Star Shape Calculator

A regular star shape alternates evenly spaced outer tips and inner valleys around one center. Three values completely determine this symmetric outline: the number of points, the outer radius, and the inner radius. The outer radius runs from the center to each tip. The inner radius runs from the center to each valley. Together they create twice as many boundary vertices as the number of visible points. The calculator treats the outline as a simple, non-self-intersecting star polygon. It divides the shape into identical triangles meeting at the center. The angle between each outer vertex and the next inner vertex is pi divided by the number of points. Adding the areas of all 2n triangles simplifies to area = n times outer radius times inner radius times sin(pi/n). This method is exact for any regular alternating-radius star. Each boundary edge joins an outer vertex to an adjacent inner vertex. The law of cosines gives its length from the two radii and their included angle: edge squared equals outer radius squared plus inner radius squared minus twice their product times cos(pi/n). Since the outline contains two edges per point, perimeter equals 2n times that edge length. Results use the same unit as the radii, while area uses the corresponding square unit. This geometry is useful for signs, badges, quilting patterns, laser-cut ornaments, logos, paving layouts, and classroom exercises. Designers can vary the inner radius to control how sharp or broad the points appear without changing the outside diameter. A small inner radius creates deep narrow valleys; a value close to the outer radius makes a shape that approaches a regular polygon with shallow notches. The point count must be a whole number of at least three. Both radii must be positive, and the inner radius must be smaller than the outer radius. This calculator does not model a self-crossing pentagram made from one continuous set of diagonals; it measures the filled star outline formed by alternating outer and inner vertices. Before manufacturing a design, keep enough precision in the displayed dimensions for the tolerances of your material and process.