Line Slope Finder

What Is Slope?

The slope of a line tells you two things: how steep the line is, and which way it goes. It is usually written with the letter m.

Slope compares how far a line goes up or down with how far it goes across. The up-or-down distance is called the rise. The across distance is called the run.

slope = rise ÷ run

−4−224−4−224run = 4rise = 3(−2, −1)(2, 2)
From (−2, −1) to (2, 2) the line rises 3 and runs 4, so its slope is 3/4.

Finding slope by counting

Pick two points on the line, ideally where it crosses the corners of grid squares. Then move from the left point to the right point:

  1. Count the squares you move across. That is the run.
  2. Count the squares you move up (positive) or down (negative). That is the rise.
  3. Divide the rise by the run, and simplify the fraction.

In the diagram above, the run is 4 and the rise is 3, so the slope is 3/4. For every 4 steps to the right, the line goes up 3.

See this line in Slope Finder →

The four kinds of slope

  • Positive slope: the line goes up from left to right, like walking uphill. Example: m = 2.
  • Negative slope: the line goes down from left to right, like walking downhill. Example: m = −1/2.
  • Zero slope: the line is flat (horizontal). The rise is 0, so the slope is 0.
  • Undefined slope: the line is vertical. The run is 0, and you can’t divide by zero, so the slope is undefined.

There is more on the last two in Zero and undefined slope.

Bigger numbers mean steeper lines

The further the slope is from zero, the steeper the line. A line with slope 3 is steeper than a line with slope 1/2. The sign only tells you the direction, so a line with slope −4 is steeper than a line with slope 1, even though it goes down.

Slope in everyday life

  • Roads: a sign showing a 10% gradient means the road rises 10 metres for every 100 metres across. That is a slope of 10/100 = 1/10.
  • Ramps: wheelchair ramp guidelines often use a ratio of 1:12, which means a rise of 1 for every 12 across, a slope of 1/12.
  • Rates: if a taxi charges £2 for every extra mile, a graph of cost against miles has a slope of 2.

Example

A line passes through (1, 2) and (4, 8). Moving from (1, 2) to (4, 8), the run is 4 − 1 = 3 and the rise is 8 − 2 = 6.

Slope = 6 ÷ 3 = 2. The line goes up 2 for every 1 step to the right.

Counting squares works well on a graph. When you only have the coordinates, use the slope formula, which does the same thing with numbers.