Torque Calculator
Solve τ = r·F·sin θ for torque, force or lever arm at any angle, and see exactly why a force applied along the arm produces no turning effect at all.
τ = r·F·sin θ
Torque (N·m)
Formula used for this direction
τ = r·F·sin θ
Your values
r = 0.3 m, F = 200 N, θ = 90
Force, Distance and Angle
Torque is what actually turns things. The same force produces wildly different rotation depending on where and how it is applied — which is why a door handle sits far from the hinges, and why pushing a door near its hinge barely moves it.
The formula τ = r·F·sin θ has three levers, and this calculator solves for any of them. Rearranged, F = τ ÷ (r·sin θ) tells you the hand force a given bolt torque demands, and r = τ ÷ (F·sin θ) tells you how long a spanner you need.
Worked example — tightening a bolt
200 N applied at right angles, 300 mm from the bolt:
τ = 0.3 × 200 × sin 90° = 60 N·m
Halve the spanner to 150 mm and the same hand force gives only 30 N·m. To restore the original 60 N·m from that shorter spanner you would need 400 N — 89.92 lbf, which is a genuinely hard pull.
A second example — pulling at an awkward angle. The same 200 N at 300 mm but angled at 45° instead of square on:
τ = 0.3 × 200 × sin 45° = 42.43 N·m — about 71% of the square-on result, lost purely to geometry.
Pull straight along the spanner instead, at 0°, and the torque is 0 N·m. You are simply trying to pull the bolt out of the engine.
How Much Angle Costs You
200 N at a 300 mm lever arm, angle varying:
| Angle | Torque | Fraction of maximum |
|---|---|---|
| 0° | 0 N·m | 0% |
| 15° | 15.53 N·m | 25.9% |
| 30° | 30 N·m | 50% |
| 45° | 42.43 N·m | 70.7% |
| 60° | 51.96 N·m | 86.6% |
| 75° | 57.96 N·m | 96.6% |
| 90° | 60 N·m | 100% |
Lever Arm Versus Required Force
Force needed to produce 60 N·m at right angles, by spanner length:
| Lever arm | Force needed | In pound-force |
|---|---|---|
| 100 mm | 600 N | 134.9 lbf |
| 150 mm | 400 N | 89.92 lbf |
| 200 mm | 300 N | 67.44 lbf |
| 300 mm | 200 N | 44.96 lbf |
| 450 mm | 133.3 N | 29.97 lbf |
| 600 mm | 100 N | 22.48 lbf |
Keep going
- Torque is a force applied at a distance from a pivot, so the straight-line force is the input; the force calculator solves for it first. Force Calculator
- Torque turning through an angle does rotational work, the rotational counterpart of force acting over a distance; the work calculator covers the linear case. Work Calculator
Frequently Asked Questions
What is torque?
Torque is the turning effect of a force about a pivot: τ = r·F·sin θ, where r is the distance from the pivot, F the force, and θ the angle between them. It is measured in newton-metres. Doubling either the force or the lever arm doubles the torque.
Why does the angle matter?
Only the component of force perpendicular to the lever arm turns anything. The sin θ term extracts that component. At 90° all of the force turns the object; at 0° or 180° the force points straight along the arm, pushing or pulling the pivot rather than rotating it, and the torque is exactly zero.
Why is a longer spanner easier to use?
Because torque scales with r. A 300 mm spanner needs half the hand force of a 150 mm one for the same torque on the bolt. This calculator makes that concrete: solve for force, halve the lever arm, and watch the required force double.
Is torque the same as work? Both are force times distance.
No, despite sharing units dimensionally. Work is force times distance moved along the direction of the force, and it is energy — joules. Torque is force times perpendicular distance from a pivot, and it is a turning tendency — newton-metres. The convention is to keep N·m for torque and reserve joules for energy precisely to avoid confusing them.
What is torque in pound-feet?
Pound-feet is the imperial torque unit, a pound-force acting one foot from the pivot. One lb·ft is about 1.356 N·m. Car engine outputs are often quoted this way, which is why a figure like 300 lb·ft corresponds to roughly 407 N·m.
Related Calculators
- Force Calculator — Forces & Rotation
- Momentum Calculator — Forces & Rotation
- Acceleration Calculator — Motion & Kinematics
- Velocity Calculator — Motion & Kinematics