Being able to calculate stair angles is an important tool in construction. Not only is it used frequently, but it also applies to other types of projects and calculations. As with most aspects of construction, it involves taking direct measurements and manipulating those measurements with math to obtain a goal.
Use the measuring tape to determine the distance from the bottom of one step to the top of one step. This will later be called the rise.
Take another measurement, on the same stair step, of how long the step is. This is the horizontal measurement, and will later be called the run. Be sure to measure to the same point on each measurement.
In the case that all steps are not equal, you may have to be a bit creative. If they are different from one step to the next, possibly try making many rise and run measurements, then averaging each type of measurement by adding them up and dividing by the number of measurements taken. If the steps are one size for a while, then change size, split up the stair into two sections and calculate the angle of each section.
Calculating the Stair Angle
Make sure your scientific calculator is in degree mode. Most calculators have a button labeled "DRG" that can change the mode they are in.
If using a calculator, take the inverse tangent of the rise over the run. Press the inverse tangent, also known as the arctan or tan^(-1), button. Enter the rise measurement. Press the division button. Enter the run measurement. Depending on your calculator, you may have to close the parentheses after entering the run measurement. Pressing enter will give you the stair angle in degrees. If the number is a decimal, you are probably in radian mode, not degree mode, and must switch.
If using a trigonometry table, calculate the quotient of the rise divided by the run, then look up the arctan value for that number in degrees. That will be the stair angle.