Regular Polygons
Professional calculators and formulas for regular and irregular polygons
Regular Polygons
Triangle (A = ¼√3×a²)
Equilateral triangle with three equal sides
Square (A = a²)
Regular quadrilateral with four equal sides
Pentagon (5 sides)
Regular pentagon - frequently found in nature
Hexagon (6 sides)
Hexagonal form - like honeycombs
Heptagon (7 sides)
Regular heptagon with complex properties
Octagon (8 sides)
Octagonal form - known from stop signs
Nonagon (9 sides)
Regular nonagon with nine equal sides
Decagon (10 sides)
Decagonal form with complex mathematical properties
Hendecagon (11 sides)
Regular hendecagon with eleven equal sides
Dodecagon (12 sides)
Twelve-sided form - often in clocks and architecture
Hexadecagon (16 sides)
Complex polygon with sixteen equal sides
Regular N-Gon (n sides)
Universal calculator for any regular polygon
Irregular Polygons
Axial Symmetric Pentagon
Symmetric pentagon with one axis of reflection
Irregular Hexagon
Hexagon with different side lengths
Irregular Octagon
Octagon with varying side lengths
Regular Polygon Ring
Ring made from regular polygons
Concave Equilateral Hexagon
Inward-curved hexagon
Star-shaped Polygons
Pentagram
Five-pointed star - classic symbol
Hexagram
Six-pointed star - Star of David
Octagram
Eight-pointed star with eight spikes
Star of Lakshmi
Hindu star with eight points
About Polygon Geometry
Polygon geometry forms a fundamental foundation of mathematics and finds practical application in:
- Architecture - Floor plans, roof construction
- Engineering - Mechanical parts design
- Art & Design - Geometric patterns
- Nature Studies - Crystal structures
- Computer Graphics - 3D modeling
- Education - Geometry learning
Fundamental Polygon Formulas
Regular Polygons
Area: A = ¼n×a²×cot(π/n)
Perimeter: P = n×a
Perimeter: P = n×a
Angles
Interior angle: α = (n-2)×180°/n
Exterior angle: β = 360°/n
Exterior angle: β = 360°/n
Special Properties
Diagonal count: d = n(n-3)/2
Central angle: γ = 360°/n
Central angle: γ = 360°/n
Star Polygons
Pentagram: Golden ratio φ
Star angle: varies by type
Star angle: varies by type
Tip: Regular polygons have all sides and angles equal.
The sum of interior angles in any n-sided polygon is (n-2)×180°.
Practical Application Examples
Architecture & Construction
- Hexagonal Tiles: Bathroom and kitchen design
- Octagonal Buildings: Unique architectural forms
- Pentagon Structures: Modern building design
Engineering & Manufacturing
- Hex Bolts: Mechanical fasteners
- Polygonal Shafts: Power transmission
- Crystal Analysis: Material science
Art & Design
- Islamic Patterns: Geometric art
- Logo Design: Corporate identity
- Mosaic Work: Decorative patterns
Science & Nature
- Honeycomb: Hexagonal efficiency
- Snowflakes: Six-fold symmetry
- Crystals: Natural polygon forms
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Quick Reference
3 sides
Triangle
4 sides
Square
6 sides
Hexagon
8 sides
Octagon
∑ angles = (n-2)×180°
Interior Angle Sum
Properties
Regular Polygons: All sides and angles equal
Constructible: 3, 4, 5, 6, 8, 10, 12, 15, 16, 17... sides
Golden Ratio: Appears in pentagon and pentagram
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