2026-08-21 12 min

Torque Calculator

Compute torque from force, distance, and angle using the fundamental rotational mechanics equation. Convert between N·m, lbf·ft, lbf·in, and kgf·m for mechanical engineering, automotive, and industrial applications.

Calculator

Force(N) Help

Magnitude of the applied force

Radius / Distance(m) Help

Distance from axis to force application point

Angle(\u00B0) Help

\u00B0

Angle between force vector and lever arm (0-180\u00B0)

Decimal Precision

Number of decimal places in results

Results update live as you type

Results

Optimal

Torque

50.00N·m

τ = 100.00 × 0.50 × sin(90.00°)

Torque

36.88lbf·ft

Imperial unit — pound-force feet

Torque

442.54lbf·in

Imperial unit — pound-force inches

Torque

5.10kgf·m

Gravitational metric unit — kilogram-force meters

Perpendicular Distance

0.50m

r × sin(θ) — the effective moment arm

Results are exact for the given inputs using the standard torque formula. Real-world applications may require additional factors for friction, material strength, and dynamic loading.

Torque Visualization

Force (F)

100.00 N

Radius (r)

0.50 m

sin(theta)

1.0000

Torque (tau)

50.00 N\u00B7m

Formula

Torque is the cross product of the position vector and the force vector. The scalar form τ = F × r × sin(θ) captures the magnitude, where θ is the angle between the force direction and the lever arm. Maximum torque occurs at 90° (perpendicular). At 0° or 180°, the force is parallel to the lever arm and produces zero torque.

τ = F × r × sin(θ)

Alternative Form

τ = F × d⊥ where d⊥ = r × sin(θ)

Variables

Symbol	Name	Unit	Description
τ	Torque	N·m	The rotational force produced about the axis
F	Force	N	The magnitude of the applied force
r	Radius / Lever Arm	m	Distance from the axis of rotation to the point of force application
θ	Angle	degrees	Angle between the force vector and the lever arm
sin(θ)	Sine of Angle	—	The perpendicular component factor (1 at 90°, 0 at 0° and 180°)
d⊥	Perpendicular Distance	m	The effective moment arm: r × sin(θ)

Explanation

The sine function extracts the component of force perpendicular to the lever arm. When θ = 90°, sin(90°) = 1 and all force contributes to torque. As θ decreases toward 0° or increases toward 180°, more of the force acts along the lever arm (producing no rotation) and less contributes to torque.

Derivation

The torque is defined as the cross product τ = r × F. In two dimensions with angle θ between r and F, the magnitude is |τ| = |r||F|sin(θ). This can be decomposed into the tangential force component F_t = F·sin(θ) acting at distance r, giving τ = r·F_t = F·r·sin(θ).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Identify the applied force F and its direction.

F = given

Force (F):100.00 N

Result: F = 100.00 N

Step 2

Identify the lever arm length r (distance from axis to force application point).

r = given

Radius (r):0.50 m

Result: r = 0.50 m

Step 3

Identify the angle θ between the force vector and the lever arm.

θ = given

Angle (θ):90.00°

Result: θ = 90.00°

Step 4

Calculate sin(θ) to find the perpendicular component factor.

sin(θ)

θ:90.00°

Intermediate: sin(90.00°) = 1.0000

Result: sin(θ) = 1.0000

Step 5

Compute the perpendicular distance (effective moment arm).

d⊥ = r × sin(θ)

r:0.50 m

sin(θ):1.0000

Intermediate: d⊥ = 0.50 × 1.0000 = 0.5000 m

Result: d⊥ = 0.50 m

Step 6

Multiply F × r × sin(θ) to get the torque.

τ = F × r × sin(θ)

F:100.00 N

r:0.50 m

sin(θ):1.0000

Intermediate: τ = 100.00 × 0.50 × 1.0000 = 50.00

Result: τ = 50.00 N·m

Step 7

Convert to other torque units.

1 N·m = 0.7376 lbf·ft = 8.8507 lbf·in = 0.1020 kgf·m

τ (N·m):50.00

Intermediate: lbf·ft = 50.00 × 0.7376; lbf·in = 50.00 × 8.8507; kgf·m = 50.00 × 0.1020

Result: 50.00 N·m = 36.88 lbf·ft = 442.54 lbf·in = 5.10 kgf·m

Final Answer

50.00

Worked Examples

See how the calculator works with different scenarios.

Scenario

A mechanic applies a 200 N force perpendicular to the end of a 0.3 m wrench handle to tighten a bolt.

Inputs

Force:200 N

Radius:0.3 m

Angle:90°

Calculation

sin(90°) = 1.0000 τ = 200 × 0.3 × 1.0000 = 60 N·m

Result

60 N·m (44.25 lbf·ft)

Interpretation

The perpendicular application produces maximum torque. This is the ideal scenario for bolt tightening — all of the applied force contributes to rotational effect.

Understanding Your Result

Your Result: Torque: 50.00 N·m

Typical range: Hand tools: 5-100 N·m. Automotive lug nuts: 80-150 N·m. Engine head bolts: 80-200 N·m. Industrial shafts: 100-10,000+ N·m. The torque value alone does not determine adequacy — it must be evaluated against the application requirements.

What This Means

The torque calculator determines the rotational force produced when a force is applied at a distance from an axis at a given angle. The result represents the twisting moment that causes or tends to cause rotation. The angle factor sin(θ) accounts for the fact that only the perpendicular component of force contributes to torque.

Why It Matters

Torque is fundamental to mechanical engineering: it determines bolt tightening specifications, motor sizing requirements, gear ratios, shaft diameters, and brake performance. Incorrect torque can cause bolt failure (too much), loose connections (too little), or structural damage. Understanding how angle affects torque is critical for proper wrench technique and force application.

Professional Interpretation

In professional practice, torque is always specified with its application context: bolt grade, lubrication state, and assembly method. The sin(θ) factor is often implicitly assumed to be 1 (perpendicular application) unless specifically noted. When forces act at angles, the perpendicular component should be calculated explicitly to avoid underestimating required force or overestimating available torque.

Component decomposition

Any force applied at an angle can be resolved into two components: the tangential component F_t = F·sin(θ) that produces torque, and the radial component F_r = F·cos(θ) that pushes or pulls along the lever arm. Only the tangential component causes rotation.

Cross product interpretation

Torque is formally the vector cross product τ = r × F. In 2D this simplifies to the scalar magnitude |τ| = rF sin(θ). The direction follows the right-hand rule: curl fingers from r toward F, thumb points along the torque vector.

Practical angle consideration

In real-world applications, perfect perpendicularity is rare. A mechanic pulling a wrench at even 10° off-perpendicular loses only about 1.5% of torque (cos(10°) = 0.985). But at 30°, the loss is 13.4%, which can be significant for precision applications.

Recommendations

•Always apply force perpendicular (90°) to the lever arm for maximum torque efficiency
•When angle is unavoidable, increase force proportionally: F_required = F_perpendicular / sin(θ)
•Use calibrated torque wrenches for critical fastener applications
•Account for the sin(θ) reduction when forces are applied at angles
•Consider the full system: torque, power (torque × speed), and structural capacity
•Apply safety factors (typically 1.5-2x) for design applications

Educational Content

Introduction

Torque (τ) is the rotational equivalent of linear force. Just as a force causes linear acceleration, torque causes angular acceleration. It is one of the most fundamental quantities in mechanical engineering, governing everything from how tightly a bolt is secured to how quickly a vehicle accelerates.

Concept Explanation

Torque arises when a force acts at a distance from a pivot point, with the effectiveness depending on the angle of application. The formula τ = F × r × sin(θ) captures all three factors: how hard you push (F), how far from the pivot (r), and how efficiently the push is directed (sin θ). Maximum torque occurs when the force is perpendicular to the lever arm.

Applications

▸Bolt and fastener tightening in automotive, aerospace, and structural engineering

▸Electric motor and engine sizing for industrial and automotive applications

▸Gear train design and power transmission systems

▸Shaft and coupling design in rotating machinery

▸Brake system design and stopping torque calculations

▸Hand tool design (wrenches, screwdrivers, levers)

▸Robotic joint actuator selection

▸Wind turbine and hydraulic motor analysis

Benefits

•Provides the fundamental calculation for all rotational mechanical systems
•Enables proper specification of bolts, motors, and mechanical components
•Accounts for force angle, which is often overlooked in simplified calculations
•Supports unit conversion for international engineering collaboration
•Essential for safety-critical applications in aerospace and automotive
•Foundation for understanding power, angular momentum, and rotational dynamics

Limitations

•Does not account for friction losses in the system
•Assumes rigid body mechanics (no deformation under load)
•Does not consider dynamic effects like angular acceleration or vibration
•Bolt torque calculations require additional factors (nut factor, friction) for accuracy
•The formula gives magnitude only; direction requires vector analysis

Best Practices

•Apply force perpendicular to the lever arm whenever possible for maximum efficiency
•Use calibrated torque wrenches for precision fastener applications
•Always specify the units and angle when communicating torque values
•Account for the angle factor when forces cannot be applied perpendicularly
•Consider both static and dynamic torque requirements in system design
•Apply appropriate safety factors based on application criticality

Common Misconceptions

Myth: Torque and force are the same thing.

Truth: Force is a linear push or pull; torque is the rotational effect of a force applied at a distance. You can apply a large force and produce zero torque if the force passes through the pivot point (θ = 0°).

Myth: A longer wrench always gives you more torque.

Truth: A longer wrench gives more torque only if you can apply the same force at the end. If your grip strength limits the force, the longer wrench may not help. Also, the angle of application matters.

Myth: Torque is only important for rotating machinery.

Truth: Torque is critical in any application involving rotation or twisting: bolt tightening, door hinges, steering systems, wrench use, and even opening a jar lid.

Myth: Higher torque always means better performance.

Truth: Torque must be evaluated in context. An engine with high torque but low RPM may have less power than one with moderate torque at high RPM. The application determines the optimal torque-speed combination.

Practical Tips

•Always try to apply force perpendicular (90°) to the wrench handle for maximum torque
•When working at an angle, compensate by increasing force: F_needed = F_perpendicular / sin(θ)
•Use a torque wrench for critical fasteners rather than estimating by feel
•Remember that doubling the wrench length doubles the torque for the same force
•Check bolt manufacturer specifications for proper torque values
•Consider lubrication effects — lubricated bolts require less torque for the same clamp load
•When converting between N·m and lbf·ft, multiply by 0.7376 or divide by 1.3558

Common Mistakes

Ignoring the angle of force application

Always account for sin(θ). A 30° off-angle application loses 13.4% of available torque. Calculate the perpendicular component explicitly.

Confusing torque with force

Torque depends on both force and distance. A small force at a large distance can produce more torque than a large force applied close to the axis.

Using hand-tight estimates for critical fasteners

Use a calibrated torque wrench for any application where proper clamping force is safety-critical. Hand estimates can vary by ±50% or more.

Forgetting unit conversions when mixing systems

Be consistent with units. If force is in N and distance in m, torque is in N·m. Do not mix N with feet or lbf with meters without converting.

Applying torque to an unlubricated bolt using a lubricated specification

Lubrication reduces friction, so the same torque produces more clamp force. Dry bolts need about 20% more torque than lubricated bolts for the same preload.

Assumptions

This calculator makes the following assumptions to provide estimates:

Rigid Body Assumption: The formula assumes both the lever arm and the point of force application are rigid. In reality, flexible wrenches and deforming components alter the effective geometry.

Point Force Application: Force is assumed to act at a single point at distance r. If force is distributed (e.g., finger pressure across a handle), use the effective centroid distance.

No Friction Losses: The calculation gives the torque produced by the applied force. It does not account for friction in bearings, joints, or thread engagement that consumes part of the input torque.

Static Analysis: The formula calculates static torque. Dynamic effects (angular acceleration, vibration, impact loading) require additional analysis using rotational dynamics equations.

Ideal Geometry: The lever arm is assumed to be straight and the angle is measured from the arm axis. Bent wrenches or offset force application points require geometric correction.

Limitations

Be aware of these limitations when using this calculator:

No Friction Modeling: The calculator does not account for friction losses at the pivot, in bearings, or at thread interfaces. Actual required torque is always higher than theoretical due to friction.

Static Only: This is a static torque calculation. Dynamic torque involving acceleration, deceleration, and impact requires rotational dynamics analysis with moment of inertia.

Material Strength Not Considered: The torque value is calculated but not compared against the material strength of the lever arm, shaft, or fastener. Exceeding material limits causes permanent deformation or failure.

Temperature and Lubrication: Temperature affects material properties and lubricant behavior. The calculator does not account for thermal effects on torque requirements or material capacity.

Frequently Asked Questions

Answers to common questions about this calculator.

What is torque?

Torque is a measure of the rotational force applied to an object about an axis. It is calculated as τ = F × r × sin(θ), where F is the applied force, r is the radius or lever arm length, and θ is the angle between the force and the lever arm. Torque is a vector quantity that causes angular acceleration.

How do you calculate torque?

Torque is calculated using τ = F × r × sin(θ). For example, applying a 50 N force at the end of a 0.3 m wrench with the force applied perpendicular (θ = 90°) produces τ = 50 × 0.3 × 1 = 15 N·m. If the force is applied at an angle, sin(θ) reduces the effective torque.

What is the difference between torque and moment?

While often used interchangeably, torque typically refers to the twisting effect about an axis of a shaft or rotating member, while moment usually refers to the bending effect about a point in a beam. Both have units of N·m and are calculated using the same formula F × d × sin(θ).

What are common torque units?

Common torque units include: newton-meter (N·m) — SI standard; pound-foot (lbf·ft) — US automotive; pound-inch (lbf·in) — smaller fasteners; kilogram-meter (kgf·m) — some industrial applications; kilonewton-meter (kN·m) — heavy machinery.

How do you convert between torque units?

Key conversions: 1 N·m = 0.7376 lbf·ft, 1 lbf·ft = 1.3558 N·m, 1 lbf·in = 0.1130 N·m, 1 kgf·m = 9.8067 N·m. To convert, multiply the value by the appropriate conversion factor.

What is the relationship between torque and horsepower?

Horsepower is related to torque and rotational speed: HP = (Torque × RPM) / 5252 (for lbf·ft and RPM). In SI units: Power (W) = Torque (N·m) × Angular velocity (rad/s). This means an engine producing 300 lbf·ft at 4000 RPM generates 300 × 4000 / 5252 = 228 HP.

How does lever arm length affect torque?

Torque is directly proportional to the lever arm length. Doubling the handle length of a wrench doubles the torque for the same applied force. This is why mechanics use longer breaker bars for stubborn bolts and why cheater pipes can provide additional torque.

What is the torque required to tighten a bolt?

Bolt torque depends on bolt size, grade, and lubrication. Typical values: M8 bolt (grade 8.8) — 23 N·m; M10 bolt (grade 8.8) — 46 N·m; M12 bolt (grade 8.8) — 79 N·m; M16 bolt (grade 8.8) — 196 N·m. Always consult manufacturer specifications.

How does angle affect torque?

Torque is maximized when the force is applied perpendicular to the lever arm (θ = 90°). As the angle deviates from 90°, torque decreases proportionally to sin(θ). At θ = 0° or 180° (force parallel to the arm), torque is zero regardless of force magnitude.

What is a torque wrench?

A torque wrench is a precision tool that measures and limits the torque applied to a fastener. Types include: beam-type (deflection beam with scale), click-type (audible click at set torque), digital (electronic sensor with display), and dial-type (dial indicator).

What is the difference between static and dynamic torque?

Static torque is applied to a non-rotating object (e.g., tightening a bolt) and does not involve angular velocity. Dynamic torque is applied to a rotating object (e.g., a rotating shaft) and involves power transmission. Dynamic torque is related to power by P = τ × ω.

How do you calculate torque from power and RPM?

In imperial units: τ (lbf·ft) = (HP × 5252) / RPM. In SI units: τ (N·m) = Power (W) / ω (rad/s), where ω = 2π × RPM / 60. For example, a 100 HP motor at 1800 RPM produces τ = (100 × 5252) / 1800 = 291.8 lbf·ft.

What is torque in an engine?

Engine torque is the rotational force produced by the crankshaft. It varies with RPM and is measured on a dynamometer. Peak torque typically occurs at lower RPM than peak horsepower. The torque curve shape determines engine characteristics.

How does gear ratio affect torque?

Gear multiplication changes torque: τ_output = τ_input × gear ratio (for speed reduction). A 5:1 gear reduction multiplies input torque by 5 but reduces speed by 5. This is why lower gears provide more torque for acceleration or climbing.

What is stall torque?

Stall torque is the maximum torque produced by a motor when the shaft is prevented from rotating (zero speed). It is typically the highest torque a motor can produce but cannot be sustained indefinitely without overheating.

How do you measure torque?

Torque is measured using dynamometers (engine/motor torque), torque transducers (strain-gauge-based), torque wrenches (fastener torque), and reaction torque sensors. Measurement involves converting the mechanical twisting into an electrical signal.

What is the torque on a pulley or sprocket?

For a pulley or sprocket, torque is calculated from the net tension difference and radius: τ = (T₁ - T₂) × r, where T₁ and T₂ are belt tensions on the tight and slack sides, and r is the pulley radius.

What is the difference between torque and work?

Torque and work have the same units (N·m) but different meanings. Torque is a force that causes rotation, while work is energy transferred by a force. Work involves displacement in the direction of the force; torque involves rotation about an axis.

How does friction affect torque requirements?

Friction in bearings, gears, and seals increases the torque required to drive a system. The torque needed to overcome friction is: τ_friction = μ × F_normal × r, where μ is the friction coefficient.

What is the torque-speed characteristic of a motor?

A typical DC motor has high torque at low speed (stall torque) and decreasing torque as speed increases. The torque-speed curve is approximately linear. AC induction motors have a more complex torque-speed relationship.

How do you calculate bolt tightening torque?

Bolt tightening torque: T = K × F × d, where K is the nut factor (typically 0.2), F is the desired clamp load, and d is the bolt diameter. Approximately 50% of torque overcomes underhead friction, 40% thread friction, and only 10% creates clamp load.

What is torque ripple?

Torque ripple is the periodic variation in torque output during rotation, common in electric motors and engines. It results from cogging torque, commutation effects, and combustion cycles. Torque ripple causes vibration and noise.

How do temperature changes affect torque?

Temperature affects torque through thermal expansion of components, changes in lubricant viscosity, and changes in material properties. Bolt torque specifications often include temperature derating factors.

What is the maximum torque on a shaft?

The maximum torque a shaft can transmit is limited by its shear strength: τ_max = (π/16) × τ_yield × d³ for a solid circular shaft, where τ_yield is the shear yield strength and d is the shaft diameter.

How do you calculate the torque on a gear?

For a gear pair, the torque ratio equals the gear ratio: τ₂/τ₁ = N₂/N₁, where N is the number of teeth. The tangential force at the pitch circle is F_t = τ/r.

What is the torque due to a couple?

A couple consists of two equal and opposite parallel forces separated by distance d. The torque produced is τ = F × d, independent of the reference point. This pure torque causes rotation without translation.

How does torque relate to angular acceleration?

Newton’s Second Law for rotation: τ_net = I × α, where τ_net is the net torque, I is the moment of inertia, and α is the angular acceleration. This is the rotational analog of F = ma.

What is a torque converter?

A torque converter is a fluid coupling in automatic transmissions that multiplies engine torque during launch. It consists of a pump (impeller), turbine, and stator. The torque multiplication ratio can be 2:1 or higher at stall.

How do you select a motor based on torque?

Motor selection requires: 1) Calculate the required continuous torque. 2) Determine peak torque requirements. 3) Consider the torque-speed curve. 4) Include a safety factor (typically 1.5-2). 5) Verify thermal capacity.

What is the torque on a disc brake?

Disc brake torque is: τ_brake = 2 × μ × F_clamp × r_effective, where μ is the pad friction coefficient, F_clamp is the caliper clamping force, and r_effective is the effective radius to the pad center.

How do you verify torque calculations experimentally?

Torque can be verified using a calibrated torque wrench for static torque, a dynamometer for engine/motor torque, strain-gauge torque transducers, and reaction torque sensors. Experimental results should match calculations within measurement uncertainty.

What is the relationship between torque and bending moment in a shaft?

A shaft can experience combined torque (twisting) and bending moment. The von Mises equivalent stress combines both: σ_v = √(σ_b² + 3τ_t²), where σ_b is bending stress and τ_t is torsional shear stress.

What is the difference between torque and power?

Torque is a twisting force, while power is the rate at which work is done. They are related by P = τ × ω. An engine can have high torque but low power if it operates at low RPM, or high power with moderate torque at high RPM.

How do you calculate torque in a belt drive?

In a belt drive, the torque on a pulley is τ = (T₁ - T₂) × r, where T₁ is the tight side tension, T₂ is the slack side tension, and r is the pulley radius. Power transmitted is P = (T₁ - T₂) × v.

What is the torque required to accelerate a flywheel?

The torque required to accelerate a flywheel is τ = I × α, where I is the mass moment of inertia and α is the angular acceleration. For a solid disk flywheel, I = (1/2)mr².

How do you calculate tightening torque from clamp load?

The tightening torque is T = K × F × d, where K is the nut factor (0.15-0.25 for lubricated, 0.20-0.35 for dry), F is the desired clamp load (preload), and d is the nominal bolt diameter.

What is the difference between starting torque and running torque?

Starting torque is the torque required to begin motion from rest, typically higher than running torque due to static friction and inertia. Running torque is the torque needed to maintain motion at constant speed.

How does torque affect bolt preload?

Applied torque creates bolt preload (tension) through thread engagement. The relationship is approximately linear: F_preload = T / (K × d). However, due to friction variations, actual preload can vary by ±25% for the same torque.

What is the torque on a rotating shaft?

A rotating shaft transmitting power P at angular velocity ω experiences torque τ = P/ω. In design, the shaft diameter is sized to withstand this torque without exceeding the allowable shear stress: d = (16τ / πτ_allow)^(1/3).

How do you calculate the torque in a planetary gear?

In a planetary gear system, torque is distributed among sun, planet, and ring gears. The relationship is τ_sun / N_sun = τ_ring / N_ring = τ_carrier / (N_sun + N_ring).

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Torque (τ): A rotational force that causes an object to rotate about an axis, measured in N·m or lbf·ft.
Lever Arm: The perpendicular distance from the axis of rotation to the line of action of the force.
Moment of Inertia (I): A measure of an object’s resistance to angular acceleration, depending on mass distribution.
Angular Acceleration (α): The rate of change of angular velocity, measured in rad/s².
Angular Velocity (ω): The rate of rotation, measured in rad/s or RPM.
Newton-Meter (N·m): The SI unit of torque, equivalent to one newton of force applied at one meter distance.
Pound-Force Foot (lbf·ft): An imperial unit of torque, equivalent to one pound-force applied at one foot distance.
Kilogram-Force Meter (kgf·m): A gravitational metric unit of torque, equal to the torque produced by 1 kgf at 1 m.
Perpendicular Distance: The shortest distance from the axis of rotation to the line of force, equal to r × sin(θ).
Tangential Force: The component of force perpendicular to the radius, responsible for producing torque.
Moment: A tendency of a force to cause rotation about a specific point or axis.
Couple: Two equal and opposite forces separated by a distance, producing pure rotation without translation.
Moment Arm: Synonymous with lever arm; the distance from the pivot to the point where force is applied.
Stall Torque: The maximum torque a motor produces when the shaft is held stationary (zero speed).
Breakaway Torque: The minimum torque required to initiate motion of a stationary object.
Preload: The initial tension created in a bolt when it is tightened to a specified torque.
Nut Factor (K): A dimensionless coefficient relating applied torque to bolt tension, typically 0.15-0.35 depending on lubrication.
Polar Moment of Inertia (J): A measure of a shaft’s resistance to torsional deformation, depending on cross-sectional geometry.

Authoritative References

Trusted sources and standards referenced by this calculator.

ASME B106.1M - Shaft Design Engineering Toolbox - Torque Conversions Shigley’s Mechanical Engineering Design NIST - Torque Metrology EFunda - Torque and Power HyperPhysics - Torque Concepts Khan Academy - Torque and Angular Momentum MIT OCW - Machine Design

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Last Updated:

2026-08-21

Version:

2.0.0

References:

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Free to use for educational and professional purposes.