What Is Slope?
One line, three ways of saying the same thing
Slope is how steep something is, measured as the vertical change divided by the horizontal change over the same stretch. That is the rise over the run. A slope of 2 climbs 2 units for every 1 unit across. Related versions of the same rise over run idea are called gradient, grade, pitch or fall, depending on the field, though each carries its own conventions.
What slope means, in one line
On a line drawn on a grid, slope compares two distances: how far something rises against how far it travels sideways while doing it. More generally it compares a change in whatever the vertical axis measures with the change in whatever the horizontal axis measures.
slope = rise ÷ run
That is the whole definition, and everything else on this page is a consequence of it.
- A slope of 3 rises 3 for every 1 across. Steep.
- A slope of 0.05 rises 1 for every 20 across. Nearly flat.
- A negative slope falls instead of rising, reading left to right.
Two things follow from the definition that are worth having straight from the start. To work a straight line’s slope out from coordinates you need two distinct points, because a single point has nothing to be compared against. And slope is constant along a straight line: pick any two points on it, anywhere, and you get the same answer. That is what makes a line straight.
Why sources seem to contradict each other about slope
Look slope up in three places and you can easily come away with three different-looking answers. One says 1. One says 100 percent. One says 45 degrees. They are all describing the same line.
The same steepness gets written three ways, and which one you meet depends on the trade:
| Written as | Looks like | Who uses it |
|---|---|---|
| A number or ratio | 1, or 0.5, or 1 in 12 | Maths, and accessibility rules |
| A percentage, called grade | 100%, or 50% | Roads, railways, surveying, drainage |
| An angle in degrees | 45°, or 26.6° | Roofing, engineering drawings |
Converting between the first two is easy: a percentage is just the ratio multiplied by 100. The US Geological Survey defines percent slope exactly that way, as the elevation change divided by the horizontal distance, then multiplied by 100. So a slope of 0.5 is a 50 percent grade.
The third one does not behave like the other two, and this is one of the most common misunderstandings about slope.
A 100 percent grade is 45 degrees, not 90. Percent and degrees are different scales, and outside zero the two numbers never agree. At 100 percent the rise equals the run, which is a 45 degree angle by definition. Doubling to 200 percent gets you to about 63 degrees, not 90. As the run shrinks towards zero the percentage grows without limit, but a vertical drop has no percentage at all: the run is zero and you cannot divide by it.
So percentages above 100 are perfectly normal on steep ground, and reading a 30 percent grade as “a third of the way to vertical” is wrong: it is about 17 degrees. The two scales are joined by the tangent: angle = arctan(slope) and slope = tan(angle), with the percentage being the slope times 100. A 30 percent grade is a slope of 0.30, and arctan(0.30) is about 16.7 degrees. Convert with that rather than by instinct.
The four kinds of slope a line can have
Every straight line falls into one of four cases, and the names come straight from the number.
- Rising. A positive number. The line goes up as you read left to right, which is what a positive slope looks like on a graph.
- Falling. A negative number. Down as you read left to right.
- Flat. Zero. The rise is nothing however far you travel, so the line is horizontal.
- Vertical. Undefined, which is not the same as zero. The run is nothing, and dividing by nothing is not a thing you can do, so there is no number to give.
Flat and vertical are the pair people mix up, and the difference matters because one is a number and the other is the absence of one. A flat line has a slope, and that slope is zero. A vertical line does not have a slope at all.
What slope means in different subjects
Slope is taught in a maths lesson as a bare number, and then met everywhere else as something with units attached. That shift is where a lot of the confusion comes from, particularly in physics.
- Physics. On a graph of position against time, the slope is velocity, which is speed with a direction attached, and it comes out in metres per second because that is what the two axes were measured in. The slope of a velocity against time graph is acceleration. The steepness is a rate, and its units are the vertical axis’s units divided by the horizontal axis’s.
- Geography and surveying. The steepness of the ground, given as a percentage or a ratio. Contour lines close together mean a big rise over a short run, which is why a map can be read for steepness without a single calculation.
- Statistics. In a line fitted through scattered data, the slope is how much the outcome changes for a one unit change in the input. It is the number the whole exercise usually exists to find.
- Economics. On a straight cost line the slope is the added cost per unit. On a curved cost function it is the slope of the tangent at a point that gives the marginal cost. Same arithmetic, different name.
- Construction and trades. Called pitch on a roof, fall on a drain, grade on a road or a driveway. All of them are rise over run, and each trade has its own preferred units.
The thing worth carrying between all of them: slope is a rate of change. How much the thing on the vertical axis moves for each unit the thing on the horizontal axis moves. In a maths exercise both axes are just numbers so the answer is a bare number. Everywhere else the axes mean something, so the answer does too.
Working slope out from two points
Take the two points, subtract to get the rise and the run, and divide.
slope = (y2 − y1) ÷ (x2 − x1)
Through (2, 3) and (6, 11): the rise is 11 − 3 = 8, the run is 6 − 2 = 4, so the slope is 8 ÷ 4 = 2.
The only rule to keep is that both subtractions run in the same order. Start with the same point for the rise and for the run. Swapping one but not the other flips the sign and turns a rising line into a falling one.
Two points are what you need when all you have is the graph. If the line arrives as an equation instead, the slope is usually already sitting in it: in the form y = mx + b, the number multiplying x is the slope, so y = −3x + 7 has a slope of −3 without any arithmetic at all.
On a graph, counting is quicker than subtracting: find two places where the line passes cleanly through a corner of the grid, count the squares up and the squares across between them, and divide. If you want the arithmetic done and checked, along with the grade and the angle, the slope calculator takes two points and returns all three.
Frequently asked questions
What is slope in simple words?
How steep something is, written as a number. Walk along it and slope tells you how much you climb for each step forward. A big number is steep, a small number is gentle, zero is flat, and a negative number means you are going down rather than up.
Is slope the same as gradient?
In everyday and school use, yes. British and Commonwealth textbooks generally say gradient where American ones say slope, and they mean the same rise over run.
Two places to be careful. In surveying and civil engineering, gradient often implies the percentage or ratio form rather than a bare number. And in higher mathematics, gradient means something genuinely different: a vector describing the direction of steepest increase of a surface. If you are reading a maths lesson, they are the same word. If you are reading a physics or engineering paper, check which one is meant.
Can slope be a fraction or a decimal?
Both, and fractions are usually more useful. A slope of 1/12 tells you directly to go 12 across for every 1 up, which is a measurement you can take. Written as 0.0833 it is the same steepness and much harder to use. Keep the fraction unless you have been asked for a decimal.
Does the order of the two points change the answer?
No, as long as you are consistent. Taking the points in the other order flips the sign of both the rise and the run, and two flipped signs cancel when you divide. What does change the answer is flipping one and not the other, which is a common arithmetic mistake with slope.
What does slope mean on a graph in physics?
Whatever the vertical axis measures, per unit of whatever the horizontal axis measures. On position against time, that is velocity. On velocity against time, acceleration. The slope is not just a shape, it is a quantity with units, and you can read those units straight off the axis labels.
It is worth being careful which way round the division goes. Slope is the vertical axis divided by the horizontal one, so on a distance-time graph it is distance divided by time, not time divided by distance. Getting that backwards gives you a number that looks reasonable and means nothing.
Does slope have units?
It depends on what the two axes measure, and this is a point many short explanations omit. The rule is simple: the units of the slope are the units of the vertical axis divided by the units of the horizontal axis. A distance in metres plotted against time in seconds gives a slope in metres per second. Volume in litres against time in minutes gives litres per minute.
Slope only comes out as a plain number with no units when both axes measure the same quantity in the same units, which is exactly the case in a maths lesson where both are just numbers on a grid. That is why school slope looks unit-free and lab slope does not. If you are reading a value off a graph in physics, chemistry or economics and you have not written a unit next to it, you have probably lost half the meaning.
Why is slope called m?
Nobody knows, and the popular explanation is not supported. The usual story is that m comes from the French monter, meaning to climb. Wolfram MathWorld, summarising the symbol study by Jeff Miller, states plainly that there is no evidence for the connection and points out that Descartes, who was French, did not use m at all. The historian Howard Eves suggested it simply happened.
What is documented is when it appears rather than why. Wikipedia dates the first appearance in English to O’Brien in 1844. So the honest answer to a maths teacher is that m is a convention with no known reason behind it, and that anyone telling you it stands for monter is repeating a guess.
Why is a vertical line’s slope undefined instead of infinity?
Because there is no single number to settle on. Slope is the rise divided by the run, and on a vertical line the run is zero, so the division has no answer. Undefined is not a polite way of saying very large.
The reason infinity does not work is worth seeing. Tilt a steep line slightly to one side and its slope is a huge positive number. Tilt it slightly the other way and the slope is a huge negative number. As the line stands up the two approaches head to opposite ends, so there is no value the slope is settling towards. A horizontal line is different: its rise is zero and zero divided by something is a perfectly good zero. Flat has a slope, vertical does not.
What is the slope of a curved graph?
A curve does not usually have one constant slope. Its steepness can change from point to point, and at a few kinds of point, such as a sharp corner, there is no single slope at all. So the question has two answers depending on what you are asking for.
Between two points on the curve, draw the straight line joining them and take its slope. That is the average rate of change over that stretch. At a single point where the curve is smooth, draw the straight line that runs in the same direction as the curve at that spot, the tangent, and take its slope. That is the instantaneous rate of change, and it is what a derivative in calculus gives you.
A tangent is defined by matching the curve’s direction, not by staying on one side of it. It may touch and stay clear, or it may cross: the tangent to y = x³ at the origin runs straight through the curve and is a tangent all the same.
This is why a position against time graph that bends is a journey at changing speed: the average velocity over the whole trip and the velocity at one particular moment are different numbers, and both are slopes.