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Torque Calculator

Find torque from a force applied at a distance and angle, τ = r·F·sin θ, or solve for the force, lever arm or angle.

Level: Grades 11–12

Printable worksheet

Torque Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Solve for

Distance from the pivot (the axis of rotation) to where the force is applied.

90° when pushing square to the wrench or door; 0° or 180° when pushing along the arm, which gives no torque.

Result

Torque

40 N·m

Torque is 40 N·m.

In kilonewton meters
0.04 kN·m
In pound-force feet
29.5025 lbf·ft
In pound-force inches
354.0298 lbf·in
Force component perpendicular to the arm (F × sin θ)
160 N

See the step-by-step solution ↓

Step-by-step solution

  1. Change lever arm to m, the standard unit the formula uses (1 cm = 0.01 m)

    25 cm×0.01=0.25 m

  2. Evaluate sin θ for θ = 90°

    sin 90 °=1

  3. Multiply lever arm × force × sin θ. Only the square-on part of the push turns the object

    0.25×160×1=40 N·m

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how much twisting force you put on a bolt, a door or a wheel, or work backward to the push you need. Torque is the turning effect of a force. Push on a door near its hinges and it barely moves. Push the same amount near the handle and it swings easily.

Torque depends on three things. The first is the force, F. The second is the lever arm, r: the distance from the pivot to where you push. The third is the angle, θ (theta), between the lever arm and the force. Together they give τ = r × F × sin θ, where τ (tau) is the torque.

Only the part of the force at right angles to the lever arm turns anything. The sin θ part of the formula (the sine of the angle) measures how much of your push that is. This calculator finds torque, or works backward to the force, the lever arm length or the angle. It works in newton meters, kilonewton meters, pound-force feet or pound-force inches. It also shows the square-on part of the force, F × sin θ.

Where it's used

Why this math matters outside the classroom:

  • Tightening bolts. Mechanics reach 100 N·m on a bolt by pushing 200 N at the end of a 0.5 m wrench, square to the handle.
  • Door handles. Door handles sit far from the hinges because the same push gives more torque at a longer lever arm.
  • Pushing at an angle. Pushing at 30° to a wrench instead of 90° gives only sin 30°, or half, the torque for the same force.
  • Seesaws. Children balance a seesaw when a 30 kg child 2 m out matches a 40 kg child 1.5 m out.
  • Bike gears. Cyclists shift gears to change the torque reaching the rear wheel when climbing hills.

Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSG.SRT.C.8, HSA.CED.A.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Choose what to solve for: torque, force, lever arm or angle.
  2. Enter the values you know, with their units. The angle is between the lever arm and the direction you push or pull.
  3. Choose the unit you want and read the result. It's also shown in the other torque units.

What the inputs mean

Torque (τ)
Torque has the same base units as energy (N·m), but it's a different quantity. Here it's always written N·m, never joules.
Lever arm length (r)
Measure from the center of turning (the bolt center, the hinge, the axle) to where the force is applied. On a wrench, that's the middle of your hand.
Angle between lever arm and force (θ)
90° gives the most torque. At 0° or 180° the force points straight along the arm, toward or away from the pivot. Then it produces no turning effect at all.

Formula and method

τ = r × F × sin θ
F = τ ÷ (r × sin θ)
r = τ ÷ (F × sin θ)
θ = arcsin(τ ÷ (r × F))
1 lbf·ft = 4.4482216152605 N × 0.3048 m = 1.3558179483314004 N·m

τ (tau) is the torque, r is the lever arm length, F is the force and θ (theta) is the angle between the arm and the force. sin θ is the sine of that angle, a number from 0 to 1. arcsin is the reverse: it turns a sine back into an angle. lbf·ft means pound-force feet, N is newtons and m is meters.

The calculator converts the force to newtons and the lever arm to meters, so the torque comes out in newton meters. For US units it uses the exact official sizes of the pound-force and the foot. So the conversions add no rounding beyond the computer's own tiny limits.

Solving for the angle gives two answers, θ and 180° − θ, because both have the same sine. Pushing at 30° or 150° to the arm gives the same turning effect. The calculator shows the sharper angle (under 90°) first, and the wider one (over 90°) next to it.

Worked example

You push with 160 N on a wrench 25 cm long, square to the handle (90°).

  1. Convert the lever arm: 25 cm × 0.01 = 0.25 m.
  2. sin 90° = 1.
  3. τ = 0.25 × 160 × 1 = 40 N·m.
  4. In pound-force feet: 40 ÷ 1.3558179 ≈ 29.5025 lbf·ft.

Result: The wrench applies 40 N·m of torque, about 29.50 lbf·ft.

Understanding the result

A longer lever arm gives the same torque for less force. That's why a breaker bar (an extra-long wrench handle) loosens stubborn bolts. It's also why door handles sit far from the hinges.

For a car engine, torque describes the twisting effort at the crankshaft, the shaft that turns the wheels. Power is torque times how fast the shaft spins. So an engine can have high torque at low revs but modest power, or the reverse.

Assumptions and limitations

  • One force pushes at one point.
  • The angle is measured in the flat plane the object turns in.
  • Only sizes are given. The direction of turning (clockwise or counterclockwise) isn't tracked.
  • When several forces act, find each torque and add them up. Count one turning direction as plus and the other as minus.
  • Torque wrench settings for real fasteners depend on thread friction and lubrication. Follow the manufacturer's specification rather than a calculated value.
  • Three-dimensional problems need the full vector cross product τ = r × F, a 3D version of this formula.

Frequently asked questions

Is a newton meter of torque the same as a joule?

The units multiply out the same way, but they measure different things. Torque is a turning effect, while a joule is energy. Turning a shaft with 1 N·m of torque through one radian (about 57°) does 1 J of work.

How do I convert lbf·ft to N·m?

Multiply by 1.3558179483314004. For example, 100 lbf·ft is about 135.58 N·m. To go back, divide by the same number.

Why is the torque largest at 90°?

At 90° the whole force is square to the arm, and sin 90° = 1. At any other angle, part of the force just pushes or pulls along the arm toward the pivot. That part does nothing to turn it.

What is the difference between lbf·ft and lbf·in?

A pound-force foot is twelve times a pound-force inch, because a foot is twelve inches. Small fasteners are usually specified in lbf·in, larger ones in lbf·ft.

How this calculator is checked

Method: Definition of the moment of a force (τ = rF sin θ, the magnitude of the cross product r × F). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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