Heptagram, point angle and stroke length

Problem

A $7$-pointed star (heptagram $\{7/3\}$: each stroke joins a vertex to the third one after it, among $7$ equally spaced vertices) is inscribed in a circle of radius $5$. a) Find the angle at one of the star’s points. b) Using the chord theorem, find the length of one pen-stroke used to draw the star.

(Figure omitted: the $7$ vertices are equally spaced by $\tfrac{360^\circ}{7}$ on the circle.)

Solution

The $7$ vertices are $\dfrac{360^\circ}{7}$ apart. Each stroke is the chord subtending an arc of $3\cdot\dfrac{360^\circ}{7}$. a) The point angle is the vertex angle of the star polygon $\{7/3\}$: $\;\theta=\dfrac{(7-2\cdot 3)\cdot 180^\circ}{7}=\dfrac{180^\circ}{7}\approx25{,}71^\circ$. b) By the chord theorem the stroke length is $L=2R\sin\!\left(\dfrac{3\pi}{7}\right)=10\sin\!\left(\dfrac{3\pi}{7}\right)\approx9{,}749$. $\boxed{\,\theta=\tfrac{180^\circ}{7}\approx25{,}71^\circ,\qquad L=10\sin\tfrac{3\pi}{7}\approx9{,}749\,}$