Star Shape Calculator
Enter the number of points, outer radius, and inner radius to instantly find the area, perimeter, tip angle, inner valley angle, and spike height of any regular star polygon. Works for 5-pointed pentagrams through 12-pointed stars. Results update as you type.
Formula
Worked example
A classic 5-pointed star with R = 10 cm and r = 3.82 cm (the golden-ratio inner radius): angle = pi/5 = 36 deg. Segment s = sqrt(100 + 14.59 - 2 x 10 x 3.82 x cos(36 deg)) = sqrt(114.59 - 61.79) = sqrt(52.8) approx 7.27 cm. Area = 5 x 10 x 3.82 x sin(36 deg) = 5 x 10 x 3.82 x 0.5878 approx 112.2 cm^2. Perimeter = 2 x 5 x 7.27 = 72.7 cm. Tip angle = arccos((2 x 52.8 - (2 x 3.82 x 0.5878)^2) / (2 x 52.8)) approx 36 deg.
What is a regular star polygon?
A regular star polygon is a non-convex figure formed by connecting every nth vertex of a regular polygon, or equivalently by placing 2n vertices alternately on two concentric circles of radius R (outer tips) and r (inner valleys). The most familiar example is the five-pointed star or pentagram. Unlike a simple convex polygon, the edges of a star polygon cross each other, creating the distinctive layered look. All 2n edges are equal in length and all interior angles at the tips are equal, as are all the valley angles. The ratio r/R controls how dramatic the spikes look: a small ratio gives long, sharp spikes; a ratio close to 1 gives a nearly flat polygon.
How area and perimeter are calculated
The area formula A = n R r sin(pi/n) comes from dividing the star into 2n congruent isoceles triangles, each with two sides meeting at the centre and the opposite vertex alternating between an outer tip and an inner valley. Each triangle contributes (1/2) R r sin(2 pi / (2n)) = (1/2) R r sin(pi/n) to the total, and there are 2n of them, giving n R r sin(pi/n). The perimeter formula P = 2ns uses the Law of Cosines to find the length s of each straight edge: s = sqrt(R^2 + r^2 - 2Rr cos(pi/n)). The 2n edges are all equal by symmetry, so P = 2n x s.
Tip angle and the golden ratio pentagram
The tip angle at each outer point is found by the Law of Cosines on the isoceles triangle formed by the outer vertex and its two adjacent inner valley vertices. For a pentagram with the classical golden-ratio inner radius r = R / phi^2 (where phi = (1 + sqrt(5)) / 2 approx 1.618), the tip angle works out to exactly 36 degrees and the inner valley angle is 108 degrees. This is why the pentagram appears in Pythagorean mysticism and Renaissance design: its diagonals cross in the golden ratio at every intersection. A hexagram (Star of David) with r = R / sqrt(3) produces 60-degree tip angles, and all the inner triangles are equilateral.
Practical uses and design tips
Star shapes appear in flags (the US flag, Australia, Ethiopia), Islamic geometric art, jewellery, logo design, quilt patterns, and architectural ornament. For a laser-cut or CNC project, the key dimensions are the outer radius (sets the bounding circle), the inner radius (controls spike sharpness), and the number of points. To make the spikes more dramatic, reduce r while keeping R fixed. To scale the whole star without changing its shape, multiply both R and r by the same factor: the angles and the r/R ratio stay constant, while all lengths scale proportionally. The spike height output tells you the clear height from the inner concave edge to the tip, which is useful when nesting stars in a pattern or calculating material needed for a star-shaped frame.
Common star polygon presets
| Star type | Points (n) | Classic r/R ratio | Typical tip angle | Notes |
|---|---|---|---|---|
| Pentagram | 5 | 0.382 (1/phi^2) | 36 deg | Golden-ratio star; all diagonals intersect in the golden ratio |
| Hexagram | 6 | 0.577 (1/sqrt(3)) | 60 deg | Star of David; inner triangles are equilateral |
| Heptagram | 7 | 0.692 | ~77 deg | Seven-pointed; used in flag of Australia |
| Octagram | 8 | 0.765 | ~90 deg | Eight-pointed; right angles at inner vertices |
| Decagram | 10 | 0.851 | ~108 deg | Ten-pointed; tips become very blunt at standard ratios |
Characteristic angles and the classic golden-ratio inner radius for each standard star type. Tip angle shown for the listed r/R ratio.
Frequently asked questions
What is the difference between the outer radius and the inner radius?
The outer radius R is the distance from the centre of the star to each pointed tip. The inner radius r is the distance from the centre to each inner valley vertex, the concave point between two tips. Together, R and r, along with the number of points, completely define the shape of a regular star polygon.
What inner radius gives a classic five-pointed star?
The canonical pentagram uses an inner radius equal to R divided by phi squared, where phi = (1 + sqrt(5)) / 2 approx 1.618. This gives r approx 0.382 R. At this ratio the tip angle is exactly 36 degrees and all diagonal intersections divide each other in the golden ratio. For a star with R = 10 cm, set r to 3.82 cm.
How does the number of points affect the shape?
More points mean the spikes are packed closer together. For a fixed outer radius and the same r/R ratio, increasing n generally increases the area and perimeter up to a point, but the spikes become narrower. With very many points and a high r/R ratio, the star approaches a regular polygon. The tip angle also increases with n for a fixed ratio: a five-pointed star has sharper tips than a ten-pointed star at the same ratio.
Can I calculate the star if I know the area or perimeter instead of the radii?
Not directly, because both formulas involve two unknowns (R and r). You need to know at least the r/R ratio in addition to the area or perimeter, or you need two of the three quantities (R, r, area or perimeter). With the ratio known, you can solve A = n R r sin(pi/n) for R by substituting r = ratio x R, giving R^2 = A / (n x ratio x sin(pi/n)).
What is the tip angle of a regular six-pointed star (hexagram)?
For a hexagram with the standard inner radius r = R / sqrt(3) (ratio approx 0.577), the tip angle is 60 degrees and the inner valley angle is also 60 degrees, making all the small triangles equilateral. This is the Star of David. If you use a different r/R ratio, the tip angle will differ; enter your values in this calculator to find it.
Does this calculator work for non-standard stars, such as a 7- or 9-pointed star?
Yes. Enter any integer of 3 or more for the number of points and any positive inner radius smaller than the outer radius. The formulas are general and work for any regular star polygon. Seven-pointed stars appear in the flags of Australia and Papua New Guinea; nine-pointed stars are used in the Baha'i Faith symbol.
How does the r/R ratio change the look of the star?
A low ratio (below 0.3) produces very long, needle-like spikes. A ratio around 0.4 is the classic sharp star seen in logos and flags. A ratio above 0.7 creates blunt, wide spikes. A ratio of 1 is impossible (inner and outer circles coincide); as the ratio approaches 1 the figure approaches a regular 2n-gon with zero spike depth.