Taper Calculator - Find Taper Angle and Ratio

A taper calculator finds the angle and ratio of a cone-shaped or wedge-shaped part from three measurements: the diameter at each end and the length between them. Machinists, engineers, and woodworkers use it to check whether a tapered shaft, drill shank, or pipe fitting matches a required specification.

How to calculate taper angle

The taper angle is the full angle between the two sides of a taper, measured as if the sides were extended until they met. It is found from the two end diameters and the length between them:

θ = 2 × arctan( (D_L − D_S) / (2 × L) )

• D_L is the diameter at the large end
• D_S is the diameter at the small end
• L is the length between the two measurement points

All three values must use the same unit. The arctan step gives the angle in radians, which is then converted to degrees by multiplying by 180/π.

The formula works in two stages. Dividing by 2 first finds the half-angle, the angle between the taper's surface and its centerline. Multiplying by 2 afterward gives the full angle normally shown on a drawing.

How to calculate taper ratio

The taper ratio compares length to diameter change and is written as 1:X:

Taper ratio = L / (D_L − D_S)

A ratio of 1:20 means the diameter shrinks by 1 unit for every 20 units of length. Many machine tool tapers sit near 1:20 because that ratio is shallow enough to hold by friction alone, yet steep enough to release with a light tap.

Worked example

A shaft measures 10 mm across the large end and 5 mm across the small end, over a length of 100 mm.

Taper angle: θ = 2 × arctan((10 − 5) / (2 × 100)) = 2 × arctan(0.025) ≈ 2.86°

Taper ratio: 100 / (10 − 5) = 20, written as 1:20

How to measure a taper

Three measurements define a taper: the large-end diameter, the small-end diameter, and the length between the two points where those diameters were taken. The length must run parallel to the centerline. Measuring along the slanted surface instead makes the length too long and the calculated angle too small.

Calipers or a micrometer measure the diameters. Since a taper does not have one fixed diameter, both readings need to come from known, specific points along the part, not just anywhere convenient.

If both diameters come out equal, the part is a plain cylinder, not a taper. The angle is 0° and the ratio is infinite, because the diameter never changes no matter how far along the part is measured.

Standard taper ratios

Several industries use a small set of standard tapers so that tooling from different makers fits together.

Morse, Brown & Sharpe, and Jarno tapers are self-holding. Friction alone keeps the tool seated once it is tapped into place, so removing it takes a firm knock. CAT, BT, and R8 tapers are steeper and self-releasing. They rely on a drawbar to pull the tool tight during cutting, and they let go easily once the drawbar is released. The dividing line between the two behaviors sits roughly around a ratio of 1:8 to 1:10 — shallower ratios tend to lock by friction, steeper ones need a mechanical pull to stay seated.

Individual Morse taper sizes differ slightly: for example, MT1 is close to 1:20.05, MT2 is close to 1:20.02, and MT3 is close to 1:19.92. The differences are small enough that "about 1:20" describes the whole family, but they matter when machining a replacement part or a matching socket.

Other ways to express a taper

Taper per foot (TPF) states the diameter change over exactly 12 inches of length. A taper of 0.6 inch per foot equals a ratio of 1:20, since 12 ÷ 0.6 = 20.

Conicity, common on ISO drawings, is the diameter change divided by length, without converting it to a ratio: C = (D_L − D_S) / L. A conicity of 0.05 describes the same taper as 1:20.

Frequently asked questions

What is the difference between taper angle and taper ratio?

They describe the same geometry in two formats. The angle, in degrees, is the full angle between the two tapered surfaces. The ratio, written 1:X, states how much length corresponds to one unit of diameter change. Drawings and CNC programs commonly use the angle; identifying which taper standard a part follows usually relies on the ratio.

What units should be used?

Any consistent unit works, as long as all three measurements use the same one. Millimeters and inches both work; mixing the two in a single calculation produces a wrong result.

What does a 1:20 taper ratio mean in practice?

The diameter changes by 1 unit for every 20 units of length along the axis. A 200 mm long part with a 1:20 taper has a 10 mm difference between its end diameters.

Why do different Morse taper sizes have slightly different ratios?

Stephen A. Morse introduced the Morse taper system in the 1860s as a family of sizes numbered 0 to 7. Each size sits close to 1:20 but not exactly on it, so the ratios across the family range from about 1:19.0 to 1:20.05.

Can a taper calculator identify which standard a part uses?

It calculates the ratio and angle from measurements taken on the part. Matching that result to a named standard, such as Morse #2 or NPT, still requires comparing it against a reference table, since several standards use similar ratios.

Is the calculated angle the half-angle or the full angle?

It is the full included angle, the angle between the two tapered sides as normally shown on a technical drawing. Dividing it by two gives the half-angle, measured from the centerline to one side.