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A regular n-pointed star can be modeled precisely as a "bicentric" star polygon — a shape with 2n vertices total, alternating between an outer radius R (the sharp point tips) and an inner radius r (the inward notches between points), evenly spaced around the center every 360°/(2n). This construction covers everything from a simple 5-pointed star to elaborate many-pointed designs, and is used in graphic design and logo work, laser-cutting or CNC templates for decorative star shapes, and geometry coursework exploring star polygons as an extension of regular polygons.
How it works
Enter the number of points n, the outer radius R, and the inner radius r. Each pair of adjacent vertices (one outer, one inner) subtends a central angle of 180°/n at the center. The calculator treats the star as 2n congruent triangles fanning out from the center, each with two sides R and r and an included angle of 180°/n, giving a total area of Area = n·R·r·sin(180°/n) (each triangle's area is ½Rr·sin(180°/n), and there are 2n of them, so the factor of 2 cancels the ½). Each of the 2n edges connects an outer vertex to an adjacent inner vertex and, by the Law of Cosines, has length √(R² + r² − 2Rr·cos(180°/n)) — since all 2n triangles are congruent, every edge has this same length, so the perimeter is simply 2n times that edge length.
- Enter number of points.
- Enter outer radius.
- Enter inner radius.
- Click Calculate to see your results.
Examples
5-pointed star, R=10, r=4
Central angle = 180°/5 = 36°. Area = 5×10×4×sin(36°) = 200×0.5878 ≈ 117.56. Edge = √(10²+4²−2×10×4×cos(36°)) = √(116−64.72) ≈ 7.16. Perimeter = 2×5×7.16 ≈ 71.61.
6-pointed star, R=8, r=3
Central angle = 180°/6 = 30°. Area = 6×8×3×sin(30°) = 144×0.5 = 72. Edge = √(8²+3²−2×8×3×cos(30°)) = √(73−41.57) ≈ 5.61. Perimeter = 2×6×5.61 ≈ 67.27.
4-pointed star, R=5, r=2
Central angle = 180°/4 = 45°. Area = 4×5×2×sin(45°) = 40×0.7071 ≈ 28.28. Edge = √(5²+2²−2×5×2×cos(45°)) = √(29−14.14) ≈ 3.85. Perimeter = 2×4×3.85 ≈ 30.84.
Who should use it
- Designing a star shape for a logo, badge, or decorative template.
- Planning material needs for a laser-cut or CNC-routed star shape.
- Geometry coursework on star polygons.
Industry applications
- Graphic and product design
- Laser cutting / CNC fabrication
- Geometry education
Advantages
- Models any n-pointed star shape from just its two defining radii.
- Computes both area and perimeter together from consistent geometry.
Limitations
- Assumes a perfectly regular, evenly-spaced star — doesn't model irregular or asymmetric star shapes.
Common mistakes to avoid
- Entering an inner radius larger than the outer radius, which inverts the intended star shape and produces a geometrically inconsistent (self-intersecting) result.
- Forgetting that the central angle used is 180°/n, not 360°/n — a common mix-up since a full regular n-gon uses 360°/n.
- Assuming area scales linearly with the radii — it actually scales with the PRODUCT of R and r, so doubling both radii quadruples the area, not doubles it.
Best practices
- Keep the inner radius meaningfully smaller than the outer radius to get a recognizable star shape rather than a near-polygon.
- Double check which value is R (outer, point tips) and which is r (inner, notches) before entering them.
- For a very spiky star, expect a much longer perimeter relative to area than for a blunter, gear-like star with similar radii.
Tips
- Working with a simple regular polygon instead of a star? Most standard regular-polygon area/perimeter formulas apply once r equals R (which collapses this star into a 2n-sided polygon).