§1What a taper is for
A taper joins a tool or arbor to a spindle so that it runs true, centres itself, transmits torque and can still be changed quickly. A matched pair of cones does all four at once.
Push a tapered shank into a matching tapered socket and it centres automatically — the cones seat on the same axis, with none of the clearance a cylindrical fit would need — so the tool runs with little runout. That same wedging can grip hard enough to drive the tool, or, if the angle is steeper, can be made to release cleanly for fast changes. The single design variable that decides between those two behaviours is the taper’s angle: shallow tapers grip and hold themselves, steep ones let go. The rest of this page is really about that one choice and the standard systems built around it.
Contents§2Defining a taper
A taper is specified by how fast its diameter changes along its length — the taper ratio, or equivalently the taper per foot — from which the included angle follows.
The taper ratio is the change in diameter (D − d) divided by the length L over which it changes; a ratio of 1:20 means the diameter shrinks 1 mm for every 20 mm of length. The older imperial measure, taper per foot, quotes the same thing as the diameter change over a foot of length. From either, the half-angle — the angle each side makes with the axis — is the arctangent of half the taper ratio, and it is this angle, small or large, that governs everything (§4). A taper is gentle when this angle is a degree or two, steep when it is eight or more.
Contents§3Self-holding tapers
A shallow taper grips so firmly when seated that friction alone holds and drives the tool — a self-holding taper, of which the Morse taper is the classic.
The Morse taper is about ⅝ inch per foot, a taper ratio of 0.0521, giving a half-angle of arctan(0.0521/2) = 1.49° (an included angle of just 2.98°). That shallow wedge, tapped home, locks by friction tightly enough to drive a drill under full cutting load with no other retention — and it is released only by a transverse slot and drift key that break the grip. Self-holding tapers like the Morse are ideal for drills, reamers and lathe centres, where a firm, self-driving, self-centring hold matters more than a fast change. Their virtue and their vice are the same: they hold themselves, so they also resist quick release.
§4The self-locking angle
Whether a taper holds itself is not a matter of degree but a threshold: a taper self-locks if its half-angle is smaller than the friction angle of the surfaces. Cross that line and the behaviour flips.
The friction angle is the arctangent of the coefficient of friction; for typical dry steel-on-steel, with μ ≈ 0.15, it is arctan(0.15) ≈ 8.5°. A taper whose half-angle is less than this cannot push itself back out — the friction holding it exceeds the axial component trying to eject it — so it self-locks. The Morse taper’s 1.49° half-angle is far below 8.5°, which is why it grips so tenaciously. A taper whose half-angle approaches or exceeds the friction angle will slide back out on its own and must be held in by other means. This single inequality — half-angle versus friction angle — is the dividing line between the two families of taper, and it explains at a stroke why one grips and the other must be clamped.
Contents§5Self-releasing steep tapers
Milling spindles need the opposite of a Morse taper: a taper that centres accurately but lets go at once, held in only by a drawbar. The steep 7:24 taper does exactly this.
The standard milling-spindle taper is 7:24 — a taper ratio of 0.292, far steeper than a Morse. Its half-angle is arctan(0.292/2) = 8.30° (included 16.59°), close to the friction-angle limit, so it does not self-lock: the moment the drawbar releases it, the tool frees itself for a fast change. The taper still centres the tool precisely; retention is the drawbar’s job, not the taper’s. This is the whole point of a steep, self-releasing taper — accuracy of location with instant release — which is why milling machines and machining centres use the 7:24 family (and its modern face-and-taper successors) rather than a self-holding cone.
§6The standard systems
A handful of standardised taper families cover the machine tools an engineer meets, each in the self-holding or self-releasing camp.
| System | Type | Used on |
|---|---|---|
| Morse (MT) | self-holding (~⅝″/ft) | drills, reamers, tailstocks, lathe centres |
| Brown & Sharpe | self-holding (~½″/ft) | older milling collets and arbors |
| 7:24 steep (ISO/BT/CAT/NT) | self-releasing | milling spindles, machining centres |
| HSK | self-releasing, face + taper | high-speed spindles |
| Jacobs | short self-holding | drill chucks on their arbors |
| The modern high-speed systems (HSK and similar) add a hollow taper that grips on both the cone and a flat face, so the tool locates axially as well as radially and grips harder as speed rises — overcoming the tendency of a plain steep taper to pull out of its socket at high spindle speed. | ||
§7Quick reference
The working core of the page on one card rack.
Definition
ratio = (D−d)/L
half-angle = arctan[(D−d)/2L]
Self-holding
Morse ~1.49° half-angle
grips by friction
Locks if
half-angle < friction angle
arctan μ ≈ 8.5°
Self-releasing
7:24 → 8.30° half-angle
held by drawbar
Systems
Morse (drills) · 7:24/HSK (mills)