Torque Calculator
Calculate torque from force, lever arm and angle using τ = F × r × sin(θ). Get instant results in newton-metres and pound-feet with the formula and steps shown.
Updated 2026-06-14 · Free · No sign-up · Runs privately in your browser
Show the formula & steps
How the Torque Calculator Works
This calculator finds torque — the rotational turning effect a force produces about a pivot. Enter the force in newtons, the lever arm (the distance from the pivot to where the force is applied) in metres, and the angle between the force and the lever arm in degrees. It instantly returns the torque in newton-metres (N·m) and pound-feet (lb·ft).
The Formula
Torque is the cross product of the lever arm and the force, whose magnitude is:
τ = F × r × sin(θ)
where:
- τ is torque (N·m)
- F is the applied force (N)
- r is the lever arm — the straight-line distance from the axis of rotation to the point where the force acts (m)
- θ is the angle between the force vector and the lever arm
The sin(θ) factor matters because only the part of the force that is perpendicular to the lever arm actually turns the object. When the force is applied at a right angle (θ = 90°), sin(θ) = 1 and the equation simplifies to the familiar τ = F × r.
Worked Example
Suppose you push with a force of 150 N on a wrench 0.3 m long, applying the force perpendicular to the handle (θ = 90°):
- sin(90°) = 1
- τ = 150 × 0.3 × 1 = 45 N·m
- In imperial units: 45 × 0.7376 = ≈ 33.19 lb·ft
Now apply the same force at 60° instead:
- sin(60°) ≈ 0.8660
- τ = 150 × 0.3 × 0.8660 = ≈ 38.97 N·m
The angled push delivers less torque because part of the force is wasted along the lever arm rather than turning it.
How the Angle Changes the Result
The table below shows the torque produced by a 150 N force on a 0.3 m lever arm at different angles. Notice how torque peaks at 90° and falls to zero when the force lines up with the arm.
| Angle θ | sin(θ) | Torque (N·m) |
|---|---|---|
| 0° | 0.000 | 0.00 |
| 30° | 0.500 | 22.50 |
| 45° | 0.707 | 31.82 |
| 60° | 0.866 | 38.97 |
| 90° | 1.000 | 45.00 |
Where Torque Is Used
Torque appears throughout engineering and everyday life:
- Fastening — bolt specifications are given as torque values so joints are neither loose nor over-stressed.
- Engines and motors — an engine’s torque curve describes its turning force at each speed, separate from its power.
- Tools — longer wrenches and breaker bars multiply torque by extending the lever arm r.
- Structures — engineers balance torques (moments) to keep beams, levers and seesaws in equilibrium.
To boost torque without a stronger push, lengthen the lever arm or keep the force perpendicular so sin(θ) stays at its maximum of 1.
Frequently asked questions
What is the formula for torque?+
Torque equals force times the lever arm times the sine of the angle between them: τ = F × r × sin(θ). When the force is perpendicular to the lever arm (θ = 90°), sin(θ) = 1 and the formula simplifies to τ = F × r.
What are the units of torque?+
The SI unit of torque is the newton-metre (N·m), found by multiplying a force in newtons by a distance in metres. In imperial units torque is often given in pound-feet (lb·ft); 1 N·m equals about 0.7376 lb·ft.
Why does the angle matter in torque?+
Only the component of force perpendicular to the lever arm produces rotation. The sin(θ) term captures this: at 90° all the force is effective, at 0° (force along the arm) it produces zero torque, and at 45° only about 70.7% is effective.
What is the difference between torque and work?+
Although both can be written in newton-metres, they are different. Work is force applied over a distance moved in the direction of the force (a scalar measured in joules), while torque is a rotational turning effect (a vector). They are never interchangeable.
How do I increase torque without more force?+
Increase the lever arm r — apply the force farther from the pivot — or keep the force perpendicular (θ near 90°) so sin(θ) stays at its maximum. A longer wrench, for example, delivers more torque for the same hand force.