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Regular Star Polygon

A regular star has n outer points and n inner points alternating. R = outer radius (to tips), r = inner radius (to valleys). Central angle = 360°/n; point angle = 180°−360°/n.

Concept Fundamentals
n ≥ 5
Points
360°/n
Central ∠
180°−360°/n
Point ∠
n=5, golden ratio
Pentagram

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5-point star (pentagram) with r/R≈0.38 has golden ratio proportions. 6-point star (hexagram) is two overlapping triangles. Central angle = 360°/n; point angle = 180°−360°/n.

Key quantities
n ≥ 5
Points
Key relation
360°/n
Central ∠
Key relation
180°−360°/n
Point ∠
Key relation
n=5, golden ratio
Pentagram
Key relation

Ready to run the numbers?

Why: Star shapes appear in flags, logos, and sacred geometry. The pentagram (5-point) with r/R≈0.38 approximates the golden ratio.

How: Area = sum of n triangular segments. Central angle = 360°/n. Point angle = 180°−360°/n. Perimeter from edge lengths. Use Shoelace or coordinate formula for area.

5-point star (pentagram) with r/R≈0.38 has golden ratio proportions.6-point star (hexagram) is two overlapping triangles.
Sources:Wolfram MathWorld - Star PolygonWikipedia - Star Polygon

Run the calculator when you are ready.

Star Shape CalculatorEnter number of points n, outer radius R, inner radius r
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STAR POLYGON

Star Shape — Area, Perimeter, Angles

n points, outer R, inner r. Regular star polygon.

📐 Examples — Click to Load

Inputs

cm
cm

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

🧮 Fascinating Math Facts

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Regular star: n outer points, n inner points, alternating.

— Definition

★

Pentagram (5-point) approximates golden ratio when r/R≈0.38.

— Geometry

📋 Key Takeaways

  • • Regular star polygon: n outer points, n inner points, alternating
  • • R = outer radius (to tips), r = inner radius (to valleys)
  • • Central angle = 360°/n; point angle = 180° − 360°/n
  • • Area = sum of n triangular segments (Shoelace or coordinate formula)
  • • 5-point star (pentagram) with r/R ≈ 0.38 approximates golden ratio

💡 Did You Know?

⭐5-point star (pentagram) has golden ratio when r/R = 1/φ ≈ 0.618Source: Sacred Geometry
✡️Star of David is two overlapping triangles — 6-point starSource: Symbolism
🎖️Sheriff badges often use 5 or 7 points for distinct silhouetteSource: Design
🌟Christmas stars are often 8-pointed (Star of Bethlehem)Source: Tradition
⚡Ninja stars (shuriken) can have 4–8 pointsSource: Weapons
📐Star polygons are {n/k} in Schläfli notationSource: MathWorld

📖 Formulas Explained

Central & Point Angles

textCentral=frac360°n,quadtextPoint=180°−frac360°n\\text{Central} = \\frac{360°}{n}, \\quad \\text{Point} = 180° - \\frac{360°}{n}

Area

Sum of n triangles: outer vertex + two adjacent inner vertices. Uses coordinate formula.

🎯 Expert Tips

Classic 5-point

r/R ≈ 0.38–0.4 gives balanced star. Golden: r/R = 1/φ ≈ 0.618.

n ≥ 4

n=4 gives square-like star. n=5 is pentagram. n=6 can be Star of David.

r < R

Inner radius must be less than outer. r=R would collapse to polygon.

Units

R, r in same units. Area = units², perimeter = units.

⚖️ Comparison

nNameCentral °
4Square star90°
5Pentagram72°
6Hexagram60°
8Octagram45°

📊 Stats

n
Points
R
Outer r
r
Inner r
φ
Point °

❓ FAQ

What is a regular star polygon?

n outer vertices, n inner vertices, alternating. Outer at radius R, inner at r.

Minimum n?

n=4 gives a square-like star. n≥5 is more star-like.

Golden ratio star?

5-point with r/R = 1/φ ≈ 0.618. Classic pentagram.

Star of David?

6-point star. Two overlapping equilateral triangles.

Point angle formula?

180° − 360°/n. For n=5: 180−72 = 108°.

Area formula?

Sum of n triangular segments. Each triangle: outer vertex + 2 adjacent inner vertices.

r vs R?

R = distance to tips. r = distance to valleys. Must have r < R.

n=4 ninja star?

4-point star (square rotated 45°). Valid shape.

⚠️ Disclaimer: For regular star polygons. Assumes vertices on two concentric circles. Educational use.

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