Torque calculator

How do you calculate torque?

τ = r × F × sin θ, where r is the distance from the pivot to where the force acts, F the force and θ the angle between the lever arm and the force (OpenStax University Physics). A push at right angles (90°) gives the most torque.

Example: a wrench

Pushing with 20 N at right angles on a wrench 0.5 m from the bolt gives 0.5 × 20 × sin 90° = 10 N·m. Push at 30° to the handle and it drops to half: 5 N·m.

What are the units?

Newton-meters (N·m) in SI, and pound-feet (lb·ft) in US units. 1 lb·ft = 1.3558 N·m. Torque wrenches and car specs in the US usually use lb·ft or lb·in.

Why does a longer wrench help?

Torque grows in proportion to the lever arm. Doubling the distance from the bolt doubles the torque for the same push.

Is torque the same as work?

No — both can be written in N·m, but torque is a turning effect, while work (in joules) is force times distance moved.

What force do I need to reach a torque specification?

Rearrange the formula: F = τ ÷ (r sin θ). To tighten a bolt to 100 lb·ft with a handle that is 1.5 ft long and a push at right angles, F = 100 ÷ 1.5 = 66.7 lbf. A cheater bar that makes the lever arm 3 ft halves that to 33.3 lbf.

How do I add up several torques?

Give each torque a sign, positive for counterclockwise and negative for clockwise (the right-hand rule), and add them: the net torque is the sum (OpenStax University Physics, Volume 1, section 10.6, Eq. 10.24). OpenStax Example 10.15 puts three forces on a flywheel of radius 0.5 m: 20 N at 150° to the radius gives 5.0 N·m, 30 N at 90° gives −15.0 N·m, and 30 N at 0° gives 0.0 N·m, so the net torque is −10.0 N·m. The calculator finds one torque at a time.

Which angle do I enter?

The angle between the lever arm, drawn from the pivot to the point where the force acts, and the direction of the force (OpenStax, section 10.6). A push straight along the arm (0° or 180°) produces no torque and one at right angles (90°) the most; 30° and 150° give the same torque.

The formula and what it assumes

Torque is the cross product of the lever arm and the force, τ = r × F, and its size is τ = rF sin θ, where θ is the angle between the two (OpenStax University Physics, Volume 1, section 10.6, Eq. 10.22). The same thing can be written as the force times the perpendicular distance from the pivot to the line of the force, r⊥ = r sin θ (Eq. 10.23). The unit is the newton-meter, or pound-force foot in US units. The calculator gives the size of the torque; its direction follows the right-hand rule and is positive for counterclockwise turning.

Worked example: a 2 ft wrench pushed with 40 lbf

In SI the force is 40 × 4.448222 = 177.9 N and the lever arm is 2 × 0.3048 = 0.6096 m. A push at 90° gives τ = 0.6096 × 177.9 × sin 90° = 108.5 N·m, which is 80.0 lb·ft, the same as 2 ft × 40 lbf. At 60° the torque drops to 69.3 lb·ft (93.9 N·m), and at 30° to 40.0 lb·ft, half the full value.

Common mistakes

Using the wrong angle is the main one. The angle belongs between the arm and the force, not between the force and the horizontal or the bolt’s axis; a push perpendicular to the handle is 90° whatever the handle’s orientation. Torque units are the next trap: 1 lb·ft = 1.355818 N·m and 1 lb·ft = 12 lb·in, so a spec of 120 lb·in is only 10 lb·ft. And do not read N·m of torque as joules: the energy transferred depends on how far the object turns.

Sources: OpenStax University Physics, Volume 1, section 10.6 (torque, Eq. 10.22–10.24; Examples 10.14 and 10.15); NIST SP 811, Appendix B.8: 1 pound-force foot = 1.355818 N·m and 1 pound-force = 4.448222 N. Related: torque converter, moment of inertia calculator, horsepower calculator and work calculator.