Slope Angle in Degrees: Formula and Worked Example

Calculate a slope’s angle in degrees from its vertical rise and horizontal run. Enter both measurements in the same units to get the angle above or below level, along with the slope’s percentage grade.

Formula

The slope angle is the inverse tangent of rise divided by horizontal run:

Angle in degrees = arctan(rise ÷ horizontal run) × 180 ÷ π

The calculator’s atand function performs that inverse-tangent calculation and returns degrees directly, so its equivalent expression is atand(rise / run).

• Rise: vertical change in elevation, in metres, feet, inches, or another length unit.
• Horizontal run: horizontal distance between the same start and end points, using the same unit as rise.
• Angle: slope’s inclination from a horizontal, level line, in degrees.
• Grade: rise divided by horizontal run, multiplied by 100; this is a percentage, not an angle.

For example, a 1-in-12 ramp has a horizontal run of 12 units for every 1 unit of rise. Its grade is about 8.33%, and its angle is about 4.76°. Grade and degrees describe the same slope in different ways, but the numbers are not interchangeable.

Worked Example

Using the calculator’s default values, the rise is 3 m and the horizontal run is 4 m. The units match, so divide the measurements directly.

1. Calculate rise divided by run: 3 ÷ 4 = 0.75.
2. Find the inverse tangent in degrees: arctan(0.75) ≈ 36.8699°.
3. Round to two decimal places: the slope angle is 36.87°.
4. For the additional grade result, calculate 100 × 3 ÷ 4 = 75%.

The calculator therefore returns an angle of 36.87 degrees and a grade of 75.00% for its defaults.

What You’ll Need

• The vertical rise between two points.
• The horizontal run between those same points.
• A calculator with an inverse-tangent function, or this page’s calculator.
• A tape measure, level, measuring wheel, or reliable elevation and distance readings, if you still need to collect the measurements.

Step-by-Step

1. Choose the two points. Identify where the slope begins and ends. Use the same endpoints for both measurements; otherwise, the rise and run will describe different sections.
2. Measure the vertical rise. Find the difference in height between the endpoints, not the length along the sloping surface. For a small project, a level and ruler can help establish vertical change. For larger distances, use suitable elevation data or a surveying method.
3. Measure horizontal run. Measure the distance parallel to level ground between the points. This is not the distance traced over the slope. A sloped surface’s measured length is the hypotenuse of a right triangle, whereas the formula requires the horizontal leg.
4. Put both measurements in one unit. You can use metres, centimetres, feet, or inches, but convert one measurement if the units differ. For example, convert 18 inches to 1.5 feet before comparing it with a run measured in feet. The ratio is unitless once the units match.
5. Enter rise and run. Type the positive vertical change into Rise and the positive horizontal distance into Horizontal run. The defaults illustrate a rise of 3 m over a 4 m run; replace them with your measurements.
6. Read the angle. The first result is the angle measured from horizontal. A level surface is 0°. A steeper uphill slope has a larger positive angle, approaching 90° as the horizontal run gets very small.
7. Use grade as a separate description. The second result is percentage grade. It can be useful when a plan or specification expresses slope as rise per 100 units of horizontal distance. Do not report grade as degrees.
8. Keep sensible precision. The result is rounded to two decimal places. Keep more precision in intermediate calculations if working by hand, then round the final angle to suit the measurement accuracy and purpose.

Common Mistakes

• Using the sloping length as the run: Measure horizontal distance, not the surface length. If you only know the slope length and rise, first find horizontal run with the Pythagorean theorem: run = √(slope length² − rise²), provided the measurements describe a right triangle.
• Mixing units: Dividing 30 centimetres by 4 metres without conversion gives a misleading result. Convert both to centimetres or both to metres first.
• Confusing angle and grade: A 10% grade is not a 10° angle. A 10% grade corresponds to arctan(0.10) ≈ 5.71°.
• Measuring along different endpoints: The rise and run must cover the same horizontal section and the same elevation change.
• Rounding too soon: Rounding rise/run or an intermediate angle excessively can alter the final answer. Enter measurements at their available precision and round the result only at the end.

Tips

• Sketch a right triangle: rise is the vertical side, run is the horizontal side, and the slope angle is between the run and the sloping line.
• Check that the angle seems plausible. If rise is less than run, the angle is below 45°. If rise equals run, it is 45°; if rise is greater, it is above 45°.
• For a long or uneven slope, measure several sections rather than assuming one straight, constant incline. An overall angle from total rise and total horizontal run describes the average slope, not every local section.
• Take care when measuring near steep edges, roads, roofs, or unstable ground. Use equipment and a method appropriate to the site; do not put yourself in an unsafe position to obtain a measurement.
• Record whether the reported angle is relative to horizontal. Angles measured from vertical are complementary: angle from vertical = 90° − angle from horizontal for a straight slope in the same plane.

Troubleshooting

The calculator shows an error or no usable result: Check that both fields contain numbers and that horizontal run is greater than zero. A vertical change with zero horizontal distance is vertical, so this formula’s angle approaches 90° but division by zero is undefined.

The angle is unexpectedly tiny: Confirm that you entered rise rather than the total elevation at one endpoint, and that both measurements use the same units. Also check that the run is horizontal, not a much longer distance measured along a route that curves or changes direction.

The angle seems far too steep: Check whether you entered a percentage grade as though it were a rise measurement. A 12% grade means a rise/run ratio of 0.12, not 12. If the run is 100 units, enter a rise of 12 units.

Your hand calculation differs: Make sure the calculator is using degrees for its displayed result. On a scientific calculator, use inverse tangent (tan⁻¹ or atan) and set the angle mode to degrees. Radian mode produces the same mathematical angle in a different unit.

The result changes a lot with small measurement adjustments: This can happen when the run is very short or the slope is close to vertical. Recheck the measurements and report only as many decimal places as their accuracy supports.

Frequently Asked Questions

How do I calculate slope angle from rise and run?
Divide vertical rise by horizontal run, then take the inverse tangent: angle = arctan(rise/run). Convert or calculate the result in degrees.

Can I use feet and inches?
Yes. Convert both measurements to the same unit first. For instance, convert inches to feet or feet to inches before dividing.

How do I convert percentage grade to degrees?
Divide the grade percentage by 100, then take the inverse tangent. For example, a 20% grade has an angle of arctan(0.20) ≈ 11.31°.

Is slope angle the same as slope percentage?
No. Angle is measured in degrees; percentage grade is 100 times rise divided by horizontal run. Convert between them with the tangent or inverse tangent.

Can I calculate an angle if I know only the slope length?
No. You need another measurement, such as rise or horizontal run, to determine the angle. Slope length alone does not specify how steep the slope is.

What does a negative rise mean?
A negative rise indicates the second point is lower than the first. This calculator uses positive rise and returns the magnitude of the slope angle; label the direction separately as downhill or uphill.

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in rise | Rise | m | 3 | 0
in run | Horizontal run | m | 4 | 0
out angle | Slope angle | degrees | atand(rise / run) | 2
out grade | Grade | % | 100 * rise / run | 2