Torque Calculator (τ = r × F × sin θ)
Find the turning effect of a force from the lever arm, force, and angle.
τ = r × F × sin θ. Maximum when the force is perpendicular to the lever arm (θ = 90°).
Torque is the turning effect of a force: τ = r × F × sin θ, where r is the lever arm, F the force, and θ the angle between them. A 10 N force applied 2 m from the pivot at 90° gives τ = 2 × 10 × sin 90° = 20 N·m. Torque peaks when the force is perpendicular.
What torque measures
Torque (τ) is the rotational equivalent of force — it tells you how strongly a force tends to twist an object about a pivot. It grows with the size of the force, with the distance from the pivot (the lever arm r), and with how squarely the force is aimed: only the component perpendicular to the lever arm actually turns anything, which is where the sin θ comes from. The result is measured in newton-metres (N·m).
τ = torque (N·m), r = lever arm (m), F = force (N), θ = angle between r and F
Worked example
You push a wrench 2 m long with a 10 N force, applied at right angles to the handle. How much torque turns the bolt?
- 1 Write the torque formula. τ = r × F × sin θ, with r in metres, F in newtons, and θ the angle between them.
- 2 Substitute the known values. τ = 2 m × 10 N × sin 90°.
- 3 Evaluate sin θ. sin 90° = 1, so the force is fully perpendicular and none is wasted.
- 4 Compute the torque. τ = 2 × 10 × 1 = 20 N·m of turning effect.
How the angle changes torque
For a fixed r and F, sin θ scales the turning effect from zero up to its maximum.
| Angle θ | sin θ | Effect on torque |
|---|---|---|
| 0° | 0 | No torque — force points along the lever arm |
| 30° | 0.50 | Half of the maximum |
| 45° | 0.71 | About 71% of the maximum |
| 60° | 0.87 | About 87% of the maximum |
| 90° | 1 | Maximum torque — force fully perpendicular |
Why perpendicular force wins
Only the part of a force that acts at right angles to the lever arm produces rotation; any component pointing toward or away from the pivot just pulls on the axle and turns nothing. That perpendicular fraction is F × sin θ, so torque is largest at θ = 90° (sin 90° = 1) and vanishes at θ = 0° (sin 0° = 0), when you would simply be pushing the lever into its own pivot.
The lever arm matters just as much. This is the door-handle intuition: handles sit at the edge of the door, far from the hinge, so a gentle push delivers plenty of torque. Push near the hinge — a small r — and the same force barely moves it. Doubling the distance doubles the torque, which is why long wrenches and breaker bars exist. Torque is always reported in newton-metres (N·m): numerically the same units as energy, but a distinct physical quantity describing rotation rather than work done.