The regular pentagon is constructible, and the reason is that 5 is a Fermat prime: it is 2 to the power 2 to the power 1, plus one. Under the Gauss-Wantzel condition, which requires a side count equal to a power of 2 times a product of distinct Fermat primes, five qualifies directly. So do 10, 20 and 40 by repeated bisection, and 15 by combining the pentagon with the triangle, which Euclid does at Book IV Proposition 16.
Euclid's own route is indirect. He must first cut a straight line in extreme and mean ratio, so that the whole is to the greater part as the greater part is to the lesser. That is Proposition 11 of Book II, restated at Book VI Proposition 30, and it is the construction of what is now called the golden ratio, phi, equal to one plus sqrt 5 all divided by two, about 1.618. From it he builds an isosceles triangle whose base angles are each double its apex angle, the 72-72-36 triangle, inscribes that in a circle, and bisects to reach the pentagon at Book IV Proposition 11.
Shorter methods exist. The construction usually attributed to H. W. Richmond, published in 1893, inscribes a pentagon in a given circle in a handful of steps: take two perpendicular radii, bisect one of them, use that midpoint to bisect an angle, drop a perpendicular, and the resulting chord is the pentagon's side.
The ratio is not decorative. In a regular pentagon the diagonal divided by the side is exactly phi. Draw all five diagonals and they cut one another in the golden ratio, producing a pentagram whose interior is a smaller inverted pentagon, within which the same thing happens again, without end. That endless nesting is itself a proof that phi is irrational: if it were a ratio of whole numbers the regress would have to terminate, and it does not.