Where these designs come from
In 1976 Peter Viezbicke of the US National Bureau of Standards published NBS Technical Note 688, Yagi Antenna Design. It gives element lengths for six boom lengths that were optimised for maximum gain by measurement on a 400 MHz antenna range, not just by calculation. The tables are still reproduced in the ARRL Antenna Book, and they're a reliable starting point because each design was built and tested.
The original tables are for elements 0.0085 wavelengths thick, which is 17.5 mm at 145 MHz, with the reflector 0.2 λ behind the driven element. NBS gave correction curves for other element and boom diameters. The calculator applies those corrections, fitted to NBS's own worked examples, so expect them to be accurate to a few millimetres at VHF. NBS designs have fairly narrow bandwidth, so set the frequency to where you'll use the antenna most.
Element and boom corrections
Element diameter: thicker elements resonate at a shorter length, and the directors are more sensitive to this than the reflector. Corrections are most reliable for diameters between 0.002 and 0.02 wavelengths.
Metal boom: if the elements pass through a metal boom and touch it, the boom shortens them electrically, so each parasitic element needs to be longer. The correction grows with the boom's thickness, reaching about 0.005 λ for a boom 0.0085 λ thick. Elements mounted on insulators above the boom, or a wooden or plastic boom, need no correction.
The driven element
NBS didn't specify the driven element, because it's adjusted for the best match rather than for gain. The calculator gives a starting length for a resonant dipole. Long Yagis have a feed impedance well below 50 Ω, often 20 to 30 Ω, so they're usually fed with a folded dipole and 4:1 balun, a gamma match or a hairpin match. Make the driven element adjustable and set it for the lowest SWR with an antenna analyser, with the antenna clear of the ground and other metal.
Building tips
Measure element positions from the reflector to the centre of each element, and cut elements to within about a millimetre at UHF and a couple of millimetres at VHF. Keep the feedline at right angles to the elements for at least a quarter-wave behind the driven element, and fit a choke at the feedpoint. For horizontal polarisation, mount the elements horizontally. For FM and repeaters, which use vertical polarisation, turn the whole antenna on its side and keep the mast behind the reflector.
Questions
How long is a 3-element Yagi for 2m?
With the NBS 0.4 λ design at 144.3 MHz and 6 mm elements, the reflector is about 1.01 m, the director about 0.93 m and the boom 0.42 m. The driven element starts at about 1.0 m and is trimmed for the best match.
How much gain does a Yagi have?
About 7 dBd for a 3-element design on a 0.4 λ boom, rising to about 14 dBd for a 15-element design on a 4.2 λ boom. Gain grows with boom length, not just element count.
Do I need to correct for a metal boom?
Only if the elements pass through the boom and touch it. Elements on insulators, or a non-metal boom, need no correction.
Why is my Yagi's SWR high?
Long Yagis have a low feed impedance, often 20 to 30 Ω. Use a folded dipole, gamma or hairpin match, and adjust the driven element for the lowest SWR.