Consider the torque from a force exerted on a hinged rod about the hinge, as shown in the diagram. The torque points out of the screen, corresponding to counterclockwise motion.

There are three equivalent ways to determine the magnitude of the torque:

Method 1: Measure r from the hinge along the rod to where the force is applied, multiply by the force, and then multiply by the sine of the angle between the rod and the force. This gives:   τ = r F sin θ

Method 2: Resolve F into components perpendicular to and parallel to the rod. The parallel component produces no torque. The perpendicular component, Fsin θ, gives a torque of magnitude:  τ = r [Fsin θ ] = r F sin θ

Method 3: Extend the line of the force and measure the lever arm r' = r sin θ , the prependicular distance from the rotation axis to the line of the force. The magnitude of the torque then is:   τ = r' F sin 90 = r' F = r F sin θ