Figure

Math and relationships

Inverse tangent

Find a slope’s angle from a tangent ratio or measured rise and horizontal run.

Inverse tangent turns a rise-to-run ratio into an angle. Both measurements must use the same units. A ratio, a percent grade and an angle describe the slope in different forms.

Find a slope’s angle

Turn a measured rise and horizontal run into an angle in degrees.

Why inverse tangent?

Rise ÷ run gives the tangent of the angle. tan⁻¹ turns that ratio into degrees.

Right triangle: a 3 foot vertical rise and 12 foot horizontal run. Theta is the angle above horizontal. Inverse tangent of 3 divided by 12 gives about 14.04 degrees.12 ft3 ftθ
Schematic, not to scale.

Try it · 3 ft rise, 12 ft run

Both measurements are in feet, so divide their numbers.

  1. Start fresh

    Press Clear entry, then Clear entry.

  2. Rise ÷ horizontal run

    Press 3, then Divide, then 12, then Equals.

  3. Find the angle

    Press Second, then Inverse tangent.
  1. Angle above horizontalabout 14.04°

Also useful for

Roof pitch, ramp slopes, stair inclination, pipe fall and setting out angled framing or ductwork.

Remember

Rise and run must use the same units. A 25% grade means a rise/run ratio of 0.25, not an angle of 25°.

Find an angle from a tangent ratio

Enter the tangent ratio to find its angle in degrees.1

Start a new calculation

  1. Press Clear entry, then Clear entry.

Calculate

  1. Tangent ratio

    Press .87421, then Second, then Inverse tangent.
  1. Example result41.16028°

Display assumptions

  1. Angle decimal places: Variable. ↩

Keep the guide beside your calculation.

These examples are built into Figure. Open Help while you calculate, then return to the same guide and reading position.

See Help in Figure →