Radius of a circumscribed circle of a regular pentagon given its side, no trigonometry

Question

I'm going through early chapters of a geometry textbook and one of the exercises is:

Find the radius of the circumscribed circle of a regular pentagon given its side (10 cm).

Now, I could just use trigonometry, but I'm clearly supposed to rely on the straightedge & compass methods and polygon properties. I've been staring at it for a while and cannot get past calculating the angles which is of no use. The answer states "≈8,5 cm" and the approximation makes me even more confused about how am I supposed to approach this problem.

Any help is appreciated.

Answer 1

You should be able to do it by first proving that the ratio of the side length to distance between far vertices is $1:\varphi$, where $\varphi = \frac{1+\sqrt5}2$.

Then, from a vertex $A$, let $B$ be a far vertex with base $M$. Since the side length is $10$, then $AM=5$ and $AB = 10\varphi$. Consider the triangles $ABM$ and $AOM$, where $O$ is the centre of the circle. We have $AO=BO=r$ and $MO=h$, where $r$ and $h$ are the circumradius and apothem ("altitude"), respectively. Using Pythagoras, you have two equations in two unknowns.

$\hskip3cm$