Threestar Calculator

Name: Threestar - Geometry Calculator
Author: Jürgen Kummer

Calculations at a threestar or three-pronged star. At an angle of α<60° this is a concave, equilateral hexagon. This star with three spikes is formed by attaching three isosceles triangles with legs length a and base length b to an equilateral triangle 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 of course be smaller than 180°. If the angle α is 60 degrees, then the threestar is not a star, but an equilateral triangle with side length 2a. With a smaller angle, the threestar is concave; with a larger angle, it is convex and therefore actually no longer a star.

Formulas:

β = 120 ° + α

b = \sqrt{2 \cdot a^{2} \cdot (1 - \cos (α))}

i = \sqrt{\frac{4 a^{2} - b^{2}}{4}}

l = \sqrt{2 \cdot a^{2} \cdot (1 - \cos (β))}

h = \frac{\sqrt{3}}{2} \cdot l

p = 6 \cdot a

A = \frac{3}{2} \cdot i \cdot b + \frac{\sqrt{3}}{4} \cdot b^{2}

Lengths, heights and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The threestar is the 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 three. The threestar is axially symmetrical to each of the three bisectors through the vertex of a triangle and its base. It is also rotationally symmetrical at angles of 120 degrees and multiples thereof. This shape is not point symmetrical. If the vertices of a concave threestar are connected, an equilateral triangle with side length l is obtained. On this triangle, vertices with interior angle α can be added. This new threestar is geometrically similar to the original, but rotated by 60 degrees.
Threestars are frequently found as graphic symbols. This shape is probably most well-known from the Mercedes logo, the Mercedes star, where the threestar sits within a circle or annulus.

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