Star Shape
Calculate star polygon area and perimeter from outer radius, inner radius, and point count. Free tool for pentagram, hexagram, heptagram, and octagram shapes.
About This Calculator
This free online star shape calculator computes the area and perimeter of any regular star polygon -- from the classic 5-pointed pentagram to hexagrams, heptagrams, octagrams, and beyond. Simply enter the outer radius (distance from center to the tips), inner radius (distance from center to the indentations), and the number of points, and get instant geometric results. Whether you are a student studying polygon geometry, a teacher preparing classroom materials, a graphic designer creating star logos and icons, or a crafter planning star-shaped decorations, this tool provides fast and accurate calculations for any star polygon.
A regular star polygon is a self-intersecting polygon that forms a symmetrical star shape. Unlike convex polygons, star polygons have edges that cross each other, creating the characteristic pointed appearance. The most famous star polygon is the pentagram (5-pointed star), which has deep mathematical connections to the golden ratio φ ≈ 1.618. In a pentagram, the ratio of outer to inner radius equals φ^2 ≈ 2.618, and the pentagram's intersections divide each segment in the golden ratio -- a proportion found in the Parthenon, the Mona Lisa, and countless examples in nature and architecture.
The area is calculated using the formula: Area = n x R x r x sin(pi/n), where n is the number of points, R is the outer radius, and r is the inner radius. The perimeter is computed as: P = 2n x sqrt(R^2 + r^2 - 2Rr cos(pi/n)). These formulas work for any regular star polygon with 3 or more points. For a 6-pointed star (hexagram), also known as the Star of David or hexagram, the star consists of two overlapping equilateral triangles. The 7-pointed heptagram is the first star polygon with two distinct forms: 3.5 and 2.3333333333333335. The 8-pointed octagram appears frequently in Islamic art and architecture, most notably on the flag of Azerbaijan.
This calculator handles edge cases gracefully -- zero or negative values return validation errors, and very large inputs are processed without overflow. Inputs use practical step increments for easy adjustment. Star polygons appear in flags around the world: the US flag contains 50 pentagrams, the flag of Israel features a hexagram, Jordan's flag depicts a heptagram, and Azerbaijan's flag includes an octagram. From geometry homework to national flags, star shapes are everywhere -- and now you can calculate their properties instantly.
Frequently Asked Questions
What is a regular star polygon?
A regular star polygon is a self-intersecting, equilateral, and equiangular polygon that forms a star shape. Common examples include the pentagram (5-pointed star), hexagram (6-pointed Star of David), heptagram (7-pointed star), and octagram (8-pointed star). Star polygons are denoted by their Schläfli symbol {p/q}, where p is the number of vertices and q is the density or starriness.
How do you calculate the area of a star polygon?
The area of a regular star polygon with n points, outer radius R, and inner radius r is calculated using the formula: Area = n x R x r x sin(pi/n). This formula accounts for the total area enclosed by the star shape including the central region and the triangular points.
How do you calculate the perimeter of a star polygon?
The perimeter of a regular star polygon is calculated as: P = 2n x sqrt(R^2 + r^2 - 2Rr cos(pi/n)), where n is the number of points, R is the outer radius, and r is the inner radius. This represents the total boundary length of the star traced along its outer edges.
What is the difference between outer radius and inner radius?
The outer radius is the distance from the center of the star to its outermost points (tips), while the inner radius is the distance from the center to the innermost indentations between points. The ratio of outer to inner radius determines how sharp or shallow the star points appear -- a larger ratio produces more pointed stars.
What is a pentagram and how is it related to the golden ratio?
A pentagram is a 5-pointed star polygon {5/2} with deep mathematical connections to the golden ratio φ ≈ 1.618. In a pentagram, the ratio of outer radius to inner radius equals φ^2 ≈ 2.618, and every diagonal intersection divides segments in the golden ratio. The pentagram appears in art, architecture, nature, and sacred geometry worldwide.
What is the minimum number of points for a star polygon?
The minimum number of points for a star polygon is 3 (a triangle cannot form a true star, but a 3-pointed star shape is possible). Most common star polygons have 5 or more points. A 4-pointed star (like a classic star shape) is technically a concave octagon. This calculator supports any number of points from 3 upward.
Is this star shape calculator free to use?
Yes, all calculators on Calculy are completely free to use with no registration or download required. Simply enter your values and get instant results. The calculator works on any device including desktop, tablet, and mobile browsers.