The following illustrates construction of a pentagon in a circle using a compass
and straight edge. Although the circle can be of arbitrary units, the length
references presume a unit circle.

Start by drawing a unit circle with a compass keeping track of the center.
Use a straight edge to draw a line through the center.

Strike a perpendicular bisector of the horizontal diameter.

Bisect the right horizontal radius and connect a line from the point
of bisection to the top of the circle.

Use the length of the line just drawn above to strike an arc back down to the
horizontal diameter. If the circle is a unit circle, then this radius has a length of
$\frac{\sqrt{5}}{2}=\mathrm{1.11803...}$

Connect the point at the top of the circle and the point on the horizontal diameter.
If the circle is a unit circle, then this new line has a length of
$\sqrt{\frac{5-\sqrt{5}}{2}}=1.17557$

A pentagon inscribed in a unit circle has a side with a length of $\sqrt{\frac{5-\sqrt{5}}{2}}=1.17557$.
Use this side to strike a new arc intersecting the circle on both sides.

Three of the vertices of the pentagon are now defined.

Use a compass to complete the pentagon by striking arcs along the circle from
the points already determined to find the remaining two pentagon vertices. This
construction is sensitive to small errors in construction.
Try striking arcs from both directions along the circle and average out the
errors.

Using the pentagon to produce a pentagram, one can calculate the Fibonacci
ratio phi $\frac{1+\sqrt{5}}{2}$
by dividing the length of a pentagram crossbeam (length to point A) by the star-tip to
star shoulder distance (length b).

Phi

The ratio of A/B is equal to the Fibonacci ratio phi or
$\frac{1+\sqrt{5}}{2}$.