How would I find the lengths of the sides of a regular pentagram if given the perimeter of the circle that the pentagram is inscribed to?

Question

For a Turtle program in Python that I am working on, I will need to draw out a star. This star is a regular pentagram, meaning that each of the sides are of equal length. As well, the pentagram is inscribed into a circle with the perimeter of 15.625. However, I don't know the lengths of the sides of the star, and I am not sure how to use the circle's perimeter to get the sides of the star. So, what would the length of the sides of the star be, and what steps would I take to get it?

Answer 1

The chord that is one side of the pentagram subtends an angle of $\frac {4\pi}5$. If you draw the radii to the endpoints of the chord and a radius that bisects the chord you get two right triangles. The hypotenuse is the radius of the circle, $\frac {15.625}{2\pi}$. The angle of the triangle at the center is $\frac {2\pi}5$. The side opposite the angle at the center, which is half the chord, is $\frac {15.625}{2\pi}\sin \left(\frac {2\pi}5\right).$ The chord is twice this, $\frac {15.625}{\pi}\sin \left(\frac {2\pi}5\right)\approx 4.73.$

Answer 2

The circumradius $r$ of a regular $\{n/d\}$-gon with unit sides, i.e. a regular polygon with $n$ vertices and being wrapped up $d$ times, equals $$r = \frac 1 {2\ \sin(\pi\frac dn)}$$ Your case would be $n=5$ and $d=2$.

The inradius $\rho$ similarily would be obtained as $$\rho = \frac 1 {2\ \tan(\pi\frac dn)}$$

Cf. https://bendwavy.org/klitzing/dimensions/polygons.htm

--- rk

Answer 3

Easy, between adjacent vertices there's 72 degrees. Between any two connected vertices, that makes 144 degrees. $\frac{180-144}{2} =18^{\circ}$ for every line from a vertex,to a line straight to the center(radius of circle). $r = \frac{C}{2pi}$ and $2r\cos18 =\frac{C}{\pi}\cos18$ is the length of the line connecting the vertices.