The chart is hiding one formula
The tap drill chart taped inside every shop toolbox looks like a long list of arbitrary pairings — M10×1.5 takes 8.5 mm, 1/4-20 takes a #7, 3/8-16 takes 5/16 — but the whole document is a single formula evaluated over and over: drill = major − 1.29904 × pitch × (% ÷ 100). The major diameter and pitch describe the thread; the percentage describes how much of the full theoretical thread height you want the tapped hole to engage; and the odd-looking constant 1.29904 is just (3/4)√3, a fixed property of the 60-degree thread form that both ISO metric and Unified inch threads share.
Once you see the formula, the chart stops being a lookup ritual and becomes a dial. Drill exactly at the thread’s minor diameter and the tap must carve a 100% thread; drill larger and the crests of the internal thread are born slightly truncated — less material to cut, less torque on the tap, a hole that threads more easily. The percentage of full thread is the knob that trades thread height against tapping effort, and the entire chart is that knob frozen at one convenient setting.
Percentage of full thread: what 75% buys and what 100% costs
The frozen setting is 75%, and it is not a compromise so much as a discovery. Tapping tests reported in Machinery’s Handbook show that pushing engagement above roughly 75% adds very little joint strength — threads strip through roughly the same failure surface either way — while the torque required to drive the tap climbs steeply and the odds of snapping it in the hole climb with it. A 100% thread is only marginally stronger than a 75% thread and several times harder to tap. Published charts standardized on 75% because it keeps nearly all the strength and loses most of the grief.
The dial turns in both directions with intent. In hard or gummy materials — stainless, tool steel, titanium — and in small taps or deep holes where chip packing multiplies torque, dropping to 60–65% engagement is standard practice and costs little. Below about 50% the thread genuinely gets too shallow and can strip under ordinary preload, which is why calculators clamp the range there. And when a joint in soft material needs more strength, the effective move is not a tighter drill but a deeper hole: added length of engagement — up to about one nominal diameter — buys real strength where added percentage does not.
The metric shortcut: major minus pitch
Metric threads carry a shortcut so clean it feels like a coincidence: the tap drill is the major diameter minus the pitch. M10×1.5 drills at 10 − 1.5 = 8.5 mm; M8×1.25 at 6.75 — call it 6.8; M6×1.0 at exactly 5.0. It is not a coincidence. Run "major minus pitch" through the engagement formula backwards and it corresponds to a percentage of exactly 100 ÷ 1.29904 = 76.98% — a whisker above the 75% target, safely inside the comfortable range for ordinary work. The shop rule is the master formula wearing overalls.
The same identity explains the two printed forms handbooks use. For metric threads: drill = major − pitch × % ÷ 76.98, which collapses to "major minus pitch" at 76.98%. For inch threads, where pitch is 1/TPI: drill = major − 0.01299 × % ÷ TPI, the 0.01299 being 1.29904 ÷ 100. Same constant, same 60-degree geometry, different unit clothing — one more reminder that metric and Unified threads are near-identical triangles that simply disagree about how to write down their spacing.
Number, letter, fractional: why inch drills come in three alphabets
Metric shops answer the formula with a drill index in clean 0.1 mm steps. Inch shops answer it with three interleaved families: fractional drills in exact 64ths of an inch, number (wire-gauge) drills #80 through #1, and letter drills A through Z. The gauges exist because 64ths are too coarse where holes are small — between 1/16″ and 5/64″ yawns a 0.0156 in gap into which five number drills (#52–#48) fit — and tap drills demand exactly that density. The numbering even runs backwards, a habit inherited from wire gauge, where more drawing operations meant thinner wire: #80 is the tiny one at 0.0135 in, #1 the big one at 0.228 in, and the letters carry on from A (0.234 in) to Z (0.413 in).
The gauge diameters are standardized decimal equivalents published in ANSI/ASME B94.11M rather than any arithmetic series, which is why tap drill callouts sound like bingo — #29 for 8-32, F for 5/16-18, U for 7/16-14. A few curiosities fall out of the system: letter E is exactly 0.250 in, a duplicate of the 1/4″ fractional, because the letter run was defined continuously. None of it changes the math; the alphabets are simply the set of standard answers available when the formula hands you a theoretical diameter and the index must supply a real drill.
One M10 and one 1/4-20, drilled and tapped
The running example, worked in both systems. Metric first: M10×1.5 at the 75% target gives drill = 10 − 1.29904 × 1.5 × 0.75 = 8.539 mm. No such drill exists; the nearest standard is 8.5 mm, and feeding it back through the engagement relation — (10 − 8.5) × 76.98 ÷ 1.5 — shows the hole it makes engages 76.98% of full thread. Slightly tighter than target, entirely healthy, and precisely the "major minus pitch" answer. Now suppose the part is stainless and the tap is precious: aim at 65% instead, get a theoretical 8.733 mm, pick the 8.7 mm drill, and land at 66.7% actual — an easy-driving hole that gives up almost nothing in strength.
Inch next: 1/4-20 UNC at 75% wants 0.25 − 0.01299 × 75 ÷ 20 = 0.2013 in. The index answers with the #7 number drill at 0.2010 in, which works out to 75.4% actual engagement — the pairing every printed chart lists. Worth noticing what the drill is not: the bolt’s external minor diameter for 1/4-20 is 0.1887 in, well below the tap drill, because the hole sizes an internal thread with its engagement allowance, not the external root. And if the #7 has walked off, the metric side of the index covers it — a 5.1 mm drill is 0.2008 in, nearly the same hole. The tap drill size calculator runs all of this in one pass: theoretical diameter at any engagement from 50 to 85%, the nearest standard drill from the metric, fractional, number, and letter tables, the engagement that drill actually delivers, and the closest cross-system substitute.
The rule in one pass
Recap the pair: M10×1.5 wanted 8.539 mm at 75%, took the 8.5 mm drill, and tapped at 76.98% — the metric shortcut of major minus pitch made rigorous; 1/4-20 wanted 0.2013 in, took #7 at 0.2010, and tapped at 75.4% — with 8.7 mm and 66.7% waiting as the hard-material variant. That is the entire discipline: one formula, drill = major − 1.29904 × pitch × %/100, evaluated at a percentage you choose on purpose — 75% as the default the charts assume, 60–65% when the tap’s survival is worth more than phantom strength, never below 50%, and rarely above 75% because the strength is not there to collect. The chart on the toolbox lid is the answer key; the formula is the understanding that lets you leave the key behind.