units. The apothem (distance from center to side midpoint) is 8.66 units, and the circumradius (distance from center to vertex) is 10 units.
What is a Regular Polygon?
A regular polygon is a polygon that is both equilateral (all sides equal) and equiangular (all angles equal). Regular polygons are highly symmetric shapes found throughout mathematics, nature, and human design. As the number of sides increases, a regular polygon approaches a circle.
Regular Polygon Formulas
From Side Length
Most common formula
Requires side length
From Apothem
Using apothem length
Half perimeter × apothem
From Circumradius
Using circumradius
Circle-related formula
Perimeter
Total boundary length
Sum of all sides
Common Regular Polygons
Polygon Area Formulas
1. From Side Length (Standard Formula)
Where:
- A = area of regular polygon
- n = number of sides
- s = side length
- π = mathematical constant (≈ 3.14159)
2. From Apothem (Simpler Formula)
Where:
- A = area of regular polygon
- P = perimeter = n × s
- a = apothem (distance from center to side midpoint)
- n = number of sides
- s = side length
3. From Circumradius (Circle Relationship)
Where:
- A = area of regular polygon
- n = number of sides
- R = circumradius (distance from center to vertex)
- 2π/n = central angle in radians
Common Polygon Area Calculations
| Polygon | Sides (n) | Side Length (s) | Area | Formula Used |
|---|---|---|---|---|
| Equilateral Triangle | 3 | 10 | 43.30 | A = (√3/4)s² |
| Square | 4 | 10 | 100.00 | A = s² |
| Regular Pentagon | 5 | 10 | 172.05 | A = (1/4)√(5(5+2√5))s² |
| Regular Hexagon | 6 | 10 | 259.81 | A = (3√3/2)s² |
Polygon Properties and Relationships
| Property | Formula | Description | Example (n=6) |
|---|---|---|---|
| Interior Angle | 180° × (n-2)/n | Angle between adjacent sides | 120° |
| Central Angle | 360°/n | Angle at center between vertices | 60° |
| Apothem | s / (2 × tan(π/n)) | Distance from center to side | 8.66 (s=10) |
| Circumradius | s / (2 × sin(π/n)) | Distance from center to vertex | 10.00 (s=10) |
Real-World Applications
Architecture & Construction
- Structural design: Hexagonal and octagonal floor plans for optimal space usage
- Tile patterns: Regular polygons in tiling (triangles, squares, hexagons)
- Dome construction: Geodesic domes using triangular polygons
- Window design: Octagonal and hexagonal windows in architecture
Nature & Biology
- Honeycomb structure: Hexagonal cells for optimal space efficiency
- Crystal formations: Regular polygonal patterns in mineral crystals
- Cell structures: Polygonal shapes in biological cell arrangements
- Snowflakes: Hexagonal symmetry in ice crystal formation
Engineering & Design
- Bolt heads: Hexagonal and octagonal shapes for tools
- Gear design: Regular polygonal teeth for smooth rotation
- Computer graphics: Polygonal modeling for 3D objects
- Packaging design: Optimal space utilization with polygonal shapes
Mathematics & Education
- Geometry education: Fundamental shapes for teaching mathematics
- Tessellation studies: Regular polygons in plane filling patterns
- Symmetry groups: Dihedral groups in abstract algebra
- Trigonometry: Applications in trigonometric calculations
Step-by-Step Calculation Examples
Example 1: Regular Hexagon with side = 10
- Number of sides: n = 6
- Side length: s = 10
- Calculate π/n: π/6 ≈ 0.5236 radians
- Compute tangent: tan(π/6) = tan(30°) = 1/√3 ≈ 0.5774
- Square side length: s² = 10² = 100
- Multiply: n × s² = 6 × 100 = 600
- Divide denominator: 4 × tan(π/n) = 4 × 0.5774 = 2.3094
- Final calculation: 600 / 2.3094 ≈ 259.81
- Result: Area ≈ 259.81 square units
Example 2: Square with side = 10 (Simplified)
- Number of sides: n = 4
- Side length: s = 10
- Use simplified formula for square: A = s²
- Calculate: 10² = 100
- Result: Area = 100 square units
- Verification using general formula: tan(π/4) = tan(45°) = 1, A = (4 × 100) / (4 × 1) = 400 / 4 = 100
Related Calculators
Frequently Asked Questions (FAQs)
Q: What's the difference between regular and irregular polygons?
A: Regular polygons have all sides equal and all angles equal (e.g., equilateral triangle, square). Irregular polygons have sides and/or angles of different lengths/measures. Regular polygons are symmetric, while irregular polygons lack symmetry.
Q: How do I find the apothem if I only know the side length?
A: For a regular polygon with n sides and side length s, the apothem is: a = s / (2 × tan(π/n)). For example, for a regular hexagon (n=6): a = s / (2 × tan(30°)) = s / (2 × 0.5774) ≈ s / 1.1547.
Q: What happens to the area as the number of sides increases?
A: As n → ∞, the regular polygon approaches a circle. The area approaches πR² where R is the circumradius. For a fixed circumradius, area increases with n and approaches the circle area. For a fixed side length, area increases dramatically with n.
Q: Can all regular polygons tessellate (tile the plane)?
A: Only three regular polygons can tessellate by themselves: equilateral triangles (n=3), squares (n=4), and regular hexagons (n=6). Other regular polygons cannot tile the plane alone because their interior angles don't divide 360° evenly.
Accurately calculate polygon areas with Toolivaa's free Regular Polygon Area Calculator, and explore more geometric tools in our Geometry Calculators collection.