The formulas
A quarter-wave vertical is half a dipole, with the ground or a set of radials acting as the other half. The vertical element is a quarter of a wavelength, shortened a little for the end effect:
Elevated radial (metres) = 75 ÷ f (MHz) (246 ÷ f in feet)
The shortening factor of 0.95 gives the 71.3 figure. Elevated radials are cut to a full free-space quarter-wave, which is about 5% longer than the vertical, because they're tuned conductors in their own right.
Elevated radials
With the feedpoint a few metres up, two to four tuned radials make a ground plane. With horizontal radials the feed impedance is about 35 Ω, an SWR of about 1.4:1 on 50 Ω coax. Sloping the radials down at about 45° raises it to about 50 Ω. The radials carry high RF voltage at their ends, so keep them out of reach. Elevated radials must be the right length, so trim them along with the vertical. Fit a choke on the coax at the feedpoint to stop it acting as an extra radial.
Ground-mounted radials
Radials lying on the ground or buried a few centimetres down are detuned by the soil, so their length isn't critical. What matters is how many there are, because they collect the return current that would otherwise flow through lossy soil. A single ground stake is a poor substitute. The table follows the classic study by Brown, Lewis and Epstein of RCA, as reproduced in the ARRL Antenna Book. A handful of short radials is better than one or two long ones, and 16 radials of 0.1 λ loses only about 3 dB against a near-perfect system. The feed impedance is about 35 Ω with a large radial field, rising to about 50 Ω with 16 radials as ground loss adds to it. That makes a sparse radial system look like a good match, even though it's wasting power in the soil.
Trimming the vertical
Make the vertical slightly long and trim it with an antenna analyser, watching the frequency of lowest SWR. Shortening the vertical moves resonance up, and lengthening it moves resonance down. With elevated radials, trim the radials by the same proportion if the SWR minimum is in the right place but not low enough.
Questions
How long is a quarter-wave vertical for 20m?
About 5.03 m at 14.175 MHz, with elevated radials of about 5.29 m each.
How long should the radials be?
Elevated radials should be a quarter-wave, about 75 ÷ f metres. Ground-mounted radials can be any length from about 0.1 λ upward. More radials matter more than longer ones.
How many radials do I need?
Two to four tuned radials are enough for an elevated ground plane. On the ground, 16 is a good start and 30 to 60 gets you within a dB or so of an ideal system.
Why slope the radials down?
It raises the feed impedance from about 35 Ω towards 50 Ω, for a better match to coax, and makes the antenna easier to support.