Free Torque Calculator
This torque calculator finds the rotational moment τ produced when a force F acts at a distance r from a pivot, using τ = F × r × sin(θ). Enter the force in newtons (N), the lever arm length in metres (m) and the angle θ between the force and the lever arm in degrees, and the calculator returns the torque in newton-metres (N·m) along with handy conversions. Torque, also called the moment of a force, is what makes things rotate: it is the turning effect behind tightening a bolt, spinning a wheel, swinging a door or balancing a see-saw. The angle matters because only the component of the force perpendicular to the lever arm contributes to turning the object.
Enter the force, lever arm and angle to calculate torque τ = F · r · sin(θ).
τ = F × r × sin(θ). Use newtons (N) and metres (m) for a result in N·m. Torque is maximised when θ = 90° (sin 90° = 1); a force along the arm (θ = 0°) gives zero torque. 1 N·m ≈ 0.7376 lbf·ft.
Quick answer
Torque is calculated with τ = F × r × sin(θ), where F is the applied force in newtons, r is the lever-arm distance from the pivot in metres, and θ is the angle between the force and the lever arm. The result is in newton-metres (N·m). For example, a force of 50 N applied perpendicularly (θ = 90°, sin 90° = 1) at a 0.3 m lever arm gives τ = 50 × 0.3 × 1 = 15 N·m. Maximum torque occurs when the force is applied at 90° to the lever arm.
Formula & method
τ = F × r × sin(θ)
- τ — torque (moment of force), in newton-metres (N·m)
- F — applied force, in newtons (N)
- r — lever arm length — distance from the pivot to where the force acts, in metres (m)
- θ — angle between the force vector and the lever arm, in degrees
sin(θ) selects the component of force perpendicular to the lever arm. At θ = 90° the force is fully perpendicular (sin 90° = 1) and torque is maximised; at θ = 0° or 180° the force is parallel to the arm (sin = 0) and produces no torque. 1 N·m ≈ 0.7376 lbf·ft.
Examples
- Input
- F = 50 N, r = 0.3 m, θ = 90°
- Result
- τ = 15 N·m
- Why
- With the force applied at a right angle, sin(90°) = 1, so τ = F × r × sin(θ) = 50 × 0.3 × 1 = 15 N·m. This is the default case and gives the maximum torque for that force and lever arm.
- Input
- F = 200 N, r = 0.25 m, θ = 60°
- Result
- τ ≈ 43.30 N·m
- Why
- Here sin(60°) ≈ 0.866025, so τ = 200 × 0.25 × 0.866025 = 50 × 0.866025 ≈ 43.30 N·m. Because the force is not perpendicular, only about 86.6% of it contributes to turning the object.
- Input
- F = 120 N, r = 0.4 m, θ = 90°
- Result
- τ = 48 N·m
- Why
- A 120 N push on a 0.4 m breaker bar at 90° gives τ = 120 × 0.4 × 1 = 48 N·m. Doubling the bar length to 0.8 m would double the torque to 96 N·m for the same force — which is why longer wrenches make loosening easier.
- Input
- F = 80 N, r = 0.5 m, θ = 30°
- Result
- τ = 20 N·m
- Why
- With sin(30°) = 0.5, τ = 80 × 0.5 × 0.5 = 20 N·m. The same 80 N applied perpendicularly (θ = 90°) would produce 80 × 0.5 × 1 = 40 N·m — twice as much — showing how a poor angle wastes effort.
When to use this tool
- Sizing or checking the torque needed to tighten a bolt, nut or fastener with a wrench, spanner or breaker bar.
- Working out the turning effect of a force in physics or engineering coursework, including statics and equilibrium problems.
- Comparing how lever-arm length or the angle of pull changes the rotational effect of the same applied force.
- Estimating the moment about a pivot for see-saws, doors, levers, cranks, pedals or any system that rotates about an axis.
- Converting a known force-and-distance setup into newton-metres before comparing it against a tool's or motor's rated torque.
Common mistakes
- Entering the angle in radians instead of degrees. This calculator expects θ in degrees (90° for a perpendicular force); if you have radians, multiply by 180/π first (π/2 rad = 90°).
- Using the angle from the axis of the force rather than the angle between the force and the lever arm. τ uses sin(θ), so θ should be measured between the force vector and the arm — a force directly along the arm (θ = 0°) produces zero torque.
- Forgetting to use SI units. Force must be in newtons and the lever arm in metres to get torque in N·m. A length entered in centimetres (e.g. 30 instead of 0.3) inflates the result by 100×.
- Confusing torque (N·m) with energy or work, which is also measured in newton-metres. Torque is a rotational vector quantity (F × r), while work is force times displacement along the direction of motion; they are physically different despite sharing units.
- Assuming maximum torque always occurs at the longest reach. Torque is maximised when the force is perpendicular (θ = 90°); applying even a large force at a small angle to the arm gives little turning effect.
Frequently asked questions
What is the formula for torque?
Torque is calculated with τ = F × r × sin(θ), where F is the applied force in newtons, r is the perpendicular distance (lever arm) from the pivot to the line of force in metres, and θ is the angle between the force and the lever arm. The result is in newton-metres (N·m). When the force is perpendicular to the arm (θ = 90°), the formula simplifies to τ = F × r because sin(90°) = 1.
What units is torque measured in?
In SI units, torque is measured in newton-metres (N·m), obtained when force is in newtons and the lever arm is in metres. In imperial units it is often given in pound-feet (lbf·ft) or pound-inches (lbf·in). One newton-metre is approximately 0.7376 lbf·ft, and one lbf·ft is about 1.3558 N·m.
Why does the angle θ affect torque?
Only the component of the force that acts perpendicular to the lever arm causes rotation, and that component equals F × sin(θ). When the force is at 90° to the arm, sin(θ) = 1 and the entire force contributes, giving maximum torque. When the force points along the arm (θ = 0° or 180°), sin(θ) = 0 and there is no turning effect at all.
How do I increase the torque I can apply?
You can increase torque in three ways: apply more force (F), use a longer lever arm (r), or apply the force closer to a right angle to the arm so sin(θ) is larger. Doubling the lever-arm length doubles the torque for the same force, which is exactly why a longer wrench or breaker bar makes a tight bolt easier to turn.
Is torque the same as work or energy?
No. Although torque and work both have the units newton-metre, they are different quantities. Torque is the rotational equivalent of force (it is force times lever arm, τ = F × r) and is a vector. Work and energy measure force times displacement and are scalars; to avoid confusion, energy is expressed in joules (J) while torque keeps the N·m label.
What is the difference between torque and moment?
In everyday and engineering usage the terms are largely interchangeable: 'moment of a force' and 'torque' both describe the turning effect of a force about a point or axis, calculated as τ = F × r × sin(θ). 'Moment' is more common in structural statics, while 'torque' is favoured in mechanics involving rotating shafts, motors and fasteners, but the underlying formula is identical.
Sources & references
- Torque - Wikipedia
- Torque - HyperPhysics (Georgia State University)
- Torque | Definition, Formula, Units - Britannica
External references open in a new tab. We are independent and not affiliated with these organizations.
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Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.
Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.
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