How to Calculate Slope
Slope is the single number that describes how a line behaves. Every other property of a straight line follows from it, which is why the definition is worth knowing properly rather than memorising as a formula.
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Try the Slope CalculatorRise Over Run
The slope is the vertical change divided by the horizontal change. Rise is the change in y and run is the change in x. The order of subtraction within each is irrelevant because both change sign together when the points are swapped.
Reading the Sign
A positive slope rises as x increases, a negative slope falls, and a slope of zero is horizontal. On a conventional graph with y increasing upward, a positive slope is a line going up to the right.
The Vertical Line Problem
A vertical line has no run, so dividing by zero leaves the slope undefined. This is a genuine property of the line rather than a failure of arithmetic, and vertical lines are written as x equals a constant instead of as a slope.
Units Change Everything
A slope of 0.5 where x is metres and y is metres is a 50 percent gradient, while the same number with x in kilometres and y in metres is a gradient of 0.0005. Always state the units of both axes.
Percentage Grades
Roads quote gradients as percentages, which are slopes multiplied by 100. A 6 percent grade is a slope of 0.06, not 6. Confusing the two overstates the gradient by a factor of 100, which is the difference between a gentle hill and an unusable ramp.
Slope as a Rate of Change
Slope is a rate: y units gained per x unit consumed. This is the same idea as speed being distance per time, and it is why slope shows up wherever a quantity changes in proportion to another.
Worked Example
For 0,0 and 4,8 the rise is 8 and the run is 4, so the slope is 8 divided by 4, which is exactly 2.
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