Torque Calculator

What this torque calculator does

This calculator computes the magnitude of torque produced when a force acts at a distance from a pivot, with a specified angle between the force and the lever arm. You enter three values:

• Force F (N) – the applied force in newtons
• Lever arm r (m) – the perpendicular distance from the pivot to where the force is applied, in meters
• Angle θ (degrees) – the angle between the lever arm vector and the force vector, in degrees

The tool then returns the torque magnitude in newton‑meters (N·m) using the standard cross‑product definition. It is intended for quick engineering, physics, and mechanical design calculations where you need an approximate rotational effect of a force.

How to use the calculator

1. Enter the force F in newtons (N). If you know the mass in kilograms that gravity is acting on, you can estimate force as F ≈ 9.81 × mass.
2. Enter the lever arm r in meters (m). This is the distance from the rotation axis (hinge, bolt center, shaft, etc.) to the point where the force is applied, measured along the lever arm.
3. Enter the angle θ in degrees. This is the angle between the lever arm direction (from pivot to force application point) and the direction of the force.
4. Press the button to compute torque. The output is the torque magnitude in N·m.

For many practical cases, such as pushing perpendicular to a wrench or door, the angle is close to 90°. That is why the default angle is set to 90°, which gives the maximum torque for a given force and lever arm.

Torque definition and vector form

Torque describes how strongly a force tends to rotate an object about some axis. When you push on a door away from its hinges, you create torque that causes it to swing. The same idea applies to wrenches turning bolts, crank arms turning bicycle wheels, and many other mechanisms.

In vector form, torque is defined as the cross product

τ⃗ = r⃗ × F⃗

where:

• r⃗ is the position vector from the pivot or rotation axis to the point where the force is applied
• F⃗ is the applied force vector
• τ⃗ is the resulting torque vector, perpendicular to both r⃗ and F⃗

The direction of τ⃗ is given by the right‑hand rule, but this calculator only computes the magnitude of torque, not its direction.

Torque magnitude formula

The magnitude of the torque vector from the cross‑product definition is

τ = r F sin(θ)

where:

• τ is the magnitude of the torque (in N·m)
• r is the distance from the pivot to the point of force application (in m)
• F is the magnitude of the force (in N)
• θ is the angle between r⃗ and F⃗ (in radians in the pure formula, but the calculator accepts degrees and converts internally)

Only the component of the force that is perpendicular to the lever arm produces torque. That is why the sine of the angle appears in the formula:

• If θ = 0° or 180° (force along the lever arm), then sin(θ) = 0 and the torque is zero.
• If θ = 90°, then sin(θ) = 1 and the torque is maximized for the given r and F.

MathML representation

The same formula can be written using MathML as:

This is exactly the relationship implemented by the torque calculator.

Interpreting the results

The output of the calculator is a single number in newton‑meters (N·m), representing the magnitude of the torque about the specified pivot. Some ways to interpret the result:

• Higher torque means a greater tendency to cause rotational motion around the pivot.
• For tightening or loosening bolts, a higher torque value usually means more clamping force on the joint (within the limits of the fastener and material).
• For a given required torque, you can trade off force and lever arm: a longer lever arm lets you use a smaller force.

Remember that this calculator does not tell you the direction (clockwise or counterclockwise) of the torque, only how large it is. It also does not account for friction, deformation, or dynamic effects like acceleration or vibration.

Worked example: tightening a bolt with a wrench

Suppose you are using a wrench that is 0.25 m long to loosen a bolt, and you push straight down on the end of the wrench with a force of 200 N.

• Force F = 200 N
• Lever arm r = 0.25 m
• Angle θ = 90° (force is perpendicular to the wrench)

Using the formula τ = r F sin(θ):

• sin(90°) = 1
• τ = 0.25 × 200 × 1 = 50 N·m

If your design or procedure requires about 50 N·m of torque to loosen this bolt, the chosen wrench length and applied force are sufficient. If you needed more torque, you could either increase the force (push harder) or use a longer wrench.

Worked example: pushing a door

Imagine a door that is 0.9 m wide. You push at the outer edge of the door with a force of 50 N, but your push is not perfectly perpendicular; instead, the angle between the door (lever arm) and your force is 60°.

• Force F = 50 N
• Lever arm r = 0.9 m
• Angle θ = 60°

Compute the torque:

• sin(60°) ≈ 0.866
• τ = 0.9 × 50 × 0.866 ≈ 38.97 N·m

If you instead pushed exactly perpendicular to the door (θ = 90°), you would have:

• τ = 0.9 × 50 × 1 = 45 N·m

This shows how the angle reduces the effective torque: a non‑perpendicular push produces less rotational effect for the same force and lever arm length.

Units of torque and conversions

In the International System of Units (SI), torque is measured in newton‑meters (N·m). This unit is dimensionally equivalent to joules (J), the unit of work and energy, but torque and energy represent different physical concepts:

• Torque is a vector associated with rotation about an axis.
• Energy is a scalar representing the capacity to do work.

In some industries, particularly automotive and mechanical trades, torque is often quoted in pound‑feet (lb·ft) or inch‑pounds (in·lb). If you need to convert between these and N·m, use a dedicated torque unit converter to avoid mistakes and rounding errors.

Practical design insights

Because τ = r F sin(θ), you can quickly see how design choices affect torque:

• Doubling r doubles torque if F and θ stay the same.
• Doubling F doubles torque if r and θ stay the same.
• Optimizing θ towards 90° increases torque without changing r or F.

This is why long handles and perpendicular force application are favored in tools like breaker bars, crowbars, and bicycle pedal cranks.

Comparison of different torque setups

The table below compares several simple scenarios to illustrate how changing force, lever arm, and angle affects the resulting torque magnitude.

Assumptions and limitations

This torque calculator is intentionally simple and makes several assumptions that you should keep in mind, especially for engineering or safety‑critical applications:

• Magnitude only – The tool calculates only the magnitude of torque, using τ = r F sin(θ). It does not indicate the sense of rotation (clockwise vs. counterclockwise) or the full vector direction.
• Planar, rigid‑body model – It assumes forces act in a single plane and that the lever behaves as a rigid body, with no bending, twisting, or flexibility.
• SI units – Inputs are interpreted as newtons (N) for force and meters (m) for lever arm. The result is in newton‑meters (N·m). If your values are given in other units, convert them before using this calculator.
• Angle definition – The angle θ is the geometric angle between r⃗ and F⃗. The calculator expects this angle in degrees; it converts to radians internally. Inconsistent angle definitions are a common source of error.
• No dynamic effects – The calculator ignores acceleration, inertia, damping, vibration, and time‑varying forces. It is best suited to static or quasi‑static situations.
• No friction or losses – It does not account for friction in bearings, joints, or threads, nor for efficiency losses in gears or linkages. Real systems require higher torques than the simple theoretical value.
• Single force application – Only a single force and single lever arm are modeled. Systems with multiple forces or distributed loads require summing torques or more advanced analysis.

For critical designs, use these computations as a starting point and then consult detailed engineering references, safety factors, and relevant standards.

Related tools

If you work regularly with rotational systems, you may also find these tools useful:

• Torque unit converter – convert between N·m, lb·ft, in·lb, and other torque units.
• Force calculator – estimate forces from mass, acceleration, or pressure where needed.
• Work and energy calculator – relate torque and angular displacement to mechanical work.

Using these together with the torque calculator can help you move from simple back‑of‑the‑envelope checks to more complete preliminary designs.