I don't know the math: Hills % and degree

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8 replies [Last post]
Anonymous's picture

Does anyone know the % and degree (steepness) of a hill 1.6 miles long and ascending 700ft?

Algebra was never my thing. Then again, school was never my thing.


Anonymous's picture
michael (not verified)

I stand corrected

Anonymous's picture
Oblong (not verified)


Thanks for the timely response and thanks for showing the math. Even I can follow that. Now, the next time I ask the same question about another hill, you'll know why school wasn't my thing - too lazy.

Thanks again,

Anonymous's picture
John Z (not verified)

"But close and a very good approximation. % Gradient is rise/run, the vertical rise over horizontal run -- the ""legs"" of a triangle. Your method incorrectly employed the triangle's hypotenuse. Knowing a climb's length and vertical rise, the horizontal run can be calculated as follows:

(length^2 - vertical rise^2)^1/2

In words, the square root of the length squared minus the vertical rise squared.


(8448^2 - 700^2)1/2 = 8419 feet horizontal run

Slope = 700/8419*100 = 8.315%, whereas the ""incorrect"" formula calculated to 3 decimal places is 8.286%. On steeper grades, the error would be a bit more pronounced.

The angle in degrees is simply the inverse tangent of 0.08315 = 4.75 degrees. Not sure where you came up with idea that angle is % gradient * 90."

Anonymous's picture
bernie (not verified)
hils and % grade

you can get the same answer another way:

1.6 * 5280 = 8448 ft is the distance you travelled up the hill.

sine of angle x = 700 / 8448 = 0.08286
angle x = 4.75 degrees is the angle of the incline

tangent of 4.75 degrees = 0.083145 = 700 / y
y = 8419 feet is the horizontal distance you travelled

700 rise / 8419 run = 0.0831 or 8.31% grade or steepness.

Anonymous's picture
John Z (not verified)

I used the formula initially because you don't need a trig table or trig calculator -- not that we all don't have one. An old habit I guess.

Anonymous's picture
Robert Gray (not verified)
What is percent grade?

Percent of grade is the relationship between the vertical rise and the horizontal distance traveled. If you travel 100 feet and rise 10 feet, that is a 10% grade.
If you rise 25 feet in 100 feet, that is a 25% grade.
If you rise more than you travel horizontally, you can have a grade over 100% but you cannot travel that on a bike or motor vehicle, it would be steeper than a stair.
Most people find hills more than 5% to be difficult and modern roads are usually designed with grades not exceeding 5%.
John Z is completely correct in his calculations here but most people do not seen to understand what grade really is and why a 100% grade is only 45 degrees.

Anonymous's picture
John Z (not verified)
Good Explanation

Its a little nuance of Civil Engineering...

Anonymous's picture
Robert Gray (not verified)
nuance of Civil Engineering

Exactly, a good way to put it.
It is a term used by people who engineer roads and deal with relatively small slopes which can be safely driven on in all types of weather.

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