Torque Calculator - τ = r × F × sin(θ)

Lever Arm Length

01,000

Force

01,000,000

Angle Between Force and Lever Arm

0180

Torque

This calculator computes your Torque from the values you enter.

Inputs

Lever Arm LengthForceAngle Between Force and Lever Arm

Outputs

Torque

What is a Torque?

The Torque Calculator computes rotational torque using τ = r × F × sin(θ), the turning effect produced by a force applied at a distance from a pivot point. Enter a lever arm length, a force, and the angle between the force and the lever arm (defaulting to 90°, the most efficient angle), and the calculator instantly returns the torque in newton-meters.

Torque is the rotational analog of force, central to understanding how wrenches, engines, motors, and any rotating mechanical system work. This general-purpose calculator handles any lever-arm-and-force scenario in physics or mechanical engineering, distinct from the Bolt Torque Calculator, which is a specialized fastener-spec tool for determining correct bolt tightening torque.

If you need the linear-motion equivalents of work or power, use the Work Calculator or Mechanical Power Calculator.

Why Use a Torque Calculator?

τ = r × F × sin(θ) involves a trigonometric function that's easy to compute incorrectly by hand, especially when the angle isn't a clean 90°, a dedicated calculator removes that friction and gives instant, verifiable results.

The result updates live as you adjust lever arm length, force, or angle, and the step-by-step breakdown shows the exact substitution, useful for verifying homework, engineering calculations, or everyday questions like "how much torque am I applying with this wrench?"

Because torque scales with both lever arm length and the sine of the angle, this calculator makes it easy to see why longer wrenches and perpendicular force application both increase turning effectiveness.

Who Should Use This Calculator?

Physics students verifying homework problems involving torque, rotational motion, and the relationship between force, lever arm, and angle.

Mechanical engineers calculating torque requirements for rotating machinery, gear systems, and drivetrains.

Automotive enthusiasts and technicians estimating torque applied through wrenches, breaker bars, or other hand tools.

Teachers demonstrating how torque depends on lever arm length and force angle, including why perpendicular force at a longer lever arm is more effective.

Hobbyists and makers working on projects involving levers, gears, or rotating mechanisms who need a quick torque estimate.

What Insights Does This Calculator Give You?

Torque, the single result of the formula, expressed in newton-meters (N·m), representing the rotational turning effect produced by the given force, lever arm, and angle.

Angle sensitivity, this calculator shows how torque drops off as the angle between force and lever arm moves away from 90°, reaching zero when the force is applied directly along the lever arm (0°).

Lever arm leverage, the direct linear relationship between lever arm length and torque explains why tools like breaker bars (with longer handles) make it easier to apply large torques with modest force.

How to use this Torque calculator

Enter the lever arm length, the distance from the pivot point to where the force is applied, in meters.
Enter the force, the magnitude of the applied force, in newtons.
Enter the angle, the angle between the force direction and the lever arm, in degrees (90° for the default perpendicular case).
Read the torque result, the highlighted result shows the torque in newton-meters.
Check the step-by-step breakdown, expand the calculation steps to see the exact formula substitution, and compare with the Bolt Torque Calculator if your specific application involves fastener tightening specs.

Show formula & methodology ↓Show less ↑

Formula & Methodology

Torque formula:
τ = r × F × sin(θ)

Variable definitions:
- r, lever arm length (meters)
- F, applied force (newtons)
- θ, angle between force and lever arm (degrees)
- τ, torque (newton-meters)

Worked example:

A 0.3 m wrench has 80 N of force applied perpendicular to it (θ = 90°).

Step 1, Apply the formula: τ = 0.3 m × 80 N × sin(90°) = 0.3 × 80 × 1 = 24 N·m

This means 24 newton-meters of torque are applied, enough to tighten many standard bolts, though the exact required torque depends on the bolt's size and material specification (see the Bolt Torque Calculator for that specialized use case).

Note: This calculator computes the torque from a single applied force. Real-world systems with multiple forces or a distributed load require summing the torque contributions from each force separately.

Frequently Asked Questions

What is the formula for torque?

Torque is calculated as τ = r × F × sin(θ), where r is the lever arm length in meters, F is the applied force in newtons, and θ is the angle between the force direction and the lever arm. The result is expressed in newton-meters (N·m). This calculator applies that formula directly to whatever values you enter.

Why does the angle matter in the torque formula?

Only the component of force perpendicular to the lever arm produces rotation, the sin(θ) term accounts for this. When force is applied exactly perpendicular to the lever arm (θ = 90°), sin(θ) = 1 and the full force contributes to torque, which is the most efficient angle. When force is applied along the lever arm (θ = 0°), sin(θ) = 0 and no torque is produced at all.

Why is 90° the default angle for this calculator?

90° represents the most common and most efficient scenario, force applied perpendicular to the lever arm, such as pushing straight down on a wrench handle. This maximizes torque for a given force and lever arm length, which is why tools are typically designed and used this way.

How is this torque calculator different from the Bolt Torque Calculator?

This calculator computes general rotational torque (τ = r × F × sin(θ)) for any lever-arm-and-force scenario in physics or engineering. The [Bolt Torque Calculator](/bolt-torque-calculator/) is a specialized fastener-spec tool that determines the correct tightening torque for specific bolt sizes and grades, which is a different application entirely.

What units does this calculator use?

Lever arm length is entered in meters (m), force in newtons (N), and angle in degrees (°), producing torque in newton-meters (N·m), the SI unit of torque. If your measurements are in other units, convert them to meters and newtons first for an accurate result.

What is a typical torque value in everyday terms?

Tightening a typical bolt might require 20–50 N·m of torque, opening a stubborn jar lid might involve a few N·m applied at the lid's small radius, and a car engine might produce 200–400 N·m of torque at its crankshaft, illustrating the wide range of torque values in daily life and engineering.

How does lever arm length affect torque for the same force?

Torque increases linearly with lever arm length for a fixed force and angle, doubling the lever arm doubles the torque. This is why a longer wrench or breaker bar makes it easier to loosen a stuck bolt: the same hand force applied farther from the pivot point produces more torque.

How is torque related to work and power in rotational systems?

In rotational motion, torque plays a role analogous to force in linear motion, rotational work equals torque multiplied by angular displacement, and rotational power equals torque multiplied by angular velocity. Compare with the [Work Calculator](/work-calculator/) and [Mechanical Power Calculator](/mechanical-power-calculator/) for the linear-motion equivalents.

Can torque be applied at an angle other than 90° in practice?

Yes, many real-world scenarios involve force applied at an angle other than perpendicular to the lever arm, such as pushing on a wrench handle at a slight angle rather than straight down. This calculator's sin(θ) term correctly reduces the effective torque in those cases.

How is torque used in mechanical engineering?

Torque calculations are fundamental to motor and engine specifications (torque curves define usable power across speeds), gear system design (torque changes as gears change speed ratios), fastener tightening specifications, and any rotating machinery from wind turbines to power tools.

What is the difference between torque and force?

Force is a straightforward push or pull measured in newtons, while torque is the rotational effect of a force applied at a distance from a pivot point, measured in newton-meters. The same force can produce very different torque depending on how far from the pivot it's applied and at what angle.

How do I maximize torque for a given force?

To maximize torque for a given force, apply it at the longest practical lever arm length and at exactly 90° to that lever arm (perpendicular), since both a longer lever arm and a 90° angle increase the resulting torque, as shown directly in the τ = r × F × sin(θ) formula.

Also known as

torque formula calculatorrotational force calculatormoment of force calculatorlever arm torque calculatortau = rF sin theta calculator