The turning radius (or turning circle) of a motorcycle is one of the most crucial aspects of its low-speed performance, especially where parking and U-turns are concerned. Just as with any other motor vehicle, the turning circle has everything to do with wheelbase and the wheel's turning angle; a shorter wheelbase and higher angle equals a sharper turn, and a longer wheelbase and lower angle makes for a wider turn.
Determine your bike's steering wheel angle at full left or right lock. You can either measure it directly with a protractor or construction speed square, or you can consult your bike's owners' manual. As an example, we'll use a Honda Valkyrie, which has a 40 degree wheel angle at full lock.
Subtract your bike's turning angle from 90 and then find the sine value of that angle. Sine is a trigonomic function that describes one angle of a right triangle; if you don't have a calculator with trig functions, you can find the sine of an angle by entering the phrase "sine of (number) degrees" into your computer's search engine. So, we'll subtract 40 from 90 (equals 50), then find the sine value of 50, which equals 0.766.
Divide the wheelbase, in inches, by the sine value from Step 2, then multiply by two to get the turning circle diameter, in inches. Our Valkyrie has a 67-inch wheelbase, so we'll divide 67 by .766 to derive an 87.46-inch turning radius. Multiply by two and you've got an overall turning circle of 174.9 inches, or 14.75 feet.
Written in mathematical terms, the formula looks like this: Turning Circle = 2 x (wheelbase / sine (90 - Wheel Angle))
This formula also works for automotive applications, but you'll use the turn angle of the outside steer tire, since it dictates the turning circle. You are supposed to use a conversion factor to account for things such as tire width and suspension design, but this kind of data is vehicle-specific, hard-to-find and doesn't really make much difference with regards to the end figure.