Star shape calculator

A regular pointed star is the familiar badge, flag, and ornament outline whose tips sit on one circle and whose valleys sit on a smaller concentric circle. Designers, teachers, laser-cut operators, and game artists usually know how many tips they want and the two radii of that outline; they do not start from a Schläfli density. This star shape calculator takes those three numbers — an integer point count of three or more, a positive outer radius, and a strictly smaller positive inner radius — and returns the filled area and the boundary perimeter of the isotoxal 2n-gon that traces tip-valley-tip all the way around.

What a regular pointed star is and how it differs from {n/d}

A regular pointed star, sometimes called an isotoxal star or a star polygon with independent inner and outer radii, is built from 2n vertices that alternate between two concentric circles. The n outer vertices sit at distance R from the center and become the visible tips. The n inner vertices sit at distance r from the center, with r strictly less than R, and become the valleys between tips.

Consecutive vertices are equally spaced in angle: the step from a tip to the next valley is π over n radians, so a full circuit visits 2n vertices and returns to the start. The filled region is a simple 2n-sided polygon, not the self-intersecting Schläfli star {n/d} that you get by connecting every d-th vertex on a single circle. That distinction matters for search intent.

People who type star shape calculator, pointed star area, or star perimeter from outer and inner radius already have both radii from a drawing, a die, or a CAD sketch. They want the paint area and the cut length of that outline. They do not have a density d, and they would be misled by a {n/d} area that derives an inner radius from trigonometry instead of taking the radius they measured.

Three-pointed, four-pointed, five-pointed, and six-pointed stars are all valid here as long as n is an integer of at least three. A two-pointed figure is not a star in this model and is rejected. When r approaches R the points flatten toward a regular 2n-gon; when r is much smaller than R the tips become long and sharp.

This calculator reports the exact closed-form area and perimeter of that family so you can size badges, ornaments, collision hulls, and homework figures without tracing vertices by hand.

Where a star shape calculator earns its keep

Star-shaped badges, medals, stickers, and ornaments are specified by tip count and two radii in almost every design file, yet area and perimeter are still derived by tracing vertices or by eyeballing a bitmap. A deterministic star shape calculator turns those three numbers into a paint area for coating estimates and a cut length for laser, vinyl, or CNC paths.

Teachers and exam writers who ask for the area of a regular five-pointed star given outer and inner radius need an oracle that uses the sector formula rather than a pentagram identity that ignores the given inner radius. Craft and wrapping workflows that tile stars on paper or fabric convert the perimeter into ribbon or stitching length and the area into material yield.

Landscape and event planners who lay star-shaped beds or stages convert a measured outer radius and a chosen inner radius into turf area and border length without opening a full CAD suite. Keep requests small, pin the rounded area and perimeter in tests, and you get reproducible star metrics for homework, LMS plugins, and edge workers without standing up a geometry kernel of your own.