Consider the torque from one force exerted on a hinged rod, as shown in the diagram.

The torque is counterclockwise, although the rod does not rotate because of the balancing clockwise torque coming from the force of gravity. For now, focus on the counterclockwise torque.

There are three equivalent ways to determine the magnitude of the torque about a rotation axis passing through the hinge:

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 (the line you measure r along) and the force.

τ = r F sin(θ)

Method 2 - Split the force into components perpendicular to and parallel to the rod. The parallel component produces no torque. The perpendicular component, Fsin(θ), results in a torque with a magnitude of:

τ = r [Fsin(θ)] sin(90) = r F sin(θ)

Method 3 - Extend the line of the force and measure the distance from the rotation axis to the line of the force along a line that is perpendicular to the line of the force. The distance measured along this line is often called the lever arm - we'll use r' for this perpendicular distance. The magnitude of the torque is:

τ = r' F sin(90) = r' F

Using geometry, r' = r sin(θ), so the torque (once again) has a magnitude of:

τ = r F sin(θ)