Slope Calculator

Slope is one number that tells you how steep a line is and which way it leans. It is the rise divided by the run: how far the line climbs, over how far it travels across. This calculator takes two points, works out that number exactly, then writes it every other way you are likely to need: as an angle, as a percent grade, as a ratio, as a roof pitch and as the equation of the line.

The slope formula, and what rise over run means

Pick any two points on a line, (x₁, y₁) and (x₂, y₂). The rise is the change in y, y₂ minus y₁. The run is the change in x, x₂ minus x₁. The slope, written m, is the rise divided by the run:

m = (y₂ - y₁) / (x₂ - x₁)

Take (-2, 1) and (6, 5), the points the calculator opens with. The rise is 5 - 1 = 4 and the run is 6 - (-2) = 8, so m = 4 / 8 = 1/2. For every step of 1 to the right the line climbs half a step. The letter m for slope is old; its first known use in English is in a textbook by O'Brien from 1844, and it has stuck in y = mx + b ever since.

The sign carries the direction. Reading left to right, a positive slope climbs and a negative slope falls. A slope of 0 is a horizontal line, because the rise is zero. A vertical line has a run of zero, and since nothing can be divided by zero its slope is undefined rather than zero. The calculator says so and gives the line as x = c instead.

Which point goes first

Either one. Swapping the two points flips the sign of the rise and of the run together, and a negative over a negative is the same fraction. What you cannot do is mix the order: y₂ - y₁ on top with x₁ - x₂ underneath gives the right size with the wrong sign. The working panel prints the subtraction in full with your numbers, so this slip is easy to spot. The other classic mistake is putting the run on top. Rise always goes over run, up over across.

From slope to the equation of the line

Two points fix a line, so once you have m you can write the whole line three ways. The calculator gives all three, in exact fractions where your inputs allow it, so a slope of one third is printed as 1/3 and not only as 0.3333.

• Slope intercept form, y = mx + b. The b is the y-intercept, where the line crosses the y axis. Find it with b = y₁ - m × x₁. For the opening example, b = 1 - (1/2)(-2) = 2, so the line is y = 0.5x + 2.
• Point slope form, y - y₁ = m(x - x₁). This one needs no intercept at all, which makes it the quickest to write from a single known point: y - 1 = 0.5(x + 2).
• Standard form, Ax + By = C, with whole numbers. The same line is x - 2y = -4. Standard form is the only one of the three that can also describe a vertical line, which is why the calculator keeps it even when the slope is undefined.

The x-intercept, where the line meets the x axis, comes from setting y to zero: x = -b / m, which is -4 here. A horizontal line that is not the x axis never meets it, and the calculator says that rather than printing an error.

Parallel and perpendicular slopes

Parallel lines never meet, so they must rise at the same rate: parallel lines have equal slopes. Perpendicular lines meet at a right angle, and their slopes multiply to -1. In practice you flip the fraction and change the sign. The perpendicular to a slope of 1/2 is -2, and the perpendicular to 4/3 is -3/4. The one exception is the pair that has no number to flip: a horizontal line, slope 0, is perpendicular to a vertical line, whose slope is undefined.

Slope as an angle

Slope is the tangent of the angle the line makes with the horizontal, m = tan θ, so the angle comes back with the arctangent: θ = arctan(m). A slope of 1 is 45 degrees, a slope of 1/2 is 26.57 degrees, and 4/3 is 53.13 degrees.

For a falling line there are two honest answers, and the calculator gives both. In coordinate geometry the angle of inclination is measured anticlockwise from the positive x axis, so it always sits between 0 and 180 degrees; a slope of -0.4 has an inclination of 158.2 degrees. On a hill what you want is the steepness, the angle down from the horizontal, which for the same line is 21.8 degrees.

Slope as a percent grade, and why 10% is not 10 degrees

Road signs and site plans quote grade: the slope times 100. A 5% grade climbs 5 metres, or 5 feet, for every 100 across. The trap is that percent and degrees do not scale together, because the tangent does not grow in a straight line. A 10% grade is only 5.71 degrees, while a 10 degree slope is a 17.63% grade. At 45 degrees rise equals run and the grade is 100%, and past that the percentage keeps going: 60 degrees is a 173.2% grade, and as a line approaches vertical the grade grows without limit. So a grade above 100% is not an error; it just means steeper than 45 degrees.

Railways in Europe often use per mille instead, rise over run times 1000, so a 1% grade is 10‰. Builders in the UK and Australia like a ratio, 1 in 12 or 1:12, meaning one unit up for every twelve across. The calculator prints the ratio as rise to run, and as a roof pitch, which is the rise over a run of 12: a slope of 1/2 is a 6 in 12 roof.

Real grades, and what the limits actually are

The reference table above this guide lines up the numbers people most often need to check, and the badge under the result tells you which band your slope falls within. A few are worth knowing by heart.

• Wheelchair ramps. The 2010 ADA Standards cap a ramp's running slope at 1:12, an 8.33% grade or 4.76 degrees, and a single ramp run may rise no more than 30 inches before a landing. The cross slope, sideways across the ramp, may be no steeper than 1:48. A walking route no steeper than 1:20, which is 5%, is not treated as a ramp at all.
• Highways. US Interstate standards generally allow up to 6% in mountainous terrain, 5% in rolling country and 4% on level ground, with one extra percentage point in urban areas.
• Roofs. The 2021 International Residential Code allows asphalt shingles only on a pitch of 2 in 12 or steeper, which is 16.67% or about 9.46 degrees, and asks for a double layer of underlayment from 2 in 12 up to 4 in 12. Canadian code treats anything under 4 in 12 as a low slope roof.
• Streets and railways. Baldwin Street in Dunedin is listed at 34.8%, Canton Avenue in Pittsburgh at 37% and Bradford Street in San Francisco at 41%. The Pilatus Railway in Switzerland climbs at up to 48%, which is still under 26 degrees.

Measuring a real slope

To use the calculator on something physical, put point 1 at (0, 0) and point 2 at (run, rise), both in the same unit. A ramp that climbs 2 feet over 24 feet of floor is (0, 0) to (24, 2): exactly 1:12, 8.33%. The key word is horizontal. Rise over run uses the level distance across, taken from a plan or a level line, not the length measured along the sloping surface. Dividing by the sloped length instead is a common shortcut that is close for gentle slopes, about 1.5% out at 10 degrees, and gets worse the steeper you go. Over longer distances the same rule holds: climbing 396 feet over 3 miles, which is 15,840 feet, is a 2.5% grade, 1 in 40, or 1.43 degrees.

Reading the diagram

The drawing redraws with every keystroke. It shows both points, the full line through them, the run as a dashed blue leg and the rise as a dashed orange leg, and the angle at the first point. It is drawn to scale, one unit across the same length as one unit up, so a slope of 1 really does look like 45 degrees. Most graphing tools stretch one axis to fill the frame, which makes a gentle grade look steep and a steep one look gentle.

The run and rise form a right triangle with the segment between the points as its long side. That is why the calculator can also give the straight line distance between the points, shown as a single row. For more on that length, including distance in three dimensions and across the Earth, use the distance calculator; to solve the sides of a right triangle directly, use the Pythagorean theorem calculator. If your slope is a ratio you want to scale up, say a 1:12 ramp that must climb 21 inches, the proportion calculator finds the missing run.