Star Shape Calculator
Calculate dimensions, perimeter, and area of star polygons (pentagram, hexagram, heptagram, octagram)
Calculate Star Shape Properties
Schläfli symbol: 5/2
Distance between two contiguous points of the star
Pentagram (5-pointed star) Properties
Star Type: Pentagram (5-pointed star) (Schläfli symbol: 5/2)
Golden Ratio relationships: l/a = a/b = b/c = φ ≈ 1.618
Perimeter formula: P = 2n × b, where n = number of points
Example Calculation
Pentagram Example
Problem: Find all properties of a pentagram with side a = 5 cm.
Given: Side a = 5 cm
Golden ratio: φ = (1 + √5)/2 ≈ 1.618
Solution
Ray b = a/φ = 5/1.618 ≈ 3.090 cm
Original side c = b/φ = 3.090/1.618 ≈ 1.910 cm
Full side l = a × φ = 5 × 1.618 ≈ 8.090 cm
Perimeter = 10 × b = 10 × 3.090 = 30.90 cm
Area = 5(5-2√5)/2 × a² ≈ 1.720 × 25 ≈ 43.01 cm²
Star Types
Pentagram (5/2)
5-pointed star with golden ratio relationships
Hexagram (6/2)
6-pointed star (Star of David)
Heptagram (7/2)
7-pointed star with complex geometry
Octagram (8/2)
8-pointed star with 45° angles
Star Properties
Non-convex and self-intersecting
Equilateral (all outer sides equal)
Equiangular (equal angles in same regions)
Can be drawn without lifting pen
Identified by Schläfli symbol n/m
Understanding Star Polygons
What are Star Polygons?
Star polygons (n-grams) are fascinating geometric shapes that are non-convex, self-intersecting, equilateral, and equiangular. They're created by extending the sides of regular polygons until the extensions intersect with each other.
Schläfli Symbol
Each star polygon is uniquely identified by its Schläfli symbol n/m, where:
- •n is the number of corners/sides
- •m is the starriness (distinct boundaries)
Key Measurements
Side a
Distance between two contiguous "points" of the star
Ray b
Length of the star "rays" winding around the polygon
Original side c
Side length of the original regular polygon
Full side l
Complete edge length: l = c + 2b
Special Properties
Pentagram & Golden Ratio
The pentagram is unique among star polygons because all its proportions relate to the golden ratio φ ≈ 1.618. The relationships l/a = a/b = b/c = φ create perfect mathematical harmony.
Cultural Significance
Star shapes appear in 40+ national flags worldwide. The pentagram appears in many cultures, while the hexagram is known as the Star of David. These shapes bridge mathematics and symbolism.