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Regular Polygon

A regular polygon has all sides equal and all angles equal. Interior angle = (n−2)×180°/n. As n→∞ it approaches a circle. Only triangle, square, hexagon tessellate alone.

Concept Fundamentals
A = ns²/(4·tan(π/n))
Area
P = ns
Perimeter
(n−2)×180°/n
Interior ∠
3, 4, 6 only
Tessellate

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Only triangle, square, and hexagon tessellate the plane alone. As n→∞, a regular n-gon approaches a circle. Hexagons minimize perimeter for given area—honeycomb efficiency.

Key quantities
A = ns²/(4·tan(π/n))
Area
Key relation
P = ns
Perimeter
Key relation
(n−2)×180°/n
Interior ∠
Key relation
3, 4, 6 only
Tessellate
Key relation

Ready to run the numbers?

Why: Regular polygons appear in tiles, molecular structures, and design. Hexagons minimize perimeter for given area—why honeycombs use them.

How: Area = ns²/(4·tan(π/n)). Inradius = s/(2·tan(π/n)), circumradius = s/(2·sin(π/n)). Interior angle = (n−2)×180°/n.

Only triangle, square, and hexagon tessellate the plane alone.As n→∞, a regular n-gon approaches a circle.
Sources:Wolfram MathWorld - Regular PolygonKhan Academy - Geometry

Run the calculator when you are ready.

Regular Polygon CalculatorEnter number of sides (3–100) and side length, area, or perimeter
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GEOMETRYRegular Polygons

Regular Polygon Calculator

Calculate area, perimeter, angles, and radii for any regular polygon (3–100 sides).

Square →Hexagon →Area →

📐 Real-World Examples — Click to Load

Calculation Settings

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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Regular polygon area = ns²/(4·tan(π/n)) for n sides.

— Formula

🍯

Only triangle, square, hexagon tessellate alone—honeycombs use hexagons.

— Tessellation

📋 Key Takeaways

  • • A regular polygon has all sides equal and all angles equal
  • • Only 3 shapes tessellate alone: triangle, square, hexagon
  • • Interior angle = (n-2)×180°/n; exterior angle = 360°/n
  • • As n → ∞, a regular polygon approaches a circle
  • • Hexagons minimize perimeter for given area — why honeycombs use them

💡 Did You Know?

📐The ancient Greeks proved only 5 regular polyhedra exist (Platonic solids) — tetrahedron, cube, octahedron, dodecahedron, icosahedronSource: Euclid
🏛️The Pentagon building is a regular pentagon — 5 equal sides, each 281m longSource: Architecture
🧮A regular hexagon has the highest area-to-perimeter ratio among tessellating shapesSource: Wolfram MathWorld
🎯Stop signs are octagons (8 sides) for high visibility from any approach angleSource: Traffic Design
💻Computer graphics use polygon meshes — more sides = smoother curves (e.g. circles as 64-gons)Source: CG
🔬Carbon nanotubes and graphene form hexagonal lattices — nature's regular polygonsSource: Materials Science

📖 Regular Polygon Formulas Explained

Area

A=n⋅s24tan⁡(π/n)A = \frac{n \cdot s^2}{4 \tan(\pi/n)}

n = sides, s = side length. Also A = ½ P × r (apothem).

Perimeter & Angles

P=n⋅s,θint=(n−2)⋅180°n,θext=360°nP = n \cdot s, \quad \theta_{\text{int}} = \frac{(n-2) \cdot 180°}{n}, \quad \theta_{\text{ext}} = \frac{360°}{n}

Perimeter is n × side. Interior + exterior = 180° at each vertex.

Inradius & Circumradius

r=s2tan⁡(π/n),R=s2sin⁡(π/n)r = \frac{s}{2\tan(\pi/n)}, \quad R = \frac{s}{2\sin(\pi/n)}

r = apothem (to side midpoint). R = to vertex.

🎯 Expert Tips

💡 Tessellation

Only triangles (n=3), squares (n=4), and hexagons (n=6) tile the plane by themselves.

💡 From Area

Given area A: s = √(4A·tan(π/n)/n). Then P = ns.

💡 Large n

As n increases, the polygon approaches a circle. Use n=100 for a near-circle.

💡 Honeycomb

Bees use hexagons — least wax for maximum storage. Optimal efficiency.

⚖️ Why Use This Calculator?

FeatureThis CalculatorBasicManual
3 input modes✅❌⚠️
3–100 sides✅❌✅
Step-by-step solutions✅❌✅
Interactive charts✅❌❌
Copy & share✅❌❌
All properties✅⚠️⚠️

📊 Quick Facts

3
Min Sides
6
Tessellators
360°
Ext Angle Sum
∞
→ Circle

❓ Frequently Asked Questions

What is a regular polygon?

A polygon with all sides equal and all interior angles equal. Examples: equilateral triangle, square, regular hexagon.

How do I find the area from the perimeter?

For regular polygon: P = ns, so s = P/n. Then A = (ns²)/(4·tan(π/n)).

What is the apothem?

The apothem (inradius) is the perpendicular distance from the center to any side. Used in A = ½ P × apothem.

Which regular polygons tessellate?

Only equilateral triangles, squares, and regular hexagons can tile the plane by themselves.

Why are honeycombs hexagonal?

Hexagons use the least wax to enclose a given area — optimal efficiency for bees.

What happens as n increases?

The polygon approaches a circle. Interior angle → 180°, area/πR² → 1.

How do I find side length from area?

s = √(4A·tan(π/n)/n). Our calculator does this automatically.

What is the circumradius?

The radius of the circle that passes through all vertices. R = s/(2·sin(π/n)).

⚠️ Disclaimer: This calculator provides mathematically precise results for ideal regular polygons. Real-world shapes may have manufacturing tolerances. Verify for construction and design projects.

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