Torque Calculator

Background.

Door handles sit on the edge farthest from the hinges for a reason. Push near the hinge and the door barely moves; push at the handle with the same effort and it swings open. What changed is the torque, the turning effect of a force, which grows with the distance between the pivot and the point where the force is applied, and with how squarely the force acts across that distance. A 150 N push at a handle 0.8 m from the hinges gives 120 N·m; the same push 0.1 m from the hinges gives only 15 N·m.

In symbols, τ = r × F × sin θ. The lever arm r runs from the pivot (a bolt's axis, a hinge, the centre of a shaft) to the point where the force is applied; F is how hard you push or pull; θ is the angle between the two. At 90° the whole force goes into turning. As the angle closes toward 0° or opens toward 180°, more of the force simply pushes along the arm, toward or away from the pivot, and contributes nothing to the rotation.

Force can be entered in newtons, pound-force or kilogram-force and the arm in metres, centimetres, inches or feet, in any combination. Everything is converted to SI first, and the torque comes back in newton-metres, pound-feet and dyne-centimetres at once, so a European specification in N·m and an American wrench scale in lb-ft can be compared directly.

What you get is the size of the torque from one force about one pivot. Direction of rotation is not shown, several forces are not combined, and there is no mode that solves backwards for the force you need, although rearranging the same equation does that in one line, as the formula section shows. For bolts, the figure is the torque you apply at the wrench, which is not the same thing as the clamping force the bolt ends up with.

What is torque calculator?

Torque, also called the moment of a force, measures how strongly a force tends to rotate an object about an axis. Its size equals the force multiplied by the perpendicular distance from the axis to the force's line of action, which is why r sin θ is called the moment arm. Formally torque is a vector, τ = r × F, pointing along the axis by the right-hand rule, and its direction tells you which way the rotation goes; the calculation here returns the magnitude only. The SI unit is the newton-metre. It has the same dimensions as the joule, but the SI Brochure notes that the joule is never used for torque, because a turning effect is not an amount of energy. Torque drives angular acceleration the way force drives linear acceleration, through τ = Iα, where I is the moment of inertia: a torque of 43.47 N·m applied to a wheel with I = 0.5 kg·m² speeds it up at about 86.93 rad/s² until friction or a load pushes back.

How to use this calculator.

1. 01Enter the applied force and choose its unit. Kilogram-force suits a hanging mass or a person's weight: a 10 kg load pulls with 10 kgf.
2. 02Enter the lever arm and its unit: the straight-line distance from the centre of the pivot to where the force is applied. On a wrench that is the bolt centre to the middle of your hand, not the overall length of the tool.
3. 03Enter the angle between the lever arm and the force, anywhere from 0° to 180°. Use 90° when you push or pull straight across the handle.
4. 04Read the torque in N·m, lb-ft and dyn·cm. For inch-pounds, the unit small-fastener specifications often use, multiply the pound-feet figure by 12.

The formula.

τ = r × F × sin θ, with r in metres, F in newtons and τ in newton-metres. Inputs are converted to SI first using exact definitions: 1 lbf = 4.4482216152605 N, 1 kgf = 9.80665 N (one kilogram under standard gravity), 1 in = 0.0254 m and 1 ft = 0.3048 m. The result is then given three ways: in N·m directly, in pound-feet by dividing by 1.3558179483314004, which is exact because it is simply 0.45359237 kg × 9.80665 m/s² × 0.3048 m, and in dyne-centimetres by multiplying by 10⁷, since a dyne is 10⁻⁵ N and a centimetre 10⁻² m.

The sine factor is the fraction of the force that acts across the arm. It is 1 at 90°, 0.5 at 30° or 150°, and 0 at 0° and 180°. Because sin θ = sin(180° − θ), an angle of 105° gives exactly the same magnitude as 75°; which side of square you pulled from does not change the size of the torque. Near 90° the sine curve is flat, so small errors in angle barely register: pulling 15° off square costs 3.4% of the torque and 10° off about 1.5%. Force and lever arm, by contrast, act in direct proportion, so doubling either doubles the torque.

An equivalent reading is force times the perpendicular distance from the pivot to the line along which the force acts. A 0.3 m arm pulled at 75° has a moment arm of 0.3 × sin 75° ≈ 0.2898 m, and that shorter effective distance is what the force works through.

Rearranged, the same equation gives the force needed for a target torque, F = τ ÷ (r sin θ), or the arm needed for a given force, r = τ ÷ (F sin θ).

At exactly 0° and 180° the result is 0 in every unit. A computer's value for sin 180° is a vanishing 10⁻¹⁶ rather than 0, so the results are rounded to 10 decimal places, which clears that residue; the same rounding means a real torque smaller than about 5 × 10⁻¹¹ N·m would also show as 0.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Halliday, D., Resnick, R., and Walker, J. (2013). Fundamentals of Physics, 10th ed. John Wiley & Sons. ISBN 978-1-118-23072-5
2. [2]Thomson, J. (1884). "On the Measurement of Electric Resistance." Phil. Mag. 18(5):316–320.
3. [3]BIPM (2019). The International System of Units (SI Brochure), 9th ed.
4. [4]Shigley, J.E. and Mischke, C.R. (1989). Mechanical Engineering Design, 5th ed. McGraw-Hill. ISBN 978-0-07-056899-0
5. [5]Meriam, J.L. and Kraige, L.G. (2012). Engineering Mechanics: Statics, 7th ed. John Wiley & Sons. ISBN 978-0-470-61473-0