Star Shape Calculator
Our free linear algebra calculator solves star shape problems. Get worked examples, visual aids, and downloadable results.
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.