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.
Best Answer
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.
$\hskip3cm$