Torque Calculator

Torque is τ = rF sin(θ), where r is the lever arm (from pivot to point of force application), F is the applied force, and θ is the angle between r and F. SI unit is the newton-metre (N·m). So a 10 N force applied 0.5 m from a hinge at 90° gives τ = 0.5 × 10 × 1 = 5 N·m. At 30° instead, τ = 0.5 × 10 × 0.5 = 2.5 N·m — half as much, even though the force is the same. Calculators handle the trig automatically; you provide the geometry and force.

When the force is perpendicular to the lever arm (the most common case), τ = rF — straight multiplication. Lever arm in metres, force in newtons, torque in newton-metres. So pushing on a wrench 0.3 m from the bolt with 50 N of force gives 15 N·m of torque. Doubling the wrench length doubles the torque for the same effort, which is why long wrenches loosen stubborn bolts that short ones can't. Watch for non-perpendicular cases, where you need the sin(θ) factor; the perpendicular component of the force is what produces torque.

When force isn't perpendicular to the lever arm, only the perpendicular component matters: τ = rF sin(θ), where θ is the angle between F and r. So pulling at 60° instead of 90° reduces torque to sin(60°) = 0.866 of its maximum. Pulling along the lever arm (θ = 0°) produces zero torque, no matter how hard. The sin(θ) factor captures this geometry. You can equivalently think of it as the moment arm — the perpendicular distance from the pivot to the line of force — multiplied by F. Either viewpoint gives the same answer.

Rearrange τ = rF sin(θ) to F = τ/(r sin(θ)). For perpendicular force, this simplifies to F = τ/r. So unscrewing a 30 N·m bolt with a 0.25 m wrench needs F = 120 N applied perpendicular at the wrench's end. Use a 0.5 m wrench instead and you only need 60 N. This is exactly why mechanics carry breaker bars — long lever arms multiply available torque without needing superhuman strength. Reverse-solving like this is essential for sizing tools or estimating bolt-tightening forces from torque specifications.

The lever arm — also called the moment arm — is the perpendicular distance from the pivot to the line along which the force acts. When force and lever arm are perpendicular, lever arm equals the geometric distance. Otherwise, it's r sin(θ), which equals the projection. Some calculators ask for the moment arm directly (the perpendicular distance), in which case you skip the sin(θ) factor and just multiply F × r_perp. Either way, you get the same torque. Sketching the force and the perpendicular dropped from the pivot to the force line clarifies which distance the calculator expects.

1 N·m ≈ 0.7376 lbf·ft, and 1 lbf·ft ≈ 1.356 N·m. So a 100 N·m torque is about 73.76 lbf·ft, while 80 lbf·ft is around 108.5 N·m. American automotive specs typically come in pound-feet, while European specs use newton-metres. Engine torque ratings, bolt tightening specs, and many tools labels show one or the other depending on origin. Calculators flip between them instantly. Get this conversion wrong and you risk over-tightening (stripping threads) or under-tightening (joints loosening) by significant margins.

Because torque depends on sin(θ), and sin(θ) reaches its maximum of 1 at θ = 90°. So perpendicular force produces the most rotational effect. Any other angle gives smaller torque, with τ → 0 as θ → 0° (force along the lever arm produces no rotation). This is why doors have handles on the opposite side from the hinges and why you push perpendicular to the door, not at an angle. Pulling at 30° or 45° wastes effort. The geometry is the reason — only the perpendicular component does the rotational work; the parallel component just stretches or compresses the lever.