Torque Calculator

What the torque calculator does

Torque is the turning effect of a force, the thing that makes a wrench loosen a bolt or a motor spin a shaft. This calculator finds it from the force applied, the distance from the pivot, and the angle between them, and it can solve for any of those when you know the torque. A simple mode assumes the force pushes squarely at a right angle; an advanced mode lets you set any angle.

Below is what torque means, the equation behind it, why the angle and the lever arm matter so much, and a worked example.

How to use it

1. Choose what to find: torque, force, or distance.
2. Pick a mode. Simple mode takes the force as acting at 90 degrees; advanced mode adds an angle field.
3. Enter the known values, each with its own unit, then press Calculate, or Reset to clear it.

What torque is

Torque is to rotation what force is to straight-line motion. A force pushes things along; a torque twists them around. It is what you produce when you push on a door, turn a steering wheel, or tighten a nut, and it is what an engine delivers to turn the wheels. The more torque, the stronger the turning effect.

What makes torque more than just force is that where you push matters as much as how hard. Push on a door right at its hinge and it barely moves, however hard you shove. Push at the handle, far from the hinge, and it swings easily. Torque captures both the force and the leverage of where it is applied, which is why it, not force alone, is the right measure of turning effect.

The equation it solves

Torque, written with the Greek letter tau, is the force F times the distance r from the pivot, times the sine of the angle θ between the force and the lever arm:

τ = r F sinθ

Rearranged, the same relationship gives the force needed for a given torque, F = τ ÷ (r sinθ), and the distance, r = τ ÷ (F sinθ). In the simple mode the angle is 90 degrees, whose sine is one, so the equation reduces to torque being simply the force times the distance.

Why the angle and the lever arm matter

Two things set how much turning effect a force produces. The first is the lever arm, the distance from the pivot. Because torque is force times distance, doubling the distance doubles the torque for the same push. This is the whole reason a long wrench loosens a stubborn bolt that a short one cannot, and why door handles are placed far from the hinge.

The second is the angle, through the sine in the formula. A force does the most turning when it pushes square to the lever arm, at 90 degrees, where the sine is one. Push at a shallower angle and only part of the force turns the object, so the torque falls. Push straight along the arm, at 0 degrees, and the sine is zero, so there is no torque at all, you are simply pulling or pushing through the pivot. This is why you instinctively push a wrench at right angles to get the most out of it.

Units and precision

The SI unit of torque is the newton-metre. The calculator takes the distance and force in your choice of units and the angle in degrees or radians, converting internally, and reports the torque in newton-metres alongside other torque units such as pound-force feet. It is worth noting that the newton-metre is also the unit of energy, but torque and energy are different quantities; the shared unit is a coincidence of dimensions, which is why torque is sometimes written as joules per radian to keep them apart. Results are carried to several figures, finer than most real measurements.

A worked example: a wrench

Suppose you push with 200 N on a wrench 0.3 metres long, pushing square to it at 90 degrees.

The torque is τ = 0.3 × 200 × sin90 = 0.3 × 200 × 1 = 60 N·m. Push at 60 degrees instead and the sine drops to about 0.87, giving 0.3 × 200 × 0.87 ≈ 52 N·m. Push straight along the handle, at 0 degrees, and the torque falls to zero, no matter how hard you push.

Questions people ask

What is torque?

It is the turning effect of a force, the rotational equivalent of a push. It depends on the force, the distance from the pivot, and the angle, combined as τ = rF sinθ.

How do you calculate torque?

Multiply the force by the distance from the pivot and by the sine of the angle between them, τ = rF sinθ. When the force is at a right angle, it is simply force times distance.

Why does a longer wrench give more torque?

Because torque is force times distance from the pivot. A longer wrench increases that distance, so the same push produces more turning effect, loosening a tighter bolt.

Is a newton-metre of torque the same as a joule?

They share the same units but are different quantities. A joule is energy; a newton-metre of torque is a turning effect. Torque is sometimes written as joules per radian to avoid confusing the two.

References

A quick note on where the physics comes from. Torque as the turning effect of a force, τ = rF sinθ, and the role of the lever arm and angle are standard rotational mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The newton-metre and the other SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 10.6, Torque. https://openstax.org/books/university-physics-volume-1/pages/10-6-torque
2. HyperPhysics, Georgia State University, Torque. http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811