The term "rise" alone, when referring to stair construction, is ambiguous -- there's actually two separate rises to your stairway. The first is the total rise, or the distance the stairs travels. The second rise is the height of each step -- the distance your foot must travel from the surface of the last step to the surface of the next, in other words. To use rise correctly, you must specify which you mean: stair rise or step rise.
When it comes to your stairs, the most important rise is the step rise. Local building codes -- all of which may vary slightly -- require that the rise of each step must first be within a minimum and maximum range and second, that it not vary much from step to step. Most local building codes limit the height to either 7 3/4 or even 8 1/4 inches, and mandate at least 6 inches. The difference between the tallest step rise and the shortest can be no more than 3/8 inch to keep steps consistent.
Local building codes also discuss stair rise. Of course you can't limit the height of a flight of stairs -- it takes as many steps as it takes to reach from the bottom floor to the top. What building code does demand is that a single flight cannot extend more than, in most cases, 12 feet before you reach a landing -- a much deeper area where you are between steps -- before starting up another flight of stairs.
The Greek mathematician Pythagoras was onto something with his theorem that A-squared plus B-squared is equal to C-squared. Builders under Julius Caesar used the formula to construct stairs, and the practice remains to this day. First a stair builder determines the total rise and divides it by the preferred step rise, usually 7 3/4 inches. The result is the total steps required. Dividing the total rise by the number of steps next tells you the exact step rise.
Step run -- the depth of the step from front to back -- must be in proportion to the rise. Builders usually use the rule of thumb that two step rises and one run equals about 25 inches. Building code is more specific, generally requiring a minimum of 9 to 10 inches. Once you calculate your rise, find the run that suits it. Next, multiply the number of steps by the step run to obtain the total stair run or the distance the stairs travels horizontally. Finally, the Pythagorean theorem applies: Square the total rise and add to the square of the total run. Take the square root of the result and translate into inches to find the stringer length. All that remains is cutting your stairs. Check your specific local building code first to ensure your measurements are in compliance.