Calculations at a fourstar or tetragram. At an angle of α<90° this is a concave, equilateral octagon. This star with four spikes is formed by attaching four isosceles triangles with legs length a and base length b to a square with edge length b. Enter the edge length a and one angles α or β, choose the number of decimal places and click Calculate. Please enter angles in degrees, here you can convert angle units. The angle α must be smaller than 180°. If the angle α is 90 degrees, then the fourstar is not a star, but a square with side length 2a. With a smaller angle, the fourstar is concave; with a larger angle, it is convex and therefore actually no longer a star.
Formulas:
β = 90° + α
b = √ 2 * a² * ( 1 - cos(α) )
i = √( 4 * a² - b² ) / 4
d = 2 * i + b
l = √ 2 * a² * ( 1 - cos(β) )
p = 8 * a
A = 2 * i * b + b²
Lengths, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The fourstar is the second simplest form of a polygram, consisting of a regular polygon and a number of isosceles triangles equal to the number of sides of the polygon, in this case four. The fourstar is axially symmetrical to each of both bisectors of the opposite triangles and also to both diagonals of the central square. So it has four axes of symmetry. It is also rotationally symmetrical at angles of 90 degrees and multiples thereof. The fourstar is point symmetrical to its central point. If the vertices of a concave fourstar are connected, a square with side length l is obtained. On this square, vertices with interior angle α can be added. This new fourstar is geometrically similar to the original, but rotated by 45 degrees.
Four-pronged stars are used in decoration and as symbols for astronomical stars. Both are often used together with five-pointed stars, which then are often regular pentagrams. There is no regular shape for tetragrams, only for stars with more than four prongs.
