Star Shape Calculator

Star Shape Calculator: Area and Perimeter of Regular Star Polygons

A star polygon is a non-convex polygon formed by connecting every kth vertex of a regular n-gon, producing the familiar radiating-point silhouette used in flags, emblems, and decorative art. The five-pointed star (pentagram) and the six-pointed Star of David are the most common examples, but the same geometric framework extends to any number of points. This star shape calculator applies exact trigonometric formulas to compute area and perimeter for any regular star with n outer points, an outer radius R, and an inner radius r.

Core Formulas

Two primary measurements describe a star shape: its enclosed area and its perimeter (total edge length around the outline).

Area Formula

The area of a regular star with n points, outer radius R, and inner radius r is:

A = n · R · r · sin(π/n)

This formula treats the star as 2n congruent triangles sharing an apex at the center. Each triangle has two sides of length R and r and an included central angle of π/n radians. The area of one such triangle is (1/2)·R·r·sin(π/n), and summing all 2n triangles yields the formula above.

Perimeter Formula

The perimeter traces all outer edges of the star:

P = 2n · √(R² + r² − 2Rr · cos(π/n))

A star with n points has exactly 2n sides of equal length. Each side connects one outer tip to one adjacent inner notch. Applying the law of cosines to the triangle formed by the center, one outer tip, and one adjacent inner vertex — with the central angle equal to π/n — gives each side length as √(R² + r² − 2Rr·cos(π/n)).

Variables Explained

• n (Number of Points): The count of outer tips. A standard five-pointed star uses n = 5; the Star of David uses n = 6. Larger values of n produce stars that resemble cogwheels or starburst shapes with progressively shallower indentations.
• R (Outer Radius): The distance from the geometric center to each outer tip. This is the dominant size parameter and defines the star's maximum extent.
• r (Inner Radius): The distance from the center to each inner concave vertex (the notch between two adjacent points). A smaller r relative to R produces sharper, more dramatic points; setting r close to R yields a nearly regular polygon with barely visible indentations.

Deriving the Regular-Star Inner Radius

For a geometrically regular star polygon {n/2} — constructed by connecting every second vertex of a regular n-gon — the inner radius is not a free parameter but is determined entirely by the outer radius:

r = R · cos(2π/n) / cos(π/n)

For a five-pointed star (n = 5), this gives r ≈ 0.382 × R, placing the inner notches at roughly 38.2% of the outer radius. For a six-pointed star (n = 6), the result is r ≈ 0.577 × R. According to Wolfram MathWorld's entry on Star Polygons, these ratios arise directly from the diagonal properties of regular polygons and define the canonical star polygon shapes studied since classical antiquity.

Worked Examples

Example 1: Five-Pointed Star, Custom Proportions

Given n = 5, R = 10 cm, r = 4 cm:

• Area: A = 5 × 10 × 4 × sin(36°) = 200 × 0.5878 ≈ 117.6 cm²
• Side length: √(100 + 16 − 80 × cos(36°)) = √(116 − 64.72) = √51.28 ≈ 7.16 cm
• Perimeter: P = 10 × 7.16 ≈ 71.6 cm

Example 2: Six-Pointed Star, Regular Proportions

Given n = 6, R = 8 cm, r = 4.62 cm (regular-star value):

• Area: A = 6 × 8 × 4.62 × sin(30°) = 6 × 8 × 4.62 × 0.5 ≈ 110.9 cm²
• Side length: √(64 + 21.34 − 73.92 × cos(30°)) = √(85.34 − 64.02) = √21.32 ≈ 4.62 cm
• Perimeter: P = 12 × 4.62 ≈ 55.4 cm

Applications

Star shape calculations appear across a wide range of practical disciplines. Graphic designers use them to size star elements in logos, badges, and emblems. Architects and tile-makers apply star polygon geometry in Islamic geometric patterns and decorative flooring. Manufacturers who cut star-shaped parts — gaskets, cookie cutters, decorative metalwork — need accurate area figures for material estimation. Teachers use these formulas to connect geometry and trigonometry for students, consistent with the scope outlined in the Common Core State Standards for Mathematics. Origami artists and paper-crafters rely on radius ratios to achieve precise point angles in folded star designs.

Methodology and Sources

Both formulas derive from standard analytic geometry. The area formula follows from decomposing the star into 2n isoceles triangles with a shared apex at the center and summing their areas via (1/2)ab·sin(C). The perimeter formula applies the law of cosines to each of the 2n equal-length sides. These derivations are consistent with the formal treatment in Wikipedia's article on Star Polygons and the rigorous definitions on Wolfram MathWorld.