Torque Calculator
Solve τ = rFsinθ for any of torque, lever-arm distance, force, or the angle between them. A live 3D wrench arm rotates around a pivot as you adjust r, F and θ, and charts show how torque depends on angle and distance.
Reviewed by the ToolNestr Editorial Team — July 2026
Two ideas that trip students up
1. Torque needs a lever arm, force, and angle
The indigo arm of length r pivots at the centre. The red force arrow is applied at angle θ to the arm — only its perpendicular component actually twists the arm.
2. Torque follows a sine curve
Three lever arms (short, medium, long) traced as τ vs θ — every curve peaks at 90° and hits zero at 0° and 180°, because torque scales with sinθ.
Torque graphs
How it works
The core idea in one line: torque is how much a force twists something around a pivot — it depends on how far from the pivot you push, how hard you push, and how perpendicular that push is.
τ = r · F · sin θ
torque = lever arm × force × sine of the angle between them
τmax = r · F
at θ = 90°, the common simplified case (force perpendicular to the arm)
θ = arcsin(τ / rF)
solving for the angle given torque, r and F
Rearranged, τ = rFsinθ solves any variable: F = τ/(r sinθ), r = τ/(F sinθ), and θ = arcsin(τ/(rF)). The sinθ factor means only the perpendicular component of the force actually contributes to rotation — a push straight along the lever arm produces no torque at all.
Worked example 1 — tightening a bolt
Given: A mechanic applies 150 N perpendicular to a 0.30 m wrench handle (θ = 90°). Find the torque.
Worked example 2 — pushing at an angle
Given: A 0.25 m long door handle is pushed with 40 N, but at 30° to the handle rather than perpendicular. Find the torque.
Compare to a perpendicular push at the same distance and force: τ = 0.25 × 40 × 1 = 10 N·m — the angled push delivers only half the torque.
Torque for common tools and tasks
Approximate values, assuming force applied perpendicular to the lever arm (θ = 90°).
| Task | Lever arm rm | Force FN | Torque τN·m |
|---|---|---|---|
| Tightening a small screw | 0.05 | 20 | 1 |
| Car lug nut (hand wrench) | 0.30 | 150 | 45 |
| Opening a door | 0.80 | 25 | 20 |
| Car engine (typical peak) | — | — | ~200–400 |
| Bicycle pedal crank | 0.17 | 300 | ~51 |
τ = rF at θ = 90°; real tools rarely apply force perfectly perpendicular, so effective torque is often somewhat lower.
Where torque actually matters
🔧 Wrenches and bolts
A longer wrench handle increases the lever arm r, letting you produce the same torque with less force — or a much larger torque with the same force, which is why breaker bars exist for stuck bolts.
🚪 Door hinges
Pushing a door near the hinge (small r) takes far more force to open than pushing near the handle (large r), because torque depends on the lever arm, not force alone. This is also why door handles are placed as far from the hinge as practical.
🏎️ Engine torque specs
Car and motorcycle engines are rated by torque (N·m or lb-ft) as well as power. Torque measures the twisting force available at the crankshaft, which determines acceleration and towing capability at a given engine speed.
Common misconceptions
"More force always means more torque."
Only if the lever arm and angle stay the same. A small force far from the pivot, applied perpendicular to the arm, can out-torque a large force applied close to the pivot or at a shallow angle.
"Torque and force are the same physical quantity."
They have different units (N·m vs N) and different roles: force causes linear acceleration, torque causes angular acceleration. Torque is force times a distance (and an angle factor), not force itself.
"Pushing along the handle still turns the bolt."
A force applied exactly along the lever arm (θ = 0° or 180°) has sinθ = 0, so it produces zero torque — only the component of force perpendicular to the arm contributes to rotation.
"Doubling the wrench length always doubles the torque."
True only if the applied force and angle stay the same. In practice a longer handle changes the geometry and grip, and torque scales with r only when F and θ are held fixed.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, University Physics Volume 1 — §10.6 "Torque" (free, peer-reviewed). openstax.org
- • Halliday, Resnick & Walker, Fundamentals of Physics — Chapter 10, Rotation.
- • Serway & Jewett, Physics for Scientists and Engineers — Chapter 10, Rotation of a Rigid Object About a Fixed Axis.
Results are rounded for display. τ = rFsinθ; θ is the angle between the lever-arm and force vectors, measured in degrees here.
How to use this calculator
Choose the unknown
The calculator solves for torque, r, F, or θ from the other three.
Enter the other values
Angle defaults to 90° (force perpendicular to the arm), the most common case.
Watch the lever arm
Use the sliders to see how r, F and θ change the rotation and torque arrow in 3D.
Related tools
Frequently asked questions
What is torque?
Torque (τ) is the rotational equivalent of force — a measure of how much a force twists an object around a pivot or axis. τ = rFsinθ, where r is the lever-arm distance, F is the force magnitude, and θ is the angle between the lever arm and the force. Units are newton-metres (N·m).
Why does the angle matter?
Only the component of force perpendicular to the lever arm produces rotation. At θ = 90° (force perpendicular to the arm), sinθ = 1 and you get the maximum torque τ = rF. At θ = 0° or 180° (force along the arm), sinθ = 0 and the torque is zero — pushing or pulling straight along a wrench handle does not turn anything.
How is torque different from force?
Force causes linear acceleration (F = ma); torque causes angular acceleration. The same force produces more torque the farther it is applied from the pivot, which is why a longer wrench (or a longer handle) makes it easier to loosen a stuck bolt.
What is a "lever arm"?
The lever arm (or moment arm) is the perpendicular distance from the pivot to the line along which the force acts. In the simplified case where the force is applied perpendicular to a straight arm of length r, the lever arm equals r itself.
What does a negative or zero torque mean?
Sign indicates rotation direction — by convention, positive torque rotates counterclockwise and negative rotates clockwise. Zero torque means the force passes through the pivot (r = 0) or is applied exactly along the lever arm (θ = 0° or 180°), so it produces no rotation at all.