Physics · Mechanics · Motion & Energy

Torque calculator — force, lever arm and angle

Calculate the magnitude of torque about a fixed axis from force, distance to the axis and their included angle, or solve for force or lever-arm distance.

Created Sep 30, 2026v1.0

By Thibaut Grzelak, Mechanical Analysis Engineer

Mechanics · Motion & Energy

Results

Torque

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Moment magnitude about the selected axis.

Force

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Force magnitude required to create the specified torque.

Distance to axis

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Distance from the axis to the point of application.

Perpendicular lever arm

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Shortest distance r⊥ = r sin θ from the axis to the force line of action.

Relation and numerical substitutionτ = r × F × sin(θ)

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Single force about a fixed axis; magnitudes are used with 0° ≤ θ ≤ 180°. Inverse modes require a nonzero perpendicular lever arm.

01Formulas and symbols

Formulas used

ττ = r × F × sin(θ)Torque magnitude equals force times the perpendicular lever arm.

FF = τ / (r × sin(θ))Required force for a known torque, distance and nonzero perpendicular component.

rr = τ / (F × sin(θ))Required axis-to-application distance for a known torque, force and angle.

r⊥r⊥ = r × sin(θ)Perpendicular lever arm measured from the axis to the force line of action.

02Assumptions and limits

Scope of validity

Single force about a fixed axis; magnitudes are used with 0° ≤ θ ≤ 180°. Inverse modes require a nonzero perpendicular lever arm.

03Validation example

Reference numerical case

With r = 0.40 m, F = 250 N and θ = 30°, r⊥ = 0.40 sin 30° = 0.20 m.
τ = 0.40 × 250 × sin 30° = 50 N·m. The inverse modes recover F = 250 N and r = 0.40 m.

04References

Technical references

University Physics Volume 1, §10.6 Torque — OpenStax

FAQ

Why does torque use sin θ?

Only the component of force perpendicular to the radius produces torque about the axis. Equivalently, r sin θ is the perpendicular distance from the axis to the force line of action.

Why is torque in N·m not treated as a joule?

Torque and energy share the same base dimensions, but they represent different physical quantities. Torque is a rotational moment, while the joule is reserved for work or energy; writing torque in N·m preserves that distinction.

05How it works

At 90°, the full radius acts as the lever arm. At 0° or 180°, the force line passes through the axis and the torque is zero. The calculator also reports r⊥ so the geometry behind τ = Fr⊥ remains visible.

06Limits of the model

This scalar tool returns torque magnitude for one force about one fixed axis. It does not sum multiple torques or determine the vector sign with a right-hand-rule convention. Force and radius inverse modes are singular when sin θ = 0.

07Common mistakes

Measure r from the axis to the point where the force is applied, and θ between r and F. Do not use the full radius as the effective lever arm unless the force is perpendicular.