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In screw thread geometry, lead $L$ is the axial advance in one complete relative revolution of the screw and its matching nut. The term screw lead distinguishes this geometric quantity from the chemical element. For $n$ thread starts with axial thread pitch $p$,A single-start thread has $L=p$, whereas a three-start thread with $p=2\mathrm{mm}$ has $L=6\mathrm{mm}$.^[ref]

Assume the screw is restrained axially and the nut is guided against rotation. If the relative angular displacement is $\theta$, consistent signs giveDifferentiating gives $v=L\omega /(2\pi )$, where $\omega$ is relative angular velocity. Equivalently, rotation at $f$ revolutions per second gives axial speed $v=Lf$. With $L=6\mathrm{mm}$ per revolution and $f=5$ revolutions per second, the ideal speed is $30\mathrm{mm}{\mathrm{s}}^{-1}$; a $90^{\circ }$ turn advances $1.5\mathrm{mm}$.

Unwrapping a cylindrical reference surface of radius $R$ gives a right triangle with circumferential run $2\pi R$ and rise $L$. Its helix inclination $\alpha$, measured relative to a plane perpendicular to the axis, therefore obeysThis is a geometric statement about the chosen reference radius. It does not by itself determine the drive torque, because contact geometry and friction also matter.

A commanded turn is a prediction from geometry, rather than a guarantee of measured travel. Backlash can delay reversal motion, and mechanical compliance changes displacement under load. These effects remain relevant even when the nominal screw lead is known exactly.

A screw thread is a ridge following a helix around an external or internal surface. Cylindrical threads repeat around a cylinder; tapered forms can instead follow a conical spiral. The thread's shape and dimensions determine how it mates with another threaded component.^[ref]

A simple geometric model represents one reference helix on a cylindrical thread bywhere $R$ is the chosen radius, $\theta$ is an angle measured in radians, and $L$ is the screw lead. Increasing $\theta$ by $2\pi$ returns to the same angular position while increasing the axial coordinate by $L$. This curve describes a reference path, rather than the full three-dimensional ridge profile.

A thread start is a distinct continuous helical ridge. With $n$ equally spaced starts and thread pitch $p$, the advance of one start per turn is $L=np$. The same axial spacing can therefore coexist with different advances per turn.^[ref]

When a matching nut is guided against rotation, relative rotation can produce axial movement in a lead screw. Clearance can produce backlash, while deformation of loaded components contributes mechanical compliance. Thus ideal thread geometry predicts a motion ratio, while the actual load path also depends on contact and deformation.

Thread pitch, usually $p$, is the axial distance between corresponding points on adjacent turns of a screw thread. It is measured parallel to the thread axis, rather than along the sloping helix. For a thread with several thread starts, this spacing differs from the screw lead, which measures advance per complete relative turn.^[ref]

The distinction between points and intervals is useful in a measurement. Suppose ten successive ridge crests along an axial line span $13.5\mathrm{mm}$ from the first crest to the tenth. They delimit nine pitch intervals, soDividing by ten would give $1.35\mathrm{mm}$ and would count the endpoints incorrectly. Measuring several intervals also distributes a fixed endpoint-position error over the number of intervals, provided the thread spacing is sufficiently uniform for that average to be meaningful.

A specification in threads per inch expresses the reciprocal spacing. For $q$ threads per inch,Thus $20$ threads per inch correspond to $p=1.27\mathrm{mm}$, using the exact inch-to-millimetre conversion.

For a two-start example with $p=1.5\mathrm{mm}$, the screw lead is $L=2p=3\mathrm{mm}$. A quarter-turn then advances a suitably constrained nut by $0.75\mathrm{mm}$ in the ideal geometry. A single-start thread with the same pitch advances only $0.375\mathrm{mm}$ in a quarter-turn. Equal pitch therefore does not establish equal motion per revolution.