Calculations for a regular concave polygon or n-gon. This can be formed from a convex regular polygon with an even number of vertices, at least eight. Every second vertex of this polygon is folded inwards. This creates a star polygon, where the vertices become shorter and blunter as the number of vertices increases. Just like the convex regular polygon, the concave polygon approaches a circle more and more closely as the number of vertices increases.
Enter edge length and number of vertices and choose the number of decimal places. Then click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units.
Formulas:
p = a * n
c = a * sin( π * 2/n ) / sin( π/n )
A = n * { a² / [ 4 * tan(π/n) ] - √( 4 * a² - c² ) / 4 * c/2 }
α = 180° - 1080° / n
β = 180° - 360° / n
pi:
π = 3.141592653589793...
Edge length, chord and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The point chord c is the chord between two adjacent convex vertices, i.e., between two adjacent points of the star. This is identical to the short diagonal d2 of the regular convex polygon. The perimeter of the concave and convex polygons is the same. The area of the concave polygon is calculated as the area of the convex polygon minus n times the area of the inward-folded vertices. These are isosceles triangles with side length a and base c. Between the two shapes lie n/2 congruent rhombuses with side length a and angle β.
The circumcircle radius of the concave polygon corresponds to the incircle radius of the convex polygon. The long diagonals, i.e., those spanning n/2 edges and opposite points, are the same for both polygons. The angle β of the inward-facing vertices of the concave polygon is the same as that of the outward-facing vertices of the convex polygon (there α), but in the opposite direction.
This concave polygon is point-symmetric about its center and axis-symmetric about the n/2 axes through the vertices and n/2 axes through the inward-pointing corners. It is rotationally symmetric at an angle of 720°/n.