Star Shape Calculator — Area and Perimeter of a Star Polygon
Find the area, perimeter, individual side length, and tip angle of any regular star polygon — a 5-pointed star, a 6-pointed Star of David, or any n-pointed shape — by specifying the outer radius (tip to centre) and inner radius (notch to centre).
Total area enclosed by the star polygon (same units as R²)
- 1
sin(π ÷ n)
sin(π ÷ 5) = 0.587785The angle between adjacent inner and outer radii. - 2
n × R × r
5 × 10 × 4 = 200 - 3
Area = n × R × r × sin(π ÷ n)
200 × 0.587785 = 117.5571
How does this calculator work?
A regular n-pointed star with outer radius R and inner radius r has area n·R·r·sin(π/n) and perimeter 2n·√(R²+r²−2Rr·cos(π/n)). Each of the 2n identical sides forms a triangle with the two adjacent radii, and the tip angle sharpens as r decreases relative to R.
Formula
How this is calculated
A regular n-pointed star polygon has 2n vertices alternating between the outer radius R (tips) and inner radius r (notches), equally spaced at angles of π/n (180°/n) from each other. Applying the shoelace formula to these 2n vertices gives the elegant area formula A = n·R·r·sin(π/n). Each of the 2n edges is a chord connecting adjacent outer and inner vertices, and its length follows directly from the law of cosines: √(R² + r² − 2Rr·cos(π/n)).
The sharpness of each tip — the angle at the outermost vertex — depends on how much smaller r is relative to R. The formula 2·arctan(r·sin(π/n) / (R − r·cos(π/n))) gives this angle exactly; for a classic 5-pointed star with r ≈ 0.382R, the tip angle is exactly 36°.
The inner radius r must be strictly less than R. The natural inner radius for a regular {n/2} star (where alternate vertices of a regular 2n-gon are connected) is R·cos(π·2/n)/cos(π/n), but any value in (0, R) produces a valid star shape. The animated XYPlot below traces the star outline and shows the symmetry.
Frequently asked questions
The inner radius of a regular pentagram — the 5-pointed star formed by connecting every other vertex of a regular pentagon — is r = R · (cos 72° / cos 36°) ≈ 0.382 R. Entering n = 5, R = 10, and r ≈ 3.82 reproduces that familiar shape.
A Star of David is traditionally two overlapping equilateral triangles, not a single star polygon, so the area is computed differently. However, if you model it as a regular 6-pointed star polygon with chosen outer and inner radii, the formula applies and gives the correct area and perimeter for that particular star outline.
For a regular pentagram with r ≈ 0.382R, each tip angle is exactly 36°. This is one of the most recognisable properties of the pentagram and it is derived directly from the formula 2·arctan(r·sin(π/n) / (R − r·cos(π/n))).
Also known as
TG we-Calculate Editorial Team. (2026). Star Shape Calculator — Area and Perimeter of a Star Polygon [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/star-shape-calculator
TG we-Calculate Editorial Team. "Star Shape Calculator — Area and Perimeter of a Star Polygon." TG we-Calculate. 2026. https://we-calculate.com/calculator/star-shape-calculator.
TG we-Calculate Editorial Team, "Star Shape Calculator — Area and Perimeter of a Star Polygon," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/star-shape-calculator
@misc{wecalculate_star_shape_calculator, title = {Star Shape Calculator — Area and Perimeter of a Star Polygon}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/star-shape-calculator}}, year = {2026}, note = {TG we-Calculate} }
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