Converting Slope Angle-Percentages to Degrees

There are many reasons you might want to know how to quickly and easily convert a percentage to a degree, not the least of which is because you happen to think it’s easier to visualize a given slope in percentages rather than degrees. One reason some people might agree with this, is because when working with round, even numbers, it might seem rather natural to simply take a rise over run.

For example, say you have a 5 meter long roof and you visualize it to rise a tenth of that, which would be 50 centimeters. You can then conclude that there would be a 10 percent slope as it would rise 10 percent of its length. As mentioned above, this is especially easy when working with round numbers – which is often the case in carpentry.

Obviously, not everyone is like this, as some – namely, those who are well-versed in trigonometry – may find it more natural to think using degrees and/or radians. But I assume you’re here reading this because – regardless of why – you’re looking to perform this conversion of percentage to degrees. But before I show you how, there’s a common misconception you should know about.

d=run, (delta)h=rise, l=slope length, a=angle of inclination

There is a difference between converting slope percentages to degrees based on 90 or any degrees for that matter, and rise over run. For example, road slopes displayed on signs as a percentage are often mistakenly thought to be based on 90 degrees. I’ve heard it said that for example, a 5 percent slope can be converted to degrees by multiplying the decimal by 90, equaling 4.5 degrees. This is not true. Road slopes are based on rise over run.

To explain further, imagine the difference between taking a percentage of a given angle – say 90 degrees – and a percentage of a given rise. The former is an angle ten percent slope signand therefore curves while the latter is simply a rise in vertical height and therefore goes straight up. You can now see clearly how even matching percentages would output very different answers.

Converting Percentage to Degrees using a Scientific Calculator

To convert to degrees from a rise over run, you take the percentage and first convert it into a decimal. Let’s use the same 5 percent from the example above. Now all you have to do is take the arctangent of 0.05 and you have your answer – 2.86 degrees. The arctangent is the inverse function of tangent and may show up on your calculator as tan-1.

If you need to convert degrees to a percentage, take the tangent instead of the arctangent, and move the decimal to the right two places. Use the following equations if you need:

where m equals the slope and theta (θ) is the angle in question.

relation between slope and rise over run

Remember, the slope can be expressed in various ways, including both degrees and percentages. So depending on what you’re using in your situation, you can input a given value and convert using the method I outlined above. (The symbol Δ stands for “change” or “difference”.)

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