Star Shape Calculator
Our free linear algebra calculator solves star shape problems. Get worked examples, visual aids, and downloadable results.
Calculator
Adjust values & calculateFormula
Where n is the number of star points, R is the outer radius (to tips), and r is the inner radius (to valleys). The star consists of 2n alternating vertices on two concentric circles, creating n pointed tips.
Last reviewed: December 2025
Worked Examples
Example 1: Five-Pointed Star
Example 2: Six-Pointed Star (Star of David)
Background & Theory
The Star Shape Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Star Shape Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Sources & References
Formula
Area = n * R * r * sin(pi/n)
Where n is the number of star points, R is the outer radius (to tips), and r is the inner radius (to valleys). The star consists of 2n alternating vertices on two concentric circles, creating n pointed tips.
Worked Examples
Example 1: Five-Pointed Star
Problem: Calculate the area and perimeter of a 5-pointed star with outer radius 10 and inner radius 4.
Solution: Number of points n = 5\nArea = n * R * r * sin(pi/n) = 5 * 10 * 4 * sin(pi/5)\n= 5 * 10 * 4 * 0.5878 = 117.56 sq units\nSegment length = sqrt(100 + 16 - 80*cos(pi/5))\n= sqrt(116 - 80*0.809) = sqrt(51.28) = 7.161\nPerimeter = 2 * 5 * 7.161 = 71.61 units
Result: Area = 117.56 sq units | Perimeter = 71.61 units
Example 2: Six-Pointed Star (Star of David)
Problem: Calculate properties of a 6-pointed star with outer radius 8 and inner radius 4.
Solution: Number of points n = 6\nArea = 6 * 8 * 4 * sin(pi/6) = 6 * 8 * 4 * 0.5 = 96 sq units\nSegment length = sqrt(64 + 16 - 64*cos(pi/6))\n= sqrt(80 - 64*0.866) = sqrt(24.58) = 4.958\nPerimeter = 2 * 6 * 4.958 = 59.49 units\nTip angle = 180 - 360/6 = 120 degrees
Result: Area = 96 sq units | Perimeter = 59.49 units | Tip angle = 120 degrees
Frequently Asked Questions
What is a star polygon and how is it defined?
A star polygon (or star shape) is a non-convex polygon that has the appearance of a star. It is defined by two concentric circles: an outer circle where the points (tips) of the star lie, and an inner circle where the indentations (valleys) between points lie. The star is formed by alternating between vertices on the outer and inner circles. The number of points, outer radius, and inner radius completely determine the star shape. Regular star polygons have equal angles at all points and equal side lengths. The most familiar example is the five-pointed star (pentagram), commonly seen on flags and as a rating symbol.
How do you calculate the area of a star shape?
The area of a star shape with n points, outer radius R, and inner radius r is calculated by dividing the star into 2n congruent triangles. Each triangle has two sides equal to R and r with an included angle of pi/n radians. The total area equals n times R times r times sin(pi/n). This formula works because the star can be decomposed into n kite-shaped quadrilaterals, each consisting of two triangles. Alternatively, you can compute the area as the outer polygon area minus the areas of the n triangular notches cut from it. Both methods give the same result, confirming the formula validity.
How does the ratio of inner to outer radius affect the star shape?
The inner-to-outer radius ratio dramatically changes the star appearance. When the ratio is close to 1 (inner radius nearly equals outer radius), the star looks almost like a regular polygon with very shallow indentations. When the ratio is close to 0, the star has extremely long, thin points. For a regular five-pointed star (like on the US flag), the golden ratio determines the ideal proportions with an inner-to-outer ratio of about 0.382. As a general rule, ratios between 0.3 and 0.5 produce visually pleasing stars. The ratio also affects the area: lower ratios mean less area relative to the circumscribed circle.
What is the angle at each point of a star?
The angle at each tip of a regular star polygon depends on the number of points and the radius ratio. For a star with n points inscribed in a circle, the general tip angle for a regular star polygon (where vertices connect every second point of a regular 2n-gon) is 180 minus 360/n degrees. A five-pointed star has tip angles of 180 - 72 = 36 degrees. A six-pointed star (Star of David) has 60-degree tips. As the number of points increases, each tip angle approaches 180 degrees and the star begins to resemble a circle. These angles are important in design and manufacturing for creating precise star patterns.
How do you calculate the perimeter of a star?
The perimeter of a star shape is the total length of its boundary, consisting of 2n line segments (where n is the number of points). Each segment connects an outer vertex to an adjacent inner vertex. Using the law of cosines, each segment length equals the square root of (R squared plus r squared minus 2Rr times cos(pi/n)), where R is the outer radius and r is the inner radius. The total perimeter is 2n times this segment length. For a five-pointed star with R=10 and r=4, each segment is approximately 7.25 units, giving a perimeter of about 72.5 units. The perimeter increases with both the number of points and the radius values.
What is the Star of David and its geometric properties?
The Star of David (hexagram) is a six-pointed star formed by two overlapping equilateral triangles. It can also be viewed as connecting every other vertex of a regular 12-gon. The ratio of inner to outer radius for a regular hexagram is exactly 1/sqrt(3), approximately 0.577. Its total area is 2/3 of the circumscribed hexagon area. The Star of David has 6-fold rotational symmetry and 6 lines of mirror symmetry. It appears in religious symbolism, architecture, chemistry (benzene ring representations), and mathematics (as the simplest compound star polygon). Each of its six triangular points has a 60-degree angle.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy