Torque Calculator

Calculate torque (rotational force), applied force, or lever arm length using τ = r·F·sin(θ).

🔧 Rotation📐 τ = r·F·sinθ⚙️ Mechanics
Force (F) Newtons
Radius / Lever arm (r) m
Angle between F and r (°)
⚠️ Please enter valid positive numbers.

What Is Torque?

Torque (τ, tau) is the rotational equivalent of force. Just as a linear force causes linear acceleration (F = ma), torque causes angular acceleration (τ = Iα). It measures the tendency of a force to rotate an object about a pivot point. The formula is τ = r · F · sin(θ), where r is the distance from the pivot to the point of force application (the lever arm), F is the applied force, and θ is the angle between the force vector and the lever arm.

The SI unit of torque is the newton-meter (N·m). Note this is dimensionally the same as joules (J = N·m), but torque is not energy — it is a tendency to rotate, not an energy transfer. When θ = 90° (force perpendicular to the lever arm), sin(90°) = 1 and torque is maximized: τ = r·F. This is why engineers make wrenches long — a longer r produces more torque for the same force.

Torque is ubiquitous in engineering: engine output is rated in N·m (or lb·ft in imperial); steering systems specify rack torque; structural bolts have torque specifications; electric motors are characterized by their rated torque. The torque-speed relationship P = τ·ω (power = torque × angular velocity) governs all rotating machinery design.

When multiple torques act on an object, they sum algebraically (clockwise negative, counterclockwise positive by convention). For a rigid body in rotational equilibrium (not accelerating rotationally), the net torque = 0: Στ = 0. This principle, along with ΣF = 0, allows engineers to solve static problems involving beams, levers, and structures.

Formula Reference Table

QuantityFormulaNotes
Torque (τ)τ = r · F · sin θN·m; θ = angle between F and r
Force (F)F = τ / (r · sin θ)sin θ ≠ 0
Lever arm (r)r = τ / (F · sin θ)Perpendicular distance from pivot
Rotational 2nd Lawτ = I · αI = moment of inertia, α = angular accel
Power (rotating)P = τ · ωω in rad/s
EquilibriumΣ τ = 0No net rotation

3 Worked Examples

Example 1
Wrench on a Bolt

A 40 cm (0.4 m) wrench is turned with 75 N of force perpendicular to the handle. Find torque.

  • θ = 90° so sin(90°) = 1
  • τ = r × F × sin θ = 0.4 × 75 × 1
  • τ = 30 N·m
✓ Torque = 30 N·m
Example 2
Engine Torque — Find Force

An engine produces 250 N·m torque via a 0.05 m crankshaft radius. What tangential force is generated?

  • F = τ / (r × sin θ) = 250 / (0.05 × 1)
  • F = 250 / 0.05 = 5,000 N
  • This force drives the piston connecting rod
✓ Force = 5,000 N (5 kN)
Example 3
Angled Force on a Lever

A 0.6 m lever has a 200 N force at 30° to the bar. Find torque.

  • sin(30°) = 0.5
  • τ = 0.6 × 200 × 0.5 = 60 N·m
  • Compared to 90°: 0.6 × 200 × 1 = 120 N·m — angle halves effectiveness
✓ Torque = 60 N·m (half the perpendicular case)

Real-World Applications

🔩
Fastener Torque
Nuts and bolts are torqued to specification (e.g., 80 N·m for a cylinder head bolt). Too little = loose joint; too much = stretched/broken fastener. Torque wrenches ensure precise r·F application.
🚗
Engine Output
Car engines are rated by torque (e.g., 400 N·m at 2,000 RPM). Torque determines acceleration feel; power (P = τω) determines top speed. Diesel engines deliver high torque at low RPM.
🏗️
Structural Engineering
Beams, columns, and joints in buildings experience torque from eccentric loads. Engineers calculate Στ = 0 to ensure structural members don't rotate or fail under combined loading.
⚙️
Gears and Transmissions
Gear ratios trade rotational speed for torque. A 4:1 reduction gives 4× the torque at ¼ the speed. τ_out = τ_in × gear_ratio × efficiency.
🦾
Robotic Joints
Robot arm joints are selected by the torque required to hold and move loads at a given reach. τ = r × m × g for a mass m at distance r — critical for motor sizing.

Common Mistakes to Avoid

⚠️
Using diameter instead of radius

τ = r·F·sin θ uses r = distance from pivot to force point. If you're given a diameter (e.g., a 60 mm bolt circle), use r = 30 mm = 0.030 m. Using the diameter gives twice the torque.

⚠️
Forgetting the angle (assuming sin θ = 1)

τ = r·F only when force is perfectly perpendicular (θ = 90°). Any other angle reduces effective torque. A force at 30° gives sin(30°) = 0.5 — half the maximum torque.

⚠️
Confusing torque units with energy units

Both torque (N·m) and work (J = N·m) have the same SI units, but they are different quantities. Torque is a rotational tendency; work is energy transferred. Never add them or treat them as equivalent.

⚠️
Sign errors in rotational equilibrium

Convention: counterclockwise torques are positive, clockwise negative. Mixing this up when applying Στ = 0 gives wrong answers for unknown forces or distances.

⚠️
Using force magnitude instead of perpendicular component

Only the component of force perpendicular to the lever arm produces torque. A force parallel to the lever arm (θ = 0°) produces zero torque regardless of magnitude.

Frequently Asked Questions

What is the difference between torque and force? ▾
Force (F, newtons) causes linear acceleration per F = ma. Torque (τ, N·m) causes rotational (angular) acceleration per τ = Iα. A force applied far from a pivot produces more torque than the same force close to the pivot — torque is force amplified by lever arm length.
Why is a longer wrench easier to turn a bolt? ▾
A longer wrench has a larger lever arm r. Since τ = r·F·sin θ, more r means more torque for the same applied force F. Doubling wrench length doubles torque — this is the principle of mechanical advantage in levers.
What is the unit of torque and why not joules? ▾
Torque is measured in N·m (newton-meters). While J = N·m dimensionally, joules measure energy (a scalar), while torque measures rotational tendency (technically a pseudovector). Using J for torque is incorrect and confusing — torque is written as N·m to distinguish it clearly.
How does torque relate to angular acceleration? ▾
τ = I·α where I is the moment of inertia (kg·m²) and α is angular acceleration (rad/s²). This is the rotational analog of F = ma. A larger I (mass farther from the axis) requires more torque to achieve the same angular acceleration.
What is net torque and torque balance? ▾
Net torque = Στ (sum of all torques, with signs). For rotational equilibrium (no angular acceleration), Στ = 0. For a seesaw: if τ_left = τ_right, it balances. Engineers use Στ = 0 with Στ = Iα to design balanced structures and rotating systems.
How is torque used to rate engines? ▾
Engine torque at the crankshaft measures the rotational force the engine produces. At a given RPM, torque × angular velocity gives power: P(W) = τ(N·m) × ω(rad/s), or P(kW) = τ(N·m) × RPM / 9,549. A car with 300 N·m at 4,000 RPM produces 300 × 4000/9549 = 125.7 kW ≈ 169 hp.
Can torque be zero even when a force is applied? ▾
Yes — if the force is directed through the pivot point (r = 0) or is parallel to the lever arm (θ = 0°, sin θ = 0), the torque is zero. This is why a door handle placed at the pivot would be impossible to open: no lever arm = no torque.
How does torque apply to bike pedaling? ▾
When you press a pedal, torque = r × F × sin θ where r is the crank length (~175 mm) and F is your leg force. At the top and bottom of the stroke, the force is nearly parallel to the crank (θ ≈ 0°), giving minimal torque. Maximum torque occurs at 3 and 9 o'clock positions where θ ≈ 90°.

Related Physics Calculators

🔄 Angular Velocity ⚙️ Moment of Inertia 🔧 Newton's 2nd Law 💡 Power 🌀 Angular Momentum

Formula Explorer connections

Interpretation: This formula is the rotational counterpart of linear mechanics, relating angle, angular motion, torque, inertia or rotational energy. Assumption: Define the rotation axis and sign convention. Rigid-body behavior, no slipping, steady rotation or negligible bearing losses may be assumed.

Banked Curve Calculator →Circular Motion Calculator →Gyroscope Precession Calculator →Physics Formula Explorer →