March 26, 20263 min read

Torque Calculator — τ = rF sin(θ)

Calculate torque from force and lever arm distance. Covers rotational mechanics, moment of force, units in Nm and ft-lb, with engineering examples.

torque rotational mechanics lever arm physics calchub

Torque is rotational force — the tendency of a force to cause an object to rotate around a pivot point. You use torque every time you turn a wrench, open a door, or ride a bike. The further your force is from the pivot (longer lever arm), the more torque you generate for the same effort. That's the mechanical advantage of a long wrench.

Use the CalcHub torque calculator to find torque, force, or lever arm length in any rotation problem.

The Formula

τ = r × F × sin(θ)

τ (tau) = torque (Newton-meters, N·m)
r = lever arm length (distance from pivot to force application, in meters)
F = force applied (Newtons)
θ = angle between force vector and lever arm

When force is perpendicular to the lever arm (θ = 90°), sin(90°) = 1, so τ = rF — the maximum torque for that force and distance. Pushing at an angle reduces effective torque.

Torque Units

Unit	Equivalent	Common Use
N·m	—	Engineering, physics
ft·lb	1.356 N·m	US automotive
in·lb	0.113 N·m	Small fasteners
kgf·m	9.81 N·m	Older standards

Most car engine specs quote peak torque in N·m or ft·lb at a specific RPM.

Practical Examples

Door handles are placed far from hinges for a reason. A 20 N push applied 0.8 m from the hinge:

τ = 0.8 × 20 × sin(90°) = 16 N·m

If you pushed at only 30 cm from the hinge with the same force:
τ = 0.3 × 20 = 6 N·m — much harder to open the door.

For a wrench: applying 150 N of force at the end of a 0.25 m wrench:
τ = 0.25 × 150 = 37.5 N·m of bolt-tightening torque.

What's the relationship between torque and power?

For rotating machinery: P = τ × ω, where ω is angular velocity in radians per second. For engine specs: P (W) = τ (N·m) × RPM × 2π / 60. This is why high-torque engines at low RPM and high-revving engines at low torque can produce the same power output.

How does torque cause angular acceleration?

Newton's second law for rotation: τ_net = I × α, where I is moment of inertia and α is angular acceleration. This is the rotational analog of F = ma. More torque, more angular acceleration. More inertia, less angular acceleration for the same torque.

Why do torque wrenches matter in automotive work?

Over-tightening fasteners strips threads or warps components. Under-tightening allows loosening under vibration. Torque specs ensure proper clamping force. Cylinder head bolts, wheel lug nuts, and suspension fasteners all have critical torque values specified by manufacturers.