Torque Calculator

Torque is the turning effect of a force, found with τ = F × d × sin(θ) - force times the lever-arm distance times the sine of the angle between them. A 100 N force applied 0.5 m from the pivot at a right angle produces 50 N·m of torque (about 36.9 lb-ft). This torque calculator computes that turning effect in newton-meters or pound-feet instantly from your force, distance and angle.

Working through a physics problem set? Pair this with the Trigonometry Calculator to resolve the sin(θ) term for odd angles, the Unit Converter to line up newtons-with-meters or pounds-with-feet before you enter them, and the Scientific Calculator for the surrounding arithmetic - this page stays focused on the single τ = F × d × sin(θ) relationship.

The torque formula

Torque - also called the moment of force - is the rotational equivalent of a push or pull. The calculator uses the standard equation:

where τ (tau) is the torque, F is the applied force, d is the distance from the axis of rotation to the point where the force acts (the lever arm), and θ (theta) is the angle between the force vector and the lever arm. When you push at a right angle - the most common case - sin(90°) = 1 and the formula simplifies to τ = F × d.

A worked example, step by step

Suppose you apply F = 100 N at d = 0.5 m from the pivot, at an angle of θ = 90°:

• sin(90°) = 1.000
• τ = 100 × 0.5 × 1.000 = 50 N·m
• Convert: 50 N·m × 0.73756 = 36.88 lb-ft

Now repeat at θ = 45°. Here sin(45°) ≈ 0.7071, so τ = 100 × 0.5 × 0.7071 ≈ 35.36 N·m (about 26.1 lb-ft). Same force, same distance - but you lose nearly 30% of the turning effect simply by pulling at an angle. The effective lever arm has shrunk from 0.5 m to 0.5 × 0.7071 = 0.354 m.

Why the angle matters

Only the component of force perpendicular to the lever arm produces rotation. The sine term in the formula extracts that perpendicular component. At 90° the entire force turns the lever and torque is maximized. At 0° or 180° the force points directly along the arm - straight toward or away from the pivot - so it cannot rotate anything and the torque is zero, no matter how hard you push. Between those extremes the torque scales with sin(θ): about 50% of maximum at 30°, 71% at 45°, and 87% at 60°. This is why pushing a wrench straight out from the bolt does nothing, while pulling at a clean right angle works best.

How much torque do you lose at each angle?

Every value below is computed for a constant 100 N force on a 0.5 m lever, so the only variable is the angle. Scale the last column by your own force and distance - the sin(θ) percentage is universal.

Note the curve is symmetric past 90°: 120° matches 60°, 135° matches 45° and 150° matches 30°, because sin(θ) = sin(180° − θ).

Newton-meters vs pound-feet

Torque shows up in two unit systems. In SI, it is the newton-meter (N·m): one newton applied one meter from the pivot. In US customary units it is the pound-foot (lb-ft): one pound-force applied one foot from the pivot. The exact relationship is 1 lb-ft = 1.3558179 N·m, so to convert N·m to lb-ft you multiply by 0.73756, and to go the other way you multiply by 1.35582. The calculator always shows both, so a 200 N·m engine spec instantly reads as about 147.5 lb-ft. Use the unit toggle to enter your force and distance in whichever system matches your data - newtons with meters, or pounds-force with feet.

Newton-meters to pound-feet conversion table

Both columns are exact conversions using 1 lb-ft = 1.3558179 N·m. Read left-to-right to turn an N·m figure into lb-ft, or use the right pair to go from lb-ft back to N·m.

How to use this torque calculator

You only need three numbers. Work through the fields in order:

1. Unit system: choose Metric (N·m) to enter force in newtons and distance in meters, or Imperial (lb-ft) to enter pounds-force and feet. Keep the two consistent - mixing newtons with feet gives a meaningless result.
2. Force: type the magnitude of the push or pull applied to the lever.
3. Distance: enter the lever-arm length - the distance from the pivot or axis to where the force is applied.
4. Angle: drag the slider to the angle between the force and the lever arm, or tap a preset (30°, 45°, 60°, 90°). Leave it at 90° if the force is perpendicular, which is the usual assumption.

Press Calculate torque and read the result in your chosen unit, with the other unit, sin(θ), and the effective perpendicular lever arm shown below it. The calculation is instant and deterministic - the same inputs always give the same answer.

Who this calculator is for

Torque turns up across physics, engineering and the garage. This tool helps:

• Physics & engineering students checking homework on moments, levers and rotational equilibrium.
• DIY mechanics and hobbyists who want to understand how lever length and angle affect the force on a bolt or beam.
• Makers and machinists sizing a motor, arm or clamp where a known force acts at a known radius.
• Teachers demonstrating why sin(θ) caps the turning effect and why longer wrenches win.
• Anyone converting a torque value between newton-meters and pound-feet.

A second worked example: a longer wrench

Imagine a stuck lug nut. You pull with 50 lbf at the end of a 1 ft wrench, perpendicular to it: τ = 50 × 1 × sin(90°) = 50 lb-ft. Swap to a 2 ft breaker bar and pull with the same 50 lbf: τ = 50 × 2 × 1 = 100 lb-ft. Doubling the lever arm doubles the torque for the very same effort - the practical reason a breaker bar frees bolts a stubby ratchet cannot. Note this calculator gives the torque you apply; whether it exceeds the bolt's break-away torque is a separate question.

Scenario comparison: same force, different setups

Using a constant 100 N force, here is how distance and angle move the torque:

• 0.5 m, 90°: 50.0 N·m (36.9 lb-ft) - the perpendicular baseline.
• 1.0 m, 90°: 100.0 N·m (73.8 lb-ft) - double the arm, double the torque.
• 0.5 m, 45°: 35.4 N·m (26.1 lb-ft) - same arm, but only 71% of the turning effect.
• 0.5 m, 30°: 25.0 N·m (18.4 lb-ft) - half the maximum, because sin(30°) = 0.5.
• 0.5 m, 0°: 0 N·m - force in line with the arm produces no rotation.

The takeaway: lever length and angle are levers in their own right. Two of them combined often change torque more than the force itself.

Torque chart: force × lever-arm length at 90°

Each cell is τ = F × d at a right angle (sin(90°) = 1), shown in N·m with the pound-feet equivalent. Notice torque doubles whenever you double either the force or the distance.

For any angle other than 90°, multiply these figures by the sin(θ) value from the angle table above - for example, at 45° every cell drops to 70.7% of what is shown.

Key torque terms explained

• Torque (τ): the turning effect of a force about an axis, in N·m or lb-ft.
• Lever arm / moment arm: the perpendicular distance from the axis to the line of action of the force - in the formula, d × sin(θ).
• Pivot / axis of rotation: the fixed point or line the object turns about.
• Force component: the part of the force perpendicular to the arm; only this part contributes to torque.
• Moment of force: a synonym for torque, common in static structural analysis.
• Newton-meter (N·m) / pound-foot (lb-ft): the SI and US customary units of torque, related by 1 lb-ft = 1.35582 N·m.

What changes the result the most

Adjust the inputs and you will see three drivers:

• Lever-arm distance: torque scales linearly with d - twice the arm is twice the torque.
• Force: also linear - twice the force is twice the torque, at a fixed arm and angle.
• Angle: nonlinear through sin(θ); the effect is strongest near 0° and 180° and flattens near 90°, where small angle changes barely matter.
• Units: the magnitude depends on whether you read N·m or lb-ft, but the physical torque is identical.

Torque vs work: why both can be N·m

A common point of confusion is that energy and torque both reduce to force times distance and can both be written in newton-meters. Physically they are different. Work (energy) is force times the distance moved along the force, and its unit is given the special name joule (J). Torque is force times the perpendicular lever arm and describes rotation, so its unit is deliberately kept as N·m rather than joules. This calculator computes torque - the turning effect at a single instant - not the work done as something rotates through an angle. If something turns through an angle, the work done is torque times the angle in radians, a separate calculation.

Tips for accurate results

• Keep force and distance in matching units: newtons with meters for N·m, or pounds-force with feet for lb-ft.
• Remember that pounds-force (lbf) is a force, not a mass. If you have a weight in pounds it already acts as a force; if you have a mass in kilograms, multiply by 9.81 m/s² to get newtons first.
• Measure the lever arm from the actual axis of rotation, not from the handle or an arbitrary point.
• If you are unsure of the angle, use 90° - it is the standard assumption and gives the maximum torque for the given force and distance.

Limitations and assumptions

This is a planning and learning estimate, not an engineering sign-off. Keep these in mind:

• It computes static torque from a single force - not net torque from multiple forces, which you would sum (with signs) separately.
• It does not model angular acceleration; that requires the moment of inertia in τ = Iα.
• It does not handle the engine torque-power-RPM relationship (power = torque × angular speed).
• It ignores friction, preload and material limits, so it is not a fastener torque specification - always follow the manufacturer's spec for real bolts.
• It returns the magnitude of torque; direction (clockwise vs counter-clockwise) is left to you.

About this formula

The equation τ = F × d × sin(θ) is a deterministic, universally accepted result of classical mechanics - the cross-product definition of the moment of a force, τ = r × F. There is no proprietary data or regional standard behind it, so no external source is needed; the only convention applied here is the unit-conversion factor 1 lb-ft = 1.3558179483 N·m, which follows directly from the definitions of the pound-force, the foot, the newton and the meter.

How it compares to related calculators

This page answers "what torque does this force produce?" For neighboring questions, a sister tool fits better:

• To work with the angles and sides of triangles, use the Triangle Calculator or the Trigonometry Calculator.
• To solve for the sides of a right triangle, use the Pythagorean Theorem Calculator.
• To switch between metric and imperial values, use the Unit Converter.
• For trig, logs and powers in one place, use the Scientific Calculator.