Slope Calculator

How to use the slope calculator

Slope measures how steeply a line climbs: the change in y divided by the change in x, usually said as rise over run. From (1, 2) to (4, 8) the line climbs 6 while moving across 3, so the slope is 2 — every step right lifts it two steps up. A negative slope falls instead, and a slope of zero is flat.

The order of the two points does not matter as long as it is consistent. Taking the second point first flips the sign of both the rise and the run, and a negative divided by a negative is positive again. What does matter is not mixing them: subtracting y in one order and x in the other gives the slope with the wrong sign, and nothing about the answer will look suspicious.

That last distinction is worth being firm about. A vertical line has a run of exactly zero, and division by zero is undefined rather than infinite. Calling it "infinite slope" is informal limiting language: as a line tips towards vertical its slope grows without bound, but at vertical there is no value at all. The line still exists and is written x = 3 instead.

The angle of inclination is the one figure with a convention attached. It is measured counterclockwise from the positive x axis and runs from 0 up to but not including 180 degrees, so a line falling at 45 degrees has an inclination of 135 rather than −45. Taking arctan of the slope directly gives the wrong answer for every negative slope.

Slope outside the coordinate plane

Every real gradient is a slope: a roof pitch, a wheelchair ramp, a railway grade, a road sign warning of a descent. Road signs use percentages, which is slope multiplied by a hundred — an eight percent grade rises eight metres over a hundred travelled. Roofs are quoted as a rise over a twelve-unit run, so a 6:12 pitch is a slope of 0.5.

Ramp and access requirements are usually written as ratios for this reason — a ratio is checkable with a tape measure and an angle is not. Where a specific figure is required by regulation, take it from the regulation rather than from a conversion, because permitted maxima vary by application and by whether the work is new or existing.

Parallel, perpendicular, and the negative reciprocal

Two lines are parallel when their slopes are equal — same steepness, different position. They are perpendicular when the product of their slopes is exactly −1, which means each slope is the negative reciprocal of the other. A line of slope 2 is crossed at a right angle by a line of slope −0.5.

The perpendicular rule has one exception, and it is the same one as everywhere else on this page: a vertical and a horizontal line are perpendicular, but their slopes are undefined and zero, and no product of those two equals −1. The rule works for every pair except the one that includes a vertical line.

The formula, worked line by line

One division for the slope, and five more formulas that fall out of the same two points without any further measurement.

That silence is what makes it worth a deliberate habit: write both subtractions the same way round every time, even when the numbers are easy enough to do in your head.

slope m = (y₂ − y₁) ÷ (x₂ − x₁)
y-intercept b = y₁ − m × x₁
equation: y = mx + b
distance = √((x₂ − x₁)² + (y₂ − y₁)²)
midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

Why the sign of the slope tells you the direction

A positive slope rises left to right; a negative slope falls. The magnitude says how fast: a slope of 5 is five times steeper than a slope of 1, and a slope of 0.2 is five times shallower. Values between −1 and 1 describe lines closer to horizontal than to vertical, which is a useful sense check when a figure looks surprising.

Gradient and slope

The one that reads differently is pitch, because its run is fixed at twelve by convention rather than being whatever the horizontal distance happens to be. A 6:12 pitch is a slope of one half, and a 12:12 pitch is a slope of one, which is 45 degrees.

Questions people ask