Torque (Moment of Force) Calculator — τ = rFsin(θ)

Enter any 3 of force, lever arm length, and angle to instantly solve for the 4th (torque, force, lever arm, or angle) — with automatic N·m to ft·lb unit conversion.

Last updated: 2026-09-09

Typical tightening torque reference values

Application	Typical torque
Car wheel lug nuts	100 - 140 N·m（74 - 103 ft·lb）
Spark plugs	20 - 30 N·m（15 - 22 ft·lb）
Bicycle stem bolts	5 - 6 N·m（3.7 - 4.4 ft·lb）
Laptop case screws	0.2 - 0.3 N·m（0.15 - 0.22 ft·lb）

What Is Torque (Moment of Force)?

Torque, also called the moment of force, describes how strongly a force tends to rotate an object around a pivot point. The same amount of force produces a much bigger rotational effect when it's applied farther from the pivot, or when it's applied in a direction closer to perpendicular to the lever arm. This relationship is captured by the formula τ = rFsinθ, where torque τ is the product of force F, lever arm length r, and the sine of the angle θ between them.

This tool lets you pick any one of the four variables — torque τ, force F, lever arm length r, or angle θ — as the one you want to solve for, and it automatically derives that value from the other three inputs you provide. Rather than just computing torque in the forward direction, it also handles cases where you need to figure out how much force is required, or what angle would produce a target torque, all on the same screen. Results are always shown in both N·m (newton-meters) and ft·lb (foot-pounds) at once.

How to Use the Torque Calculator

Choose the variable to solve for Use the buttons at the top to pick which one of torque τ, force F, lever arm length r, or angle θ you want to calculate. That variable's input field will disappear automatically.
Enter the remaining three values Fill in the three input fields that remain visible with the values you already know. Force and lever arm length must be zero or greater, and angle must fall between 0° and 180°.
Read the result panel Once all required inputs are filled in, the result panel shows all four values — torque, force, lever arm length, and angle — together, with torque given in both N·m and ft·lb.
Watch for the angle note when solving for θ Because of how sinθ behaves, two mathematical solutions exist for the angle, but this tool always shows the primary, acute-angle solution between 0° and 90°.

Tips for getting more out of it

Choose which variable you want to solve for (τ, F, r, or θ) first — the other three input fields will appear automatically.
Torque is maximized when the angle is 90°, and drops to zero at 0° or 180°, since the force becomes parallel to the lever arm and can no longer produce rotation.
When solving for the angle, two mathematical solutions exist because sinθ = sin(180° − θ); this tool always shows the primary value between 0° and 90°.
Automotive torque wrenches are often calibrated in ft·lb (foot-pounds), so results are always shown in both N·m and ft·lb for convenience.
The lever arm length is the perpendicular distance from the pivot to the line of the applied force — if the force is parallel to the arm (θ = 0°), it produces no rotational effect at all.

When You Might Use This Tool

Physics homework and exam prep

Check your own step-by-step calculation of a moment-of-force problem using τ = rFsinθ before submitting an answer.

Checking mechanical and automotive torque specs

When a bolt's tightening torque is given in ft·lb, convert it to N·m to match the setting on the tool you actually have on hand.

DIY bolt and screw tightening

Estimate in advance whether the force you can apply by hand, combined with your wrench's length, is enough to reach a specified torque value.

Lever and seesaw balance problems

Work out the numeric relationship between distance from the pivot and the magnitude and direction of an applied force.

Torque wrench unit conversion

Quickly convert between N·m and ft·lb when you're working with a mix of metric and imperial tools on the same job.

Glossary

Torque (moment of force): A measure of how strongly a force tends to rotate an object around a pivot. Measured in N·m (newton-meters) and calculated as τ = rFsinθ.
Lever arm length (r): The perpendicular distance from the pivot point to the line of action of the force — not necessarily the physical length of the arm, which can be shorter depending on the force's direction.
N·m (newton-meter): The SI unit of torque. It represents the torque produced when a force of 1 newton is applied perpendicular to a 1-meter lever arm.
ft·lb (foot-pound): The imperial unit of torque, most commonly used in American automotive work. 1 N·m is approximately equal to 0.7376 ft·lb.
Angle of application (θ): The angle between the direction of the applied force and the lever arm. The rotational effect is maximized at 90° and drops to zero at 0° or 180°.
Force (F): The magnitude of the force applied to an object, measured in newtons (N). Both its magnitude and angle affect the resulting torque.
Effective lever arm: The lever arm length r multiplied by sinθ (r × sinθ) — the portion of the arm's length that actually contributes to rotation when the force is applied at an angle.

Frequently Asked Questions

Both share the same unit, N·m (equivalent to a joule), but they measure different things: work is force acting over a distance an object moves, while torque measures the rotational effect around a pivot. To avoid confusion, engineers conventionally write torque as 'N·m' and work/energy as 'J (joules)', even though they're dimensionally identical.

Not necessarily. r is technically the perpendicular distance from the pivot to the line of action of the force. If the force is applied at an angle, the effective lever arm becomes shorter than the arm's physical length — specifically, r × sinθ.

Because the imperial system remains widely used, especially in the United States, torque wrenches calibrated in ft·lb are the standard in much automotive work there. 1 N·m is approximately equal to 0.7376 ft·lb.

sinθ reaches its maximum value of 1 at θ = 90° and then decreases again beyond that point (for example, θ = 150° gives the same sine value as θ = 30°). Since the force direction approaches the opposite of the rotational direction, it is most common in practice to work within the 0°–90° range.

Torque is zero whenever force, lever arm length, or sinθ equals zero. In particular, at θ = 0° (force parallel to the arm) or θ = 180°, applying force produces no rotational effect on the object whatsoever.

Side Note — Why a Longer Wrench Makes Loosening Bolts Easier

The common piece of workshop wisdom — 'the longer the wrench handle, the less force it takes to loosen a stubborn bolt' — is a direct, practical expression of the torque formula τ = rFsinθ. To produce the same torque τ, the required force F is inversely proportional to the lever arm length r. That's exactly why the trick of slipping a length of pipe over a wrench handle (a so-called 'cheater bar') works so well — though exceeding a tool's design limits this way risks breakage or injury, so staying within recommended torque specifications matters for safety.

Torque wrenches are considered essential tools in auto mechanics because under-tightening a bolt risks loosening or falling out, while over-tightening risks yielding the bolt or stripping the threads. Manufacturers specify precise target torque values (usually in N·m or ft·lb) for components like wheel lug nuts and cylinder head bolts, and hitting that number accurately is central to repair quality.

The concept of torque extends far beyond engineering and construction — it shows up in everyday design too. A doorknob is placed near the edge of a door rather than at its center specifically so the force is applied as far as possible from the hinge (the pivot), maximizing torque for a given amount of push and making the door easy to open with minimal effort.

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