Torque Calculator – Force, Lever Arm & Unit Converter

Solve torque, required force, or required lever arm with |τ| = rF sin(θ), then convert among SI and imperial torque units. Everything runs locally in the browser and the page does not track your inputs.

Inputs

Angle between the lever arm vector and the force direction.

Tip: use 90° for a perpendicular force. At 0° or 180°, torque is zero because the force points along the arm instead of across it.

Results

Torque
15 N·m
Perpendicular application gives the maximum turning effect for a given force and arm length.
Supporting values
    |τ| = rF sin(θ)
    Educational use only. This page handles idealized statics and unit conversion. It does not evaluate bolt preload, thread friction, wrench calibration, fatigue, material limits, or design-code compliance.

    How torque works

    Torque, also called the moment of a force, measures how strongly a force tends to rotate an object about a pivot or axis. The full vector definition is τ = r × F, and the magnitude used in most introductory mechanics problems is |τ| = rF sin(θ). Here, r is the distance from the pivot to the point where the force is applied, F is the force magnitude, and θ is the angle between the lever-arm vector and the force vector. When the force is perfectly perpendicular to the arm, sin(θ)=1, so the equation simplifies to |τ| = rF.

    The important geometric idea is the perpendicular lever arm. Instead of thinking only about the physical bar length, it is often clearer to ask: how far is the pivot from the force's line of action, measured at a right angle? That perpendicular distance is r⊥ = r sin(θ), which means torque can also be written as |τ| = F r⊥. This is why pushing straight toward a hinge does not rotate a door much, while pushing sideways at the same point does.

    This page keeps the calculations intentionally basic and transparent. Use it when you need to estimate the turning effect from a known force, back-calculate the force required to hit a target torque, or determine how long a lever arm must be to reach that target. The unit converter is included because workshop and engineering references often mix N·m, lbf·ft, lbf·in, and smaller units like ozf·in or N·cm. For real hardware, the number on paper is only part of the story: friction, tool calibration, joint design, preload method, and safety factors all matter.

    Formulas used on this page

    • Torque from force and lever arm: |τ| = rF sin(θ)
    • Perpendicular lever arm: r⊥ = r sin(θ)
    • Required force: F = |τ| / (r sin(θ))
    • Required lever arm: r = |τ| / (F sin(θ))

    Worked example

    Suppose you push on a wrench with 50 N of force at a point 0.30 m from the bolt, and the force is perpendicular to the wrench. With θ = 90°, the torque is |τ| = 0.30 × 50 × 1 = 15 N·m. If the force were applied at 30° instead, the same wrench and force would produce only 0.30 × 50 × sin(30°) = 7.5 N·m. That reduction is exactly why force direction matters in real setups.

    5 practical torque facts

    Longer handles reduce force

    Doubling the perpendicular lever arm halves the force needed for the same torque target. That is the entire logic behind breaker bars and long-handled wrenches.

    Tool design

    Angle errors matter fast

    A force applied at 60° produces only about 86.6% of the torque you would get at 90°. At 30°, you are down to 50%.

    Geometry

    10 N·m is about 7.38 lb·ft

    That conversion shows up constantly when switching between SI specifications and workshop references written in imperial torque units.

    Unit conversion

    Door hinges teach torque well

    Pushing near the hinge barely rotates the door because the perpendicular distance is tiny. The same push near the handle creates much more moment.

    Everyday physics

    Specified torque is not the whole joint story

    For bolted joints, the same wrench torque can produce different clamp loads if lubrication, thread condition, or fastener finish changes.

    Engineering caution

    Torque calculator FAQ

    What is the difference between arm length and lever arm?

    The physical arm length is the straight-line distance from the pivot to the point where the force is applied. The lever arm used in torque is the perpendicular distance from the pivot to the force's line of action. They are equal only when the force is perpendicular to the arm.

    Why does the calculator ask for an angle?

    Because only the component of force perpendicular to the arm creates turning effect. The angle controls that perpendicular component through sin(θ).

    Can torque be negative?

    Yes in signed conventions, where clockwise and counterclockwise torques have opposite signs. This page reports magnitudes because that is the most common requirement for quick engineering and workshop calculations.

    When should I use N·m instead of lb·ft?

    Use the units required by your drawing, standard, or tool. SI engineering work usually defaults to N·m, while many automotive and workshop references still use lb·ft or lb·in.

    Can I use this for torque wrench settings?

    You can use it for basic checks and conversions, but final settings should come from the equipment manual or engineering specification. Real torque transfer depends on the specific joint, not only on the simple statics equation.

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