There are many variables to consider when constructing a staircase, one of which is the stair angle. The angle of your stairs will determine how steep or comfortable they are to walk on. Steeper angles are going to make stairs difficult to walk, however. If it is too shallow, they get too much space. A Stairs Angle Calculator comes in handy in those cases!

What is Stair Rise and Run?

Before you can calculate the stair angle, there are two important terms you should know.

• Rise: The vertical distance of a single step (inches, feet, or cm.)
• Run: The horizontal depth of a single step (in inches, feet, or cm).

Applying basic trigonometry using the rise and run establishes the angle of inclination of the staircase.

Formula for Stair Angle

The formula to calculate the angle of a staircase is:

$\text{Angle} = \arctan\left(\frac{\text{Rise}}{\text{Run}}\right)$

Here:

• Rise = the total vertical height of the staircase.
• Run = the total horizontal length of the staircase.

This formula uses the arctangent function (inverse tangent) to calculate the angle.

Why Stair Angle is Important?

• Safety: Very steep staircases can be hazardous while shallow read staircases can be challenging to walk on.
• Building Code Compliance: The vast majority of building codes require a stair angle that falls between 30° and 37° in order to be guaranteed safe for public use.
• Space Saving — The angle goes a long way in determining, how much of a space the staircase will cover in your home or building.

Example 1

Suppose the staircase has:

• Rise = 120 inches
• Run = 144 inches

The angle is calculated as:

$\text{Angle} = \arctan\left(\frac{120}{144}\right)$

$\text{Angle} \approx 39.8^\circ$

Example 2

Let’s say we are building a staircase with the following measurements:

• Rise = 7 inches (vertical height per step)
• Run = 10 inches (horizontal depth per step)

Step 1: Apply the Formula

$\theta = \arctan\left(\frac{\text{Rise}}{\text{Run}}\right)$

$\theta = \arctan\left(\frac{7}{10}\right)$

Step 2: Use a Calculator

$\theta = \arctan(0.7)$

$\theta \approx 34.99^\circ$

Final Result

The stair angle is approximately 35°, which falls within the recommended range of 30° to 37° for safe stair design.