How this calculator works
Rise is the signed vertical difference and run is the signed horizontal difference. Their ratio is slope; the same ratio displayed as a percentage is grade. Use horizontal run when comparing a ramp grade with an angle. The diagonal segment length is a different measurement, calculated with the Pythagorean theorem. Point mode also anchors the line in the coordinate plane, allowing the y intercept to be recovered. With changes alone, its position and intercept are not determined.
Formula
m = rise/run; grade = 100m percent; rise = m run; run = rise/m; angle = atan(m); diagonal = sqrt(rise^2+run^2).
Worked example
A rise of 2 units over a horizontal run of 8 gives slope 0.25, grade 25%, angle about 14.036243 degrees and a diagonal length of sqrt(68), about 8.246211 units. A 100% grade has equal rise and run and therefore an angle of 45 degrees.
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Frequently asked questions
Is run the diagonal length?
No. Run measures the horizontal change. Using the diagonal in the denominator gives the sine of the angle rather than its tangent.
Does reversing both points change the slope?
No. Both rise and run change sign, so their quotient stays the same.
Can a grade exceed 100%?
Yes. It means the magnitude of rise is greater than the magnitude of horizontal run.
Why is no intercept shown for changes alone?
Rise and run give a direction. A point is also needed to fix where the line crosses the y axis.