How to Find the Slope of a Line (and What the Number Means)

Slope wears three different outfits and confuses people every time it changes clothes. In algebra it is m. On a motorway sign it is a percentage. On a set of building drawings it is a ratio like 1:12. Same measurement, different scaling.

Rise over run, in order

Pick two points on the line. Subtract the y values to get the rise. Subtract the x values, in the same order, to get the run. Divide one by the other.

Using (2, 3) and (6, 11): the rise is 11 minus 3, which is 8. The run is 6 minus 2, which is 4. Slope is 8 divided by 4, so 2. The line climbs two units for every one unit it travels to the right.

Order is where nearly every mistake happens. Calculate 11 minus 3 on top but 2 minus 6 underneath and you get negative 2, a line tilting the wrong way. Decide which point is first, then use it first in both subtractions.

Positive, negative, zero and undefined

A positive slope rises as you read left to right. A negative slope falls. A slope of zero is a flat horizontal line, because the rise is 0 and zero divided by anything is zero. A vertical line has no slope, because the run is 0 and you cannot divide by zero. Calculators report that as undefined rather than infinity, and undefined is the correct answer, not an error message.

The same number as a percentage or a ratio

Multiply a slope by 100 and you have a percent grade. A 6% hill climbs 6 feet for every 100 feet of horizontal travel, so its slope is 0.06. Going the other way, a slope of 2 is a 200% grade, steeper than any road you have driven, which makes it a useful sanity check that you have not flipped rise and run.

Ratios run backwards from what most people expect. A 1:12 accessibility ramp rises one unit for every twelve along, so the slope is 0.083 and the grade is 8.3%, not 12%. Builders use the same rise and run language for steps, which is why the arithmetic behind stair rise and run is identical to the arithmetic on a graph. If the percent conversion is the part that keeps slipping, the rules in our guide to calculating percentages apply here unchanged.

Using the slope calculator

The calculator has four boxes: x1, y1, x2, y2. Put the coordinates of your first point in the first pair and the second point in the second pair, then press calculate. You get the slope as a fraction and as a decimal, the angle of inclination, the distance between the two points, and the equation of the line in y = mx + b form.

The order you enter the points in does not matter. The calculator subtracts consistently, so (6, 11) first and (2, 3) second still returns a slope of 2.

Where the two points come from

In homework they are handed to you. In real work you read them off something. On a graph, choose two places where the line passes cleanly through a grid intersection instead of eyeballing a point in the middle of a square. On a survey drawing, the run is the horizontal distance and the rise is the difference in elevation between the two ends. On a roof, run is the horizontal distance from wall to ridge and rise is the height gained, which is why roof pitch gets quoted as something like 6:12.

Common questions

What is the slope of a vertical line? Undefined. The run is zero, and dividing by zero has no answer. Do not write infinity on a test paper; the expected answer is that the slope does not exist.

How do I turn a slope into an angle? Take the inverse tangent. A slope of 1 is exactly 45 degrees. A slope of 0.5 is 26.6 degrees and a slope of 2 is 63.4 degrees. Notice that doubling the slope does not double the angle.

Is a 1:12 ramp a 12% slope? No, and having it backwards is how ramps fail inspection. 1:12 means one unit up per twelve along, which is 1 divided by 12, or 8.3%.

What does the b in y = mx + b mean? It is the y intercept, the value of y where the line crosses the vertical axis. The calculator finds it by taking one of your points and subtracting the slope multiplied by that point's x value.