Torque Calculator

Torque is τ = r × F × sin θ: the force, the distance from the pivot, and the angle between them. Choose which to solve for above and the calculator rearranges the equation and converts between newton metres, pound-feet and pound-inches.

What torque measures

Torque is the turning effect of a force. Pushing on a door does nothing if you push at the hinge, and a great deal if you push at the handle, even though the force is identical. What changed is the distance from the pivot.

That distance is the lever arm, and torque grows in direct proportion to it. Doubling the length of a spanner halves the force needed for the same torque, which is why a breaker bar exists and why it works.

A worked example

Take a force of 200 N applied 0.3 m from the pivot, at right angles.

• Multiply force by distance: 200 × 0.3 = 60
• Multiply by sin 90°, which is 1: 60 N·m

That is 44.25 pound-feet. Apply the same force at 45 degrees instead and sin 45° is 0.7071, giving 42.426 N·m. The force has not changed, but almost 30% of its turning effect has gone.

Why the angle matters

Only the component of force perpendicular to the lever arm turns anything. The component along the arm pulls or pushes the pivot and achieves nothing rotational.

At 90 degrees all the force is perpendicular, sin θ is 1, and the torque is at its maximum. At 0 or 180 degrees the force acts straight along the arm, sin θ is 0, and the torque is nothing at all, however hard you pull. This is why pulling a spanner straight toward the bolt does not loosen it.

Torque and work are different things

Both are measured in newton metres, which causes confusion, but they are not the same quantity. Work is force applied along a distance moved and is an energy, so it is written in joules. Torque is force applied across a distance from a pivot and is a turning effect.

The convention is to keep torque in newton metres and energy in joules, even though the units are dimensionally identical, precisely to avoid mixing them up. A torque applied through an angle does produce work, which is where the two connect.

Tightening fasteners

Torque specifications exist because the point of tightening a bolt is to stretch it by a controlled amount, and torque is the practical proxy for that stretch.

Fastener	Typical torque	Force on a 0.4 m bar
Car wheel nut	110 N·m	275 N
Bicycle stem bolt	5 N·m	12.5 N
Cylinder head bolt	90 N·m	225 N

The bicycle figure explains why torque wrenches matter on lightweight components. Five newton metres is easy to exceed by a large margin with an ordinary allen key, and a carbon part will crack long before it feels tight.

Torque and power in engines

An engine's torque is how hard it twists the crankshaft; its power is how quickly it does work. They are linked by power = torque × angular velocity, so an engine producing high torque at low revolutions can make the same power as one producing less torque at high revolutions.

This is why a diesel engine with modest peak power pulls a heavy load well: it makes its torque low in the rev range, where a load actually needs it. It is also why gearing exists, since a gearbox trades rotational speed for torque without changing the power passing through.

Balancing a lever

A lever balances when the torques on each side are equal. A child of 25 kg sitting 2 m from a seesaw pivot produces 25 × 9.80665 × 2 = 490 N·m. An adult of 80 kg balances that at 490 / (80 × 9.80665) = 0.625 m from the pivot.

The same principle sets crane counterweights, decides where a load can safely sit on a forklift, and explains why a wheelbarrow with the load close to the wheel is so much easier to lift.

Where it gets used

Mechanics and assembly. Every bolt on a vehicle has a specified torque, and both under and over tightening cause failures.

Structures. Bending moments in beams are torques, and they determine the size of the section needed.

Motors and drives. Starting torque decides whether a motor can move its load at all, separately from its rated power.

Everyday tools. Spanners, door handles, taps and pedals are all lever arms chosen to make a task possible with human force.

Gears trade torque against speed

A gearbox cannot create power, but it can exchange rotational speed for torque. Halve the output speed and the torque roughly doubles, minus losses.

A bicycle makes this visible. In a low gear the pedals turn several times for each turn of the wheel, so the torque at the wheel is multiplied and a steep hill becomes possible at a slow speed. In a high gear the ratio reverses, giving speed at the cost of torque. The rider's legs supply much the same power in both cases.

The same trade explains why a car pulls hardest in first gear and why a winch turns slowly. Power is fixed by the engine; how it is divided between force and speed is a choice made by the gearing.

Static equilibrium

An object stays still only when the forces balance and the torques balance. Both conditions are needed, and the second is the one usually forgotten.

A ladder leaning against a wall has its weight acting downward at its centre, the ground pushing up and inward with friction, and the wall pushing horizontally. The forces can balance while the torques do not, in which case the ladder rotates and slides away. Taking moments about the foot of the ladder is what determines the minimum safe angle, which for most ladders works out near 75 degrees.

Cranes, shelves, brackets and cantilevers are all analysed the same way: sum the torques about a chosen pivot and require the total to be zero.

Direction, and why torque is a vector

Torque has a direction as well as a size: clockwise or anticlockwise about a given axis. When several torques act, they add with sign, and the object turns whichever way the total points.

Formally the direction is given by the right-hand rule and points along the axis of rotation rather than round it, which seems strange until you need to add torques about different axes. For a single pivot in a plane, treating anticlockwise as positive and clockwise as negative is enough for almost every practical problem.

Tightening to a specified torque

Bolts are tightened to a torque figure because that is a usable stand-in for the tension in the bolt, which is what actually holds the joint. Too little and the joint works loose; too much and the bolt stretches past its yield point or the thread strips.

A typical car wheel nut calls for about 110 N·m, or 81 lbf·ft. On a wrench 0.4 m long that means pulling with 275 N, roughly the weight of a 28 kg mass. Standing on the bar of a long breaker bar easily exceeds the figure, which is why torque wrenches exist.

Friction complicates it. A dry or corroded thread absorbs more of the applied torque, so less tension reaches the joint than the setting suggests, and published figures assume a specified thread condition.

Common mistakes

Ignoring the angle. Anything other than 90 degrees reduces the torque by the sine of the angle.

Measuring from the wrong point. The lever arm runs from the pivot, not from where it is convenient to measure.

Confusing newton metres with joules. Same units, different quantities.

Mixing units. Pound-feet and pound-inches differ by a factor of 12, and specifications are given in both.