How to measure a taper, explained

A taper is an angle wearing a diameter costume, and neither is directly measurable: calipers on a cone read somewhere-on-the-flank, and a protractor is not a shop instrument. The classic answer needs only a surface plate, gauge blocks and two equal rolls — and like the three-wire and gear-span methods, its cleverness is in what cancels.

The two-roll method

Stand the taper vertical on the plate. Rest a roll against the flank on each side, both sitting on equal gauge stacks of height h₁; mic over the rolls: M₁. Raise the stacks to h₂, measure again: M₂. The flanks converge, so the over-roll reading shrinks with height, and the rate is the angle:

tan(α/2) = (M₂ − M₁) / (2·Δh) · TPF = 24·tan(α/2)

A single over-roll reading would need a contact correction — where a cylinder touches an inclined flank depends on roll radius and the very angle being measured. The two-height difference cancels that offset exactly, because it is the same at both heights. The rolls need no calibrated size, only equality. (The dovetail calculator runs the single-height cousin of this setup, where the pin size does enter.)

Worked example

Stacks at 1.000 and 3.000 in (Δh = 2.000), readings M₁ = 2.812 and M₂ = 2.612 in:

tan(α/2) = 0.200 / 4.000 = 0.05 → α/2 = 2.862° · included 5.725° · 1.2 in/ft · ratio 1:10

Feed the equivalent diameters into the taper-per-foot calculator and the same row falls out, plus the tailstock setover if the next step is turning one. The atan itself is one keystroke chain on the machinist calculator — in degrees, as the shop expects.

Closing the loop with a sine bar

Measurement says 2.862°; verification sweeps it. Set a sine bar to that half angle — a 5 in bar wants a 0.2497 in stack — lay the taper on it, and run an indicator along the top element: flat means the taper is true to the measured angle, and every tenth of movement is angle error over the swept length. Identifying a standard? 1.2 in/ft is close to a Jarno-family ratio; Morse tapers run ~0.6 in/ft (MT2: half angle 1.431°) — compare against the published series before assuming a custom grind.

Common mistakes

• Unequal rolls or stacks. The cancellation assumes symmetry — mismatched rolls or one worn stack turns the cancelled correction back into error.
• Mic axis not square to the taper axis. Measure at the same orientation both times; a skewed mic reads a chord, and the two readings stop being comparable.
• Confusing half angle and included angle. The lathe compound and the sine bar want the half angle from centerline; the print may dimension the included angle. The factor-of-two error makes a taper exactly twice wrong.
• Quoting taper per foot without the basis. TPF is diametral (both sides); taper per side per foot is half of it. 1.2 in/ft diametral = 0.6 in/ft on a side.

Frequently asked questions

How do you measure a taper angle with rolls?

Stand the taper on a surface plate, rest two equal rolls against it on gauge-block stacks at two different heights, and mic over the rolls at each: tan(half-angle) = (M₂ − M₁) / (2·Δh). A 0.200 in spread over a 2.000 in height difference is a 2.862° half angle — 5.725° included, 1.2 in/ft.

Why does the two-height method need no roll-size correction?

Measuring over a roll on an inclined flank involves a contact offset that depends on roll radius and taper angle — fussy trig. But that offset is identical at both heights, so the subtraction M₂ − M₁ cancels it exactly. The rolls do not even need a known diameter; they only need to match each other.

How do you convert taper per foot to an angle?

half angle = atan(TPF / 24) — the 24 is twelve inches times two sides. Morse taper #2 at 0.5994 in/ft works out to a 1.431° half angle. The taper-per-foot calculator runs all the conversions (ratio, per-inch, per-100mm, setover) from any pair of diameters and a length.

How do you check a taper against its spec after measuring?

Set a sine bar to the measured half angle and sweep an indicator along the surface: a 5 in bar at 2.862° needs a 0.2497 in stack, and any indicator movement along the taper is angle error, read directly in tenths. Blue transfer against the mating part is the functional check.