Torque Calculator

Free Online Torque Calculator (τ = F × r × sin θ)

This torque calculator instantly solves for torque, force, or lever-arm radius using the standard rotational-force formula, τ = F × r × sin(θ). Enter any combination of known values — force and radius, torque and radius, or torque and force — along with the angle between the force and the lever arm, and the calculator converts every unit automatically, works through the complete step-by-step solution, and draws a labeled lever-arm diagram so the whole calculation is visible at a glance.

Whether you're a physics student solving a torque word problem, a mechanic tightening a bolt to a manufacturer's torque spec, or an engineer sizing a motor or wrench, this tool works in newtons, kilonewtons, pounds-force, dynes, and kilogram-force for force; meters, centimeters, millimeters, feet, and inches for the lever arm; and newton-meters, kilonewton-meters, foot-pounds, inch-pounds, kilogram-force meters, or dyne-centimeters for the torque result — and it always shows the full working, not just a final number.

What Is Torque? (Definition and Formula)

Torque, sometimes called the moment of force, is the rotational equivalent of a straight-line force. Instead of causing an object to speed up or slow down along a line, torque causes an object to rotate around a pivot, axis, or fulcrum. Every time you turn a steering wheel, tighten a bolt with a wrench, open a door, or pedal a bicycle, you are applying torque.

The general torque formula accounts for the fact that a force applied at an angle is less effective at producing rotation than a force applied perpendicular to the lever arm:

• Torque (τ) = Force (F) × Radius (r) × sin(θ)
• F is the magnitude of the applied force, r is the distance from the pivot to the point where the force acts (the lever arm), and θ is the angle between the force vector and the lever arm.
• The SI unit of torque is the newton-meter (N·m) — the turning effect produced by a 1 N force applied 1 m from the pivot, at a full 90° angle.
• Example: a 50 N force applied 0.3 m from a bolt, perpendicular to the wrench (θ = 90°, so sin θ = 1), gives Torque = 50 × 0.3 × 1 = 15 N·m.

Why the Angle θ Matters

Only the component of force that is perpendicular to the lever arm actually contributes to rotation — pulling or pushing straight along the lever arm (θ = 0° or 180°) produces zero torque no matter how large the force is, because sin(0°) = sin(180°) = 0. Torque is greatest when the force is applied at exactly 90° to the lever arm, since sin(90°) = 1, which is why wrenches and door handles are designed to be pushed or pulled perpendicular to their length for maximum effect with minimum effort.

This is also why this calculator asks for the angle explicitly — most textbook 'torque = force × distance' shortcuts silently assume a 90° angle, but real mechanical situations (an angled wrench swing, a slanted door push, a diagonal pedal stroke) need the full τ = F × r × sin(θ) formula to be accurate.

How to Find Force from Torque and Radius

If you already know the torque required (for example, a manufacturer's torque spec for a bolt) and the length of your wrench or lever arm, you can rearrange the formula to solve for the force you need to apply.

• Force (F) = Torque (τ) ÷ (Radius (r) × sin θ)
• Example: a bolt spec calls for 40 N·m of torque and your wrench is 0.4 m long, applied at 90° → Force = 40 ÷ (0.4 × 1) = 100 N.
• This rearrangement comes from dividing both sides of τ = F × r × sin θ by (r × sin θ).

How to Find Radius (Lever Arm) from Torque and Force

If you know the torque you need to produce and how much force you're able to apply, you can solve for how long the lever arm needs to be.

• Radius (r) = Torque (τ) ÷ (Force (F) × sin θ)
• Example: you need 60 N·m of torque and can only apply 200 N of force at 90° → Radius = 60 ÷ (200 × 1) = 0.3 m.
• This rearrangement comes from dividing both sides of τ = F × r × sin θ by (F × sin θ).

The Lever-Arm Diagram Explained

The diagram above draws the exact scenario described by your numbers: a pivot point, a lever arm of length r extending outward (shown in green), a force vector F applied at the end at your chosen angle θ (shown in blue), and a curved amber arrow showing the resulting torque τ and the direction of rotation it produces. Every number from your input — force, radius, angle, and the final torque — is labeled directly on the diagram, so you can see at a glance how each quantity fits into the physical picture, not just the equation.

Common Torque, Force, and Length Units

Torque, force, and lever-arm length are each expressed in several different unit systems depending on whether you're working in SI (metric), imperial, or CGS units. This calculator supports the units most commonly seen in physics coursework and mechanical engineering specs.

• Torque: newton-meters (N·m), kilonewton-meters (kN·m), foot-pounds (ft·lb), inch-pounds (in·lb), kilogram-force meters (kgf·m), dyne-centimeters (dyn·cm)
• Force: newtons (N), kilonewtons (kN), pounds-force (lbf), dynes (dyn), kilogram-force (kgf)
• Length / radius: meters (m), centimeters (cm), millimeters (mm), feet (ft), inches (in)
• 1 N·m = 0.737562 ft·lb = 8.850746 in·lb = 0.101972 kgf·m = 10,000,000 dyn·cm
• 1 ft·lb = 1.355818 N·m, which is why torque wrenches sold in the US are often marked in ft·lb while metric specs use N·m

How to Use This Torque Calculator

Choose what you want to find — Torque, Force, or Radius — from the dropdown at the top. Fill in the two remaining values you already know, in whichever units match your problem, and enter the angle θ between the applied force and the lever arm (use 90° if the force is applied perpendicular to the lever, which is the most common case for wrenches and door handles). The calculator instantly converts everything to a common base unit, applies the torque formula, and shows the result in your chosen unit — plus automatically converted into newton-meters, foot-pounds, and kilogram-force meters for quick reference.

Scroll down to the step-by-step solution for the complete working shown exactly the way you'd solve it by hand: the formula first, then your real values substituted in with unit conversion made explicit, then sin(θ) calculated, and finally the arithmetic carried through to the answer.

Worked Examples

Finding torque: a mechanic applies 50 N to a wrench 0.3 m long, perpendicular to the bolt → Torque = 50 × 0.3 × sin(90°) = 15 N·m.

Finding torque at an angle: a 100 N force is applied 0.5 m from a pivot at a 30° angle → Torque = 100 × 0.5 × sin(30°) = 100 × 0.5 × 0.5 = 25 N·m.

Finding force: a bolt requires 40 N·m of torque with a 0.4 m wrench at 90° → Force = 40 ÷ (0.4 × 1) = 100 N.

Finding radius: 60 N·m of torque is needed and only 200 N of force is available at 90° → Radius = 60 ÷ (200 × 1) = 0.3 m.

Unit conversion example: 120 ft·lb of torque equals 120 × 1.355818 ≈ 162.7 N·m, since 1 ft·lb = 1.355818 N·m.

Torque vs. Force: What's the Difference?

Force and torque are closely related but describe different kinds of motion. Force (measured in newtons) causes straight-line acceleration, following Newton's second law, F = m × a. Torque (measured in newton-meters) causes rotational or angular acceleration around a pivot, and depends not just on how hard you push or pull, but also on how far from the pivot and at what angle you apply that push or pull. The same force applied farther from the pivot, or applied more perpendicularly, always produces more torque — which is exactly why a longer wrench, or a wrench swung at 90° instead of a shallow angle, makes a bolt easier to turn.

Real-World Applications of Torque

The torque formula is one of the most widely used equations in mechanical and automotive engineering, as well as everyday tool use:

• Automotive engineering — engine torque ratings, wheel lug-nut torque specs, and transmission gearing all depend directly on τ = F × r × sin θ.
• Hand tools — torque wrenches are calibrated to apply an exact torque so bolts are tightened correctly without stripping threads or leaving them loose.
• Mechanical engineering — sizing motors, gears, and shafts requires knowing the torque they must deliver or withstand.
• Sports and biomechanics — analyzing the torque a pitcher's arm or a cyclist's leg generates around a joint.
• Everyday physics — opening doors, turning steering wheels, and using seesaws are all direct applications of the lever-arm torque formula.

Net Torque and Multiple Forces

Just as with linear force, more than one torque can act on an object at the same time — for example, a see-saw with people sitting at different distances from the pivot on each side. When several torques act together, the net torque is the sum of each individual torque, with torques rotating clockwise typically taken as negative and counter-clockwise as positive (or vice versa, as long as the convention stays consistent). An object in rotational equilibrium has a net torque of zero, meaning all the clockwise torques exactly balance all the counter-clockwise torques. This calculator computes a single τ = F × r × sin θ directly, so for problems involving multiple forces, calculate each torque separately and combine them by hand using the appropriate sign convention.

Tips for Solving Torque Word Problems

Most torque problems in physics courses and mechanical specs follow a predictable structure, and a few habits make them much easier to get right:

• Identify which two of the three quantities (force, radius, torque) are given, and which one you're solving for — that tells you whether to use τ = F × r × sin θ, F = τ ÷ (r × sin θ), or r = τ ÷ (F × sin θ).
• Always check the angle θ — if a problem doesn't mention an angle, it usually means the force is applied perpendicular to the lever arm, so θ = 90° and sin θ = 1.
• Convert every value to a consistent unit system (typically newtons, meters, and newton-meters) before doing any arithmetic, to avoid mixing metric and imperial units.
• Sanity-check your final answer: hand-tool torques are usually in the range of a few N·m up to a few hundred N·m; engine and industrial torques can run into the thousands. A wildly out-of-range answer usually means a unit conversion was missed.

Frequently Asked Questions

What is the formula for torque? Torque = Force × Radius × sin(θ), where θ is the angle between the force and the lever arm. When the force is perpendicular to the lever (θ = 90°), this simplifies to Torque = Force × Radius.

How do you calculate torque in newton-meters? Multiply the applied force in newtons by the lever-arm distance in meters, then multiply by sin(θ). For example, a 50 N force applied 0.3 m from the pivot at 90° gives 50 × 0.3 × 1 = 15 N·m.

What is the difference between torque and moment of force? Torque and moment of force refer to the same physical quantity — the turning effect of a force around a pivot — and the terms are used interchangeably in physics and engineering.

How do I convert newton-meters to foot-pounds? Multiply the value in newton-meters by 0.737562. For example, 15 N·m × 0.737562 ≈ 11.06 ft·lb.

Can this calculator solve for force or radius, not just torque? Yes — switch the 'What do you want to find?' dropdown to Force or Radius, and the calculator automatically rearranges the torque formula and solves for whichever value you need.

What angle gives the maximum torque? A 90° angle between the force and the lever arm gives the maximum torque for a given force and radius, since sin(90°) = 1.