The torque calculator above applies τ = rF sin θ and rearranges it for any of the four quantities involved. Given a force, a distance from the pivot and the angle between them it returns the turning effect. Given a torque it returns the force you would need at a stated radius, the radius you would need for a stated force, or the angle that would make a given combination produce that torque. The last of those has no solution outside a certain range, and the tool says so rather than quietly returning a wrong answer.
Arb Digital builds free tools that report the numbers around the answer. This one shows the result in pound-feet and pound-inches alongside newton metres, because fastener specifications are still quoted in all three, and it reports the effective lever arm — the perpendicular distance from the pivot to the line of action — which is the number that actually determines the result. It is deliberately distinct from the site's torque converter, which rescales an existing torque between units rather than deriving one.
What This Torque Calculator Does
Torque is the rotational equivalent of force: the measure of how effectively a force tends to turn something about a pivot. OpenStax University Physics Volume 1, section 10.6 on torque, gives its magnitude as rF sin θ, where r is the distance from the pivot to the point of application, F is the force and θ is the angle between the position vector and the force vector, with the unit written as newtons times metres.
Three things determine the answer and people usually think about only one of them. How hard you push is the obvious factor. How far from the pivot you push is equally important and often easier to change. The angle is the one that gets forgotten, and it is why a spanner in a cramped engine bay delivers far less than its length suggests. The share-of-maximum figure in the grid exists to make that loss visible.
Every unit is switchable independently, so a force in pounds-force applied at a radius in inches can be read out in newton metres without converting anything by hand. Everything is normalised to SI internally before the arithmetic happens, and converted back for display.
How to Use It
- Choose the unknown. The hero label renames itself, and only the three fields relevant to that mode are read.
- Measure the lever arm to the point where the force is applied. Not to the fastener head, not to the end of the tool, but to where your hand or the actuator actually acts.
- Enter the true angle between the force and the arm. If you are pulling square to the spanner, that is 90 degrees. If the tool is at an awkward angle to your pull, it is not.
- Check the effective lever arm figure. That is r sin θ, the perpendicular distance from the pivot to the force's line of action, and it is the only geometry the physics cares about.
- Use the angle mode carefully. Solving for angle asks what geometry would give a stated torque, and if the requested torque exceeds rF there is no such angle.
The Formula: How Torque Is Calculated
The magnitude is τ = rF sin θ. Rearranged, F = τ ÷ (r sin θ), r = τ ÷ (F sin θ) and θ = arcsin(τ ÷ rF). That last form is the interesting one: the argument of the arcsine cannot exceed 1, so if the torque you ask for is greater than rF, no angle exists and the tool reports that instead of producing a not-a-number. When a solution does exist there are generally two angles that give it, one acute and one obtuse, and both are reported.
Work the defaults. A force of 200 N applied 0.3 m from the pivot at 90 degrees gives 200 × 0.3 × 1 = 60 N·m. In imperial units that is 60 ÷ 1.35582 = 44.25 lbf·ft, or 531.0 lbf·in. The effective lever arm is 0.3 × sin 90° = 0.3 m, the full length, because the force is already perpendicular.
Now drop the angle to 45 degrees. Sine 45 is 0.7071, so the torque falls to 42.43 N·m and the effective lever arm to 0.2121 m. You lose nearly thirty per cent of the turning effect without changing the force or the tool at all. At 30 degrees you lose half of it, since sine 30 is exactly 0.5 — a useful landmark worth remembering when judging whether an awkward position is workable.
Why a Newton Metre Is Not a Joule
Torque and energy have identical dimensions. A newton metre of torque and a joule of work both come out as kilogram metres squared per second squared, and yet nobody quotes a bolt specification in joules. This is not sloppiness; it reflects a real physical distinction, and the SI Brochure published by the BIPM keeps the newton metre and the joule as separate named quantities for exactly this reason.
The difference lies in the geometry. Work is force multiplied by displacement along the force — a dot product, which peaks when the two are parallel. Torque is force multiplied by distance perpendicular to the force — a cross product, which peaks when the two are at right angles and vanishes when they are parallel. They are opposite cases of the same pair of vectors, which is why the sine appears in one and a cosine in the other.
The practical consequence is that a torque can exist without any work being done at all. Hold a spanner against a seized bolt and you are applying substantial torque; nothing rotates, no displacement occurs, and no work is done on the bolt. Work only appears once rotation happens, and then it is torque multiplied by the angle turned in radians. That is also where power comes from: torque times angular velocity gives watts, which is the relationship behind every engine's power curve. The work calculator handles the linear version of that arithmetic.
The Angle People Get Wrong
The angle in the formula is between the lever arm and the force, and the most common mistake is to measure something else. It is not the angle of the spanner to the ground, nor the rotation of the fastener, nor the angle to some convenient reference edge. Draw the line from the pivot to where the force is applied, draw the force as an arrow from that same point, and measure between them.
A related trap is the extension bar. Putting a torque wrench on a bolt through an extension that runs straight out from the wrench, in line with its handle, genuinely increases the torque delivered for a given wrench reading, because the effective lever arm has grown. Putting the same extension in line with the socket, along the axis of rotation, changes nothing at all, because it does not alter the perpendicular distance. Whether an adapter matters depends entirely on which direction it extends in, which is precisely what the effective lever arm figure captures.
A third case is a force that does not point where you think it does. Pulling on a rope wrapped around a pulley or a capstan produces a force along the rope's tangent at the contact point, not along the line from the axis to your hand. In those situations the honest approach is to work out the perpendicular distance from the axis to the line of action geometrically and enter that as the lever arm with an angle of 90 degrees, which is mathematically identical and much harder to get wrong.
Net Torque and Rotational Equilibrium
As with linear forces, it is the net torque that determines whether something actually rotates. Torques about the same axis add if they turn the same way and subtract if they oppose, and an object in rotational equilibrium has them summing to zero. A seesaw balances when the torque from each side matches, which is why a light child far from the pivot can balance a heavy adult close to it.
This is the basis of every lever, and it is why the mode that solves for force is useful. Enter the torque you need and the radius available, and you get the force required. Enter it again with a larger radius and the force falls in exact proportion. That trade is the whole of mechanical advantage, and the same principle drives gearing, where the ratio between shaft speeds is also the inverse ratio between torques. The gear ratio calculator handles that transformation directly.
Sign matters when torques oppose. By convention, counter-clockwise is taken as positive and clockwise as negative, matching the right-hand rule that makes torque a vector along the axis of rotation. This calculator returns magnitudes, since a single force about a single pivot has an unambiguous sense, but when you combine several you must assign signs consistently before adding them.
How This Differs From the Torque Converter
The boundary in one sentence: the torque converter rescales a torque you already have between newton metres, pound-feet and other units, while this page derives a torque from a force and a geometry. One reformats a quantity, the other computes one that was never entered.
The rest of the mechanics set divides similarly. The force calculator derives the linear force that might then be applied here, and the force converter rescales it. The angular velocity calculator supplies the rotation rate that turns torque into power, the watts to horsepower converter handles the output units of that calculation, and the angle converter moves between degrees and radians if your angle arrives in the wrong one.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Assuming the force is perpendicular — at 45 degrees you lose nearly a third of the torque, and at 30 degrees exactly half.
- Measuring the lever arm to the wrong point — it runs from the pivot to where the force is actually applied, not to the end of the tool.
- Treating newton metres and joules as interchangeable — the dimensions match but the geometry does not, and torque can exist with no work done at all.
- Misjudging extension bars — one that extends the handle changes the torque, one that extends along the rotation axis does not.
- Adding opposing torques without signs — torques that turn opposite ways subtract, and net torque is what determines rotation.
Related Free Tools From Arb Digital
Rescale results with the torque converter, the force converter or the angle converter. For the linear side of the same physics, use the force calculator and the work calculator. For rotation, the angular velocity calculator gives the rate that turns torque into power, the gear ratio calculator handles the trade between speed and torque through a drivetrain, and the watts to horsepower tool converts the result. The full free online tools hub lists everything.
Frequently Asked Questions
It is the angle between the lever arm and the force, measured from the line running from the pivot to the point of application to the direction the force acts. It is not the angle of the tool to the ground or to any other reference.
The dimensions are the same but the geometry is opposite. Work multiplies force by displacement along the force, peaking when they are parallel; torque multiplies force by distance perpendicular to it, peaking at right angles. Keeping the names separate prevents the two being confused.
The loss follows the sine. At 60 degrees you keep 87 per cent, at 45 degrees 71 per cent and at 30 degrees exactly half. At zero degrees, with the force straight along the arm, you produce no torque at all.
Yes, in direct proportion. Doubling the distance from the pivot doubles the torque for the same force. That is the whole principle of a lever, and it is why breaker bars exist.
Only if it extends the handle, lengthening the perpendicular distance from the axis. An extension that runs along the axis of rotation, between wrench and socket, does not change the lever arm and so does not change the torque.
Because the maximum possible torque for a given force and radius is their product, reached at 90 degrees. If you ask for more than that, no angle can produce it, and the calculator says so rather than returning a meaningless value.
Yes. Push on a spanner against a seized bolt and you apply real torque, but nothing rotates, so no work is done. Work only appears once there is angular displacement, and it equals torque multiplied by the angle turned in radians.
The converter rescales a torque you already have between newton metres, pound-feet and other units. This calculator derives a torque from a force, a distance and an angle, producing a number that was never entered.
This tool is provided for educational and estimating use. It computes the torque from a single force about a single pivot and does not account for friction, thread pre-load, fastener condition or any safety factor, so treat its output as a physics result rather than a tightening specification. Always follow the manufacturer's stated torque figures for fasteners.
