How this calculator works
A regular star alternates between outer points and inner valleys around a common centre. Every point is the same distance from the centre and every valley is the same distance, which gives the outline 2n vertices. The calculator treats the star as 2n identical triangles around the centre. Each triangle has two sides equal to the outer and inner radii and an included angle of 180/n degrees, so its area is half the product of the radii times the sine of that angle. Doubling for both halves of the star gives n times the radii product times sin(180/n). The edge length, perimeter and corner angles follow from the same triangle.
Formula
For n points, outer radius R and inner radius r: area = nRr sin(π/n); edge s = √(R² + r² - 2Rr cos(π/n)); perimeter = 2ns; point angle = 2 arccos((R² + s² - r²)/(2Rs)); valley angle = 2 arccos((r² + s² - R²)/(2rs)).
Worked example
A five-pointed star with outer radius 5 m and inner radius 2 m has area 5 × 5 × 2 × sin 36° ≈ 29.3893 m². Its edge length is about 3.5805 m, its perimeter about 35.8046 m and its point angle about 38.33°.
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Frequently asked questions
What inner radius gives a classic pentagram?
A regular pentagram uses an inner radius of about 0.382 times the outer radius, because the valleys sit at a golden-ratio distance from the centre. Entering inner radius 0.382 times the outer radius reproduces it.
Why does the star have twice as many vertices as points?
Each point is followed by a valley, so a five-pointed star has five outer vertices and five inner vertices, for ten in total around the outline.
Can the inner radius equal the outer radius?
Then there is no inward step and the outline is a regular polygon rather than a star. The calculator requires the inner radius to be strictly smaller.
Is the perimeter the outline or the five crossing lines of a pentagram?
It is the outline, the zigzag boundary through the points and valleys. A drawn pentagram made of five crossing lines has a different total length.