Star Shape Calculator - Star Polygon Side Lengths & Area

A star shape calculator is a geometry tool that returns the side lengths, perimeter, and area of any regular star polygon from one input. The four most common regular star polygons are the pentagram (5), hexagram (6), heptagram (7), and octagram (8), each identified by a Schläfli symbol n/m and built by extending the sides of a regular n-gon. Pick the number of points, enter a, and the calculator applies the matching formulas.

• • Geometry homework: Confirm a regular star polygon problem or verify a Schläfli symbol during a geometry class or olympiad.
• • Crafts and ornaments: Plan a five-pointed star ornament, a Star of David, a heptagonal tree topper, or an eight-pointed compass rose from one reference length.
• • Architecture and tile work: Size a star-shaped window, mosaic inset, railing panel, or precast cap from the distance between two points of the star.
• • 3D printing and CNC: Generate the polygon coordinates and area of a star-shaped badge, gear insert, or pendant for laser cutting or 3D printing.

Every regular star polygon is non-convex, equilateral, and equiangular, which is why the same panel works for all four shapes. The pentagram is governed by the golden ratio, the hexagram by equilateral-triangle ratios, and the heptagram and octagram by sin and cot of pi/n.

For a non-star regular polygon, the Polygon Area Calculator takes one side length and returns the area, apothem, and circumradius for any n.

The calculator reads the number of star points and the input distance a, then applies the matching relationships for that shape. The pentagram and the hexagram use closed-form formulas; the heptagram and the octagram use sin, cos, and cot of pi/n and 2*pi/n.

d is an auxiliary length used only in the heptagram and octagram area formulas: d = l / (2 * cos(3 * pi / 7)) - l for the heptagram, and d = b + c for the octagram. The square root gives the height of the isosceles triangles that, together with the regular n-gon of side a, fill the star.

• n: Number of star points (5, 6, 7, or 8). Every supported star has the Schläfli symbol n/2.
• a: Distance between two contiguous points of the star. This is the natural input because it is the visible side at every point.
• b, c, l: Ray length b, inner polygon side c, and long diagonal l. For the pentagram, l/a = a/b = b/c = golden ratio.
• d (heptagram, octagram only): Auxiliary length used in the area formula. Heptagram: d = l / (2 * cos(3 * pi / 7)) - l. Octagram: d = b + c. The square root sqrt(d^2 - (a/2)^2) is the isosceles triangle height that fills the star with the regular n-gon of side a.

The general perimeter P = 2 * n * b works for every star because each of the n points contributes two sides of length b. The area formulas split into a closed-form branch for n = 5, 6 and a trigonometric branch for n = 7, 8.

According to Wolfram MathWorld, the area of a regular pentagram with side length a is K * a^2 where K = sqrt(5*(5-2*sqrt(5)))/2, which equals about 0.8123027426.

According to Wikipedia (Hexagram), a regular hexagram is the star polygon formed by two overlapping equilateral triangles, and its characteristic sides satisfy l = 3*b, b = a/sqrt(3), and c = b for the rays of length b.

The hexagram is built by extending the sides of a regular hexagon, so the Hexagon Calculator page returns the apothem, circumradius, and interior angle of the base hexagon.

Four short ideas explain why the same panel works for all four shapes and why each has its own area constant.

These four ideas cover the vocabulary you will see in any star polygon reference.

Every star polygon is built from a ring of isosceles triangles around the inner n-gon, so the Triangle Area Calculator gives the matching area for one of those triangles.

Five short steps cover every common case, from a quick pentagram sketch to a precise heptagram with seven points.

1. 1 Pick the number of star points: Choose 5 for a pentagram, 6 for a hexagram, 7 for a heptagram, or 8 for an octagram.
2. 2 Enter the point distance a: Type the distance between two contiguous points. The unit is your choice and propagates to all outputs.
3. 3 Read b, c, and l: The first three result rows show b, c, and l. Pentagram follows the golden ratio; hexagram follows equilateral-triangle ratios.
4. 4 Read the perimeter and area: The last two rows show the perimeter (always 2 * n * b) and the area. The area uses a closed-form constant for n = 5 and n = 6, and a trigonometric sum for n = 7 and n = 8.
5. 5 Reset to the pentagram default: Click Reset to return to the pentagram example (n = 5, a = 5).

For a wooden pentagram with a 12 cm point distance, set Number of Star Points to Pentagram and Point Distance to 12. The panel returns b ≈ 7.42 cm, c ≈ 4.58 cm, l ≈ 19.42 cm, perimeter ≈ 74.16 cm of cut length, and area ≈ 116.97 cm² of material. Switch to Hexagram with the same 12 cm and you get b ≈ 6.93 cm, perimeter ≈ 83.14 cm, area ≈ 124.71 cm² for a Star of David silhouette.

When the next step is the area of a 10-sided polygon that frames a star layout, the Decagon Area Calculator page turns the same kind of closed-form formula into the area, apothem, and circumradius.

A dedicated star shape calculator removes the algebra from the four most common regular star polygons and keeps the result panel consistent across all of them.

• • All four common stars in one panel: Pentagram, hexagram, heptagram, and octagram share the same input, the same five outputs, and the same area unit.
• • Golden ratio and equilateral-triangle relationships built in: Pentagram applies the golden ratio; hexagram applies equilateral-triangle ratios.
• • Trigonometric formulas for heptagram and octagram: Heptagram and octagram use sin, cos, and cot of pi / n and 2 * pi / n at full double precision.
• • Unit-agnostic input and output: Enter a in any linear unit and the result panel reports the side lengths, perimeter, and area in the same unit.

The page is most useful as a check, not as a replacement for understanding the formulas. Use it to confirm a homework answer, sanity-check a craft layout, or pre-validate coordinates before 3D printing or laser cutting.

The heptagram 7/2 is built by extending the sides of a regular heptagon, so the Heptagon Area Calculator returns the area, apothem, and circumradius of the base 7-gon.

Three measurable factors control the precision of every star polygon result, and a couple of practical limits apply to the unsupported shapes.

• • The page covers the four most common regular star polygons (5, 6, 7, and 8 points). A nonagon (9 points), a decagram (10 points), or any shape with a different Schläfli step such as 7/3 is not implemented here.
• • The forward direction is supported: enter a and read b, c, l, perimeter, area. The reverse direction (enter an area and solve for a) does not always have a closed-form solution for the heptagram and the octagram.

If a real star-shaped part is curved, cut from a sheet with finite thickness, or 3D-printed with a fill pattern, the physical result will differ from the geometric area by a small amount. Add a small waste margin above the calculated area when ordering material.

According to Wikipedia (Star polygon), a regular star polygon is uniquely identified by the Schlafli symbol n/m, and its area equals the area of the inscribed regular n-gon plus the area of n identical isosceles triangles built on the polygon's sides.

If the star you need to compute is not a regular star polygon, the Irregular Polygon Area Calculator accepts a list of vertex coordinates and returns the area of the resulting non-convex outline.