Unified inch threads are named by major diameter and TPI, and each size carries two standard series under ASME B1.1: coarse (UNC) and fine (UNF) — a 1/4″ bolt is 1/4-20 UNC or 1/4-28 UNF. Numbered machine-screw sizes hide their major diameter behind a gauge number, D = 0.060 + 0.013 × gauge, so a #10 is 0.190 in. One trap worth knowing: charts that list 1″-14 as UNF are wrong — B1.1’s UNF for 1″ is 12 TPI, and 1-14 is a UNS thread.
Everything geometric flows from the reciprocal, P = 1/TPI. The pitch diameter — where the flank carries load and where go/no-go gauging happens — sits 0.649519 × P below the major (¾ of the 0.866·P fundamental triangle height), and the classic flat-root external minor diameter sits 1.226869 × P below it. Those are basic dimensions; actual product threads carry class allowances and tolerances (2A/3A external) below basic.
TPI also drives the tap-drill lookup in its handbook-printed form, drill = major − 0.01299 × %TE / TPI: a 1/4-20 tapped at the standard 75% engagement wants 0.25 − 0.01299 × 75 ÷ 20 = 0.2013 in, and the nearest standard drill is the #7 gauge at 0.2010 in — 75.4% actual engagement, exactly what the printed charts list.