Star Shape Calculator: Area, Perimeter, and Geometry of Regular Star Polygons
Calculate the area and perimeter of regular star polygons with our free tool. Enter outer radius, inner radius, and number of points to find all dimensions instantly.
Star polygons are among the most visually striking shapes in geometry, appearing on national flags, religious symbols, decorative patterns, and astronomical illustrations. From the five-pointed pentagram of ancient Greece to the six-pointed Star of David and the intricate octagrams of Islamic geometric art, stars have captivated mathematicians and artists for millennia. Calculating the area and perimeter of a regular star polygon requires understanding the interplay between its outer radius (the distance to the tips), inner radius (the distance to the valleys), and the number of points — a beautiful application of trigonometry and polygon geometry.
Key Takeaway
The area of a regular n-pointed star with outer radius R and inner radius r is A = (n/2) × R × r × sin(π/n). The perimeter equals n times the edge length, where each edge connects an outer vertex to an adjacent inner vertex.
1. What Is a Regular Star Polygon?
A regular star polygon is formed by connecting every k-th vertex of a regular n-gon. The Schläfli symbol {n/k} describes this construction. The most common star polygons include the pentagram {5/2} (five-pointed star), the hexagram {6/2} (six-pointed star or Star of David), and the octagram {8/3} (eight-pointed star).
The outer radius R is the distance from the center to the outermost points (the tips), while the inner radius r is the distance from the center to the innermost points (the valleys between tips). The ratio r/R determines how "pointy" or "fat" the star appears.
2. Area Formula
The area of a regular n-pointed star can be computed by decomposing it into n congruent triangles, each formed by the center, an outer vertex, and an adjacent inner vertex. The area of each triangle is (1/2) × R × r × sin(π/n), giving the total:
For a five-pointed star (n = 5) with R = 10 and r = 4:
3. Perimeter and Edge Length
Each edge of the star connects an outer vertex to an adjacent inner vertex. By the law of cosines, the edge length is:
Perimeter = n × e
4. Symmetry Properties
A regular n-pointed star has Dₙ dihedral symmetry: n rotational symmetries (multiples of 360°/n) and n reflection axes. The pentagram has D₅ symmetry (10 operations), the hexagram has D₆ (12 operations), and the octagram has D₈ (16 operations). This high degree of symmetry is what makes star polygons visually balanced and aesthetically pleasing.
5. Star Polygons in Culture and Nature
Five-pointed stars appear on the flags of the United States, China, Turkey, and many other nations. Six-pointed stars (hexagrams) are central to Jewish symbolism and appear on the Israeli flag. Eight-pointed stars are fundamental motifs in Islamic geometric art, appearing in mosque decorations across the Muslim world. In nature, star-shaped forms appear in starfish, flower petals, snowflakes, and certain crystal structures.
6. Frequently Asked Questions
What is the convex hull of a star?
The convex hull is the smallest convex polygon that contains the entire star. For a five-pointed star, the convex hull is a regular pentagon with vertices at the star's five outer points.
Can a star have an inner radius equal to the outer radius?
If r = R, the star degenerates into a regular polygon (all vertices are equidistant from the center). The area formula gives A = (n/2) × R² × sin(π/n), which is the area of the regular n-gon.
Using Our Calculator
Enter the number of points, outer radius, and inner radius to instantly compute the star's area, perimeter, and edge length. The calculator provides step-by-step derivations for each computed value.