Torque Calculator
Inputs — τ = F × r × sin(θ)
Examples
Solve for Unknown Variable
Force Entries
Output:
Force 1
Force 2
About This Tool
Understanding Torque: The Rotational Force
Torque — also called the moment of force — is one of the most fundamental concepts in rotational mechanics. While linear force causes objects to accelerate in a straight line, torque causes objects to rotate about a fixed point called a pivot or fulcrum. From tightening a bolt with a wrench to a car engine spinning its crankshaft, torque is at work in almost every mechanical system you encounter.
Understanding torque is essential in engineering, physics, and everyday life. This calculator helps you compute torque from force and distance, account for the angle of application, solve for unknown variables, and even determine whether a system of forces is in rotational equilibrium.
The Torque Formula
The standard formula for torque is:
τ = F × r × sin(θ)
Where:
- τ (tau) — Torque, measured in Newton-metres (N·m) or foot-pounds (ft·lb)
- F — Magnitude of the applied force (Newtons or pound-force)
- r — Length of the lever arm — the perpendicular distance from the pivot to the line of action of the force
- θ (theta) — Angle between the force vector and the lever arm direction
When the force is applied perpendicular to the lever arm (θ = 90°), sin(90°) = 1 and the formula simplifies to τ = F × r. This is the maximum torque achievable for a given force and lever arm length.
The Role of the Lever Arm
The lever arm (or moment arm) is the effective distance from the pivot point to the point where the force is applied, measured perpendicularly to the force's direction. This distance is crucial: doubling the lever arm doubles the torque for the same applied force. This principle is why a long wrench handle is easier to turn a bolt with than a short one, and why door handles are placed far from the hinges.
When a force is applied at an angle, only the component perpendicular to the lever arm contributes to rotation. The sin(θ) term in the torque formula extracts this perpendicular component. A force applied at 45° produces only about 70.7% of the torque that the same force applied perpendicularly would generate.
Clockwise and Counterclockwise Torque
Torque has both a magnitude and a rotational direction. By the standard physics convention:
- Counterclockwise (CCW) torques are treated as positive
- Clockwise (CW) torques are treated as negative
When multiple forces act on a body, the net torque is the algebraic sum of all individual torques. A net torque of zero means the body is in rotational equilibrium — it will not start rotating (or will continue to rotate at constant speed if already moving).
Rotational Equilibrium
A system is in rotational equilibrium when the sum of all torques about any chosen pivot equals zero: Σ τ = 0. This is one of the key conditions for a rigid body to be in static equilibrium (along with Σ F = 0 for linear equilibrium).
Classic examples include:
- A balanced seesaw, where the product of weight and distance is equal on both sides of the pivot
- A beam supported at both ends, where the support reactions create torques that balance the load torques
- A bicycle wheel spinning at constant speed, where driving torque equals frictional torque
Torque Units and Conversions
Torque is expressed in different units depending on the measurement system and application:
- Newton-metre (N·m) — SI unit, used in most scientific and engineering contexts
- Kilonewton-metre (kN·m) — Used for large structures like bridges and heavy machinery; 1 kN·m = 1,000 N·m
- Foot-pound (ft·lb) — Common in US and UK automotive and construction; 1 ft·lb ≈ 1.3558 N·m
- Inch-pound (in·lb) — Used for small fasteners and precision instruments; 1 in·lb ≈ 0.1130 N·m
This calculator automatically converts between all four units so you can see results in whichever system is most useful for your work.
Solving for Unknown Variables
The torque equation can be rearranged to solve for any of its variables:
- Solve for Force: F = τ / (r × sin(θ)) — find the force needed to produce a desired torque
- Solve for Distance: r = τ / (F × sin(θ)) — find the lever arm length needed
- Solve for Angle: θ = arcsin(τ / (F × r)) — find the angle at which force must be applied
These inverse calculations are valuable in design and analysis work: for example, finding the minimum wrench length needed to achieve a specified bolt torque without exceeding the force a technician can apply.
Real-World Applications of Torque
Torque appears across virtually every field of engineering and technology:
- Automotive Engineering: Engine torque curves determine acceleration performance; torque wrenches ensure bolts are tightened to specification without damage
- Structural Engineering: Beams, cantilevers, and frames are designed so internal moments (torques) stay within safe limits under load
- Robotics: Joint torques determine the payload a robotic arm can handle and the motors required for each axis
- Wind Turbines: Blade torque drives the generator shaft; gearboxes multiply torque to match generator requirements
- Everyday Tools: Spanners, screwdrivers, door handles, and bicycle pedals all exploit the torque principle to transfer rotational force efficiently
Torque vs. Work: An Important Distinction
Although torque and work (energy) share the same SI unit — Newton-metre — they are physically different quantities. Work is a scalar (it has magnitude only) and equals force times distance in the direction of motion. Torque is a vector (it has both magnitude and rotational direction) and describes a tendency to rotate, not the energy actually transferred.
Work done by a torque over a rotation is: W = τ × θ (where θ is in radians). This clarifies the relationship: torque is the rotational analogue of force, and the work done depends on how far the rotation actually proceeds.
Tips for Accurate Torque Calculations
- Always identify the exact pivot point before measuring the lever arm distance
- Use the perpendicular distance from the pivot to the force's line of action as the lever arm — not just the physical length of the arm
- Be consistent with units: convert all distances to metres and all forces to Newtons before applying the formula
- For multiple forces, assign directions carefully — a sign error will flip the equilibrium result
- Remember that an angle of exactly 0° or 180° produces zero torque, regardless of force magnitude
Frequently Asked Questions
Yes, Torque Calculator is totally free :)
Yes, you can install the webapp as PWA.
Yes, any data related to Torque Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.
Torque (also called the moment of force) is the rotational equivalent of linear force. It measures how effectively a force causes an object to rotate about a pivot point. The formula is τ = F × r × sin(θ), where F is the applied force in Newtons, r is the lever arm distance in meters, and θ is the angle between the force vector and the lever arm. When the force is perpendicular to the lever arm (θ = 90°), torque is at its maximum: τ = F × r.
Only the component of force perpendicular to the lever arm produces rotation. The factor sin(θ) captures this: at 90° (perpendicular), sin(90°) = 1 and torque is maximised. At 0° or 180° (parallel to the lever arm), sin(θ) = 0 and no torque is produced. At 45°, only ~70.7% of the force contributes to rotation. This is why you should push or pull a wrench perpendicular to its handle for maximum torque.
The direction of torque (clockwise or counterclockwise) describes which way a torque tends to rotate an object around the pivot. By convention, counterclockwise (CCW) torque is taken as positive and clockwise (CW) torque as negative. When calculating net torque from multiple forces, the directions are accounted for with sign: if the positive (CCW) torques and negative (CW) torques sum to zero, the system is in rotational equilibrium.
Torque is expressed in Newton-metres (N·m) in the SI system, which is the international standard. Other common units include: kilonewton-metres (kN·m) = 1000 N·m, foot-pounds (ft·lb) where 1 ft·lb ≈ 1.3558 N·m, and inch-pounds (in·lb) where 1 in·lb ≈ 0.11299 N·m. Car engine torque is often quoted in N·m or lb·ft; bolt tightening specifications use N·m or ft·lb; and small fasteners often use in·lb.
A rigid body is in rotational equilibrium when the net torque acting on it about any pivot point is zero. This means the sum of all clockwise torques equals the sum of all counterclockwise torques. For example, a balanced seesaw achieves rotational equilibrium: the torque from each person's weight cancels out. In engineering, rotational equilibrium is essential for designing stable structures like bridges, cranes, and beams.
In Multi-Force mode, you can add several force entries — each with its own magnitude, distance, angle, and direction (CW or CCW). The calculator assigns positive torque to counterclockwise forces and negative torque to clockwise forces, then sums them to find the net torque. It also tells you whether the system is in rotational equilibrium (net torque = 0). This is useful for analysing beams, levers, and other structures with multiple applied forces.