Torque Calculator

Find Torque, Force, or Lever Arm

Torque Formula

Formula

\[ \tau = r \cdot F \cdot \sin(\theta) \]

Where:

\(\tau\) — torque (N·m)
\(r\) — lever arm length, the perpendicular distance from the pivot to the point where the force is applied (m)
\(F\) — applied force (N)
\(\theta\) — angle between the lever arm and the force direction (degrees or radians)

The formula expresses the rotational effect of a force: torque grows with a longer lever arm, a larger force, and is greatest when the force acts perpendicular to the arm (\(\theta = 90^\circ\)).

The same relationship can be rearranged to solve for force or lever arm:

\[ F = \frac{\tau}{r \cdot \sin(\theta)} \qquad r = \frac{\tau}{F \cdot \sin(\theta)} \]

Torque – Calculation Example

Example: you push a wrench with a force of 100 N at a point 0.25 m from the bolt, perpendicular to the handle.

\[ \tau = r \cdot F \cdot \sin(\theta) = 0.25 \times 100 \times \sin(90^\circ) \]

\[ \tau = 0.25 \times 100 \times 1 = 25 \text{ N·m} \]

Result: the torque is 25 N·m (≈ 18.44 lb·ft, ≈ 2.55 kgf·m).

How to Use the Torque Calculator

In the Solve for dropdown, choose whether you want to calculate Torque (τ), Force (F), or Lever arm (r). The input for the selected quantity is hidden; you enter the other two.
Enter the known Torque (τ) value and pick its unit (N·m, lb·ft, or kgf·m) — only shown when solving for force or lever arm.
Enter the known Force (F) value with its unit (N, lbf, or kgf).
Enter the Lever arm length (r) and choose a unit (m, cm, mm, ft, or in).
Enter the Angle between force and arm (θ) in degrees. Use 90° if the force acts perpendicular to the lever.
Read the solved quantity in your chosen unit, plus equivalent values in other common units, and a summary table of all quantities.

What Is Torque?

Torque (also called the moment of force) is the rotational analogue of force. While a linear force pushes an object in a straight line, torque measures how effectively a force causes an object to rotate around a pivot point — the axis of a bolt, a door hinge, a wheel hub, or a joint in the human body. Torque is what tightens a bolt, opens a door, and turns a crankshaft.

The value of torque depends on three things: how hard you push (the force), where you push (the lever arm length), and the direction you push (the angle). The lever arm is the distance from the pivot axis to the point where the force is applied, and only the component of force perpendicular to that arm contributes to rotation — this is where the \(\sin(\theta)\) factor comes from. At 90° the full force is effective; at 0° or 180°, when the force points along the arm, no rotation is produced at all.

This is exactly why door handles are placed far from the hinges, why long breaker bars loosen stubborn bolts, and why it is easier to turn a wrench by pushing at its end rather than near the head. Doubling the lever arm doubles the torque for the same effort, and vice versa.

The SI unit of torque is the newton-meter (N·m). Note that although N·m is dimensionally the same as the joule, torque is never expressed in joules — it is a rotational quantity, not energy. In imperial practice, torque is usually given in pound-feet (lb·ft). If you need to relate torque to energy or motion, remember that the work done by a constant torque over an angular displacement is \(W = \tau \varphi\); see the Work and Power calculators for the linear equivalents.

Assumptions, Limitations, and Practical Tips

Single force, single pivot. The calculator treats one force acting at one point on a rigid lever about a fixed axis. Real systems with multiple forces or distributed loads require summing torques separately.
The angle matters. Many people assume the force is always perpendicular. If you pull a wrench at a shallow angle, the effective torque drops quickly — at 30° you get only half the torque of a 90° pull.
Direction of rotation is not shown. The result is a magnitude. By convention, counterclockwise torque is often taken as positive and clockwise as negative; assign signs yourself when balancing multiple torques.
Static vs. dynamic situations. This tool computes the torque itself, not the resulting angular acceleration. For rotating machinery, torque relates to angular acceleration through the moment of inertia, \(\tau = I\alpha\).
Unit consistency. The calculator converts all units internally, so you can mix, say, lbf with meters. When doing manual calculations, keep everything in one consistent system.
Torque wrenches and specs. Bolt tightening specifications are given in torque units (e.g. N·m or lb·ft). To hit a target torque by hand, solve for the force to find how hard to pull at a known lever length.

For problems involving forces on a rotating or constrained system — such as a rope passing over a pulley or an object on a slope — you may also find the Tension in Ropes and Inclined Plane calculators useful.

Frequently Asked Questions

How do I calculate torque by hand without a calculator?

Multiply the lever arm length by the force and by the sine of the angle between them: \( \tau = r \cdot F \cdot \sin(\theta) \). For example, 0.3 m times 200 N at a 90° angle gives \( 0.3 \times 200 \times 1 = 60 \) N·m. For other angles, find the sine first — at 45° it is about 0.707, so the same push would give only about 42.4 N·m.

How do I convert between N·m, lb·ft, and kgf·m?

1 N·m = 0.7376 lb·ft = 0.1020 kgf·m. In reverse, 1 lb·ft = 1.3558 N·m and 1 kgf·m = 9.807 N·m. To convert, multiply by the appropriate factor; for example, 50 N·m is about 36.9 lb·ft. Note that lb·ft (pound-feet, torque) is sometimes written ft·lb — both are acceptable for torque, unlike ft·lbf used for energy.

What happens if the force is applied at 0° or 180°?

The torque is zero, because \( \sin(0^\circ) = \sin(180^\circ) = 0 \). A force acting exactly along the lever arm pushes or pulls the arm toward or away from the pivot and produces no rotation at all — like trying to turn a wrench by pulling straight out along its handle. If you try to solve for force or lever arm at these angles, the calculation is mathematically impossible.

Why does the angle reduce torque, and how much does it matter?

Only the component of force perpendicular to the lever arm causes rotation, and that component is \( F \sin(\theta) \). The penalty is significant: at 60° you get 87% of maximum torque, at 45° only 71%, and at 30° exactly half. Pulling a wrench at a shallow 10° angle wastes 83% of your effort — pushing perpendicular is always the most efficient strategy.

What is the difference between lb·ft and lbf, and can I mix units?

lbf (pound-force) is a unit of force; lb·ft (pound-foot) is a unit of torque — force times distance. This calculator converts units internally, so you can safely mix, for example, lbf with a lever arm in meters. When calculating manually, convert everything to one system first: in imperial, use lbf with feet to get lb·ft; in SI, use newtons with meters to get N·m.

Is torque the same as work or power?

No. Torque is a rotational force measured in N·m; although the units look like joules, torque is never expressed in joules. Work is done only when something actually rotates: \( W = \tau \varphi \), where \( \varphi \) is the angle turned in radians. Power adds time: \( P = \tau \omega \), with \( \omega \) the angular speed in radians per second. A static bolt wrench applies torque but zero work.

How do I use this to hit a torque spec on a bolt without a torque wrench?

Choose 'Force' as the quantity to solve for, enter the required torque spec (e.g. 40 N·m), your lever arm length, and 90° for the angle. The result tells you how hard to pull at the end of the wrench. For instance, 40 N·m with a 0.5 m handle needs about 80 N of pull. Note this is an approximation — for critical or safety-related fasteners, an actual torque wrench is strongly recommended.