📐 Slope Calculator

Slope, gradient and angle — three names for the same steepness

Slope is rise over run: how far a line climbs for every unit it travels sideways. The same steepness gets reported three different ways depending on who is asking. A mathematician says the slope is 0.5, a road engineer says the gradient is 50%, and a surveyor says the angle is 26.57 degrees. All three describe one line.

m = (y₂ − y₁) / (x₂ − x₁) = rise / run

Notice that a 100% gradient is 45 degrees, not vertical. This catches people out constantly, because percentage feels like it should max out at 100. It does not: gradient is a ratio, and a ratio has no upper limit.

Converting between the three

Gradient % = m × 100  ·  Angle = arctan(m)  ·  m = tan(angle)

Going from slope to gradient is a multiplication; going to an angle needs the arctangent. The common error is to assume the relationship between gradient and angle is linear — it is not, and it diverges badly above about 20%. A 10% gradient is 5.71 degrees and a 20% gradient is 11.31, not 11.42.

The slope-intercept form, and what the intercepts mean

y = mx + c

Once you have the slope, one point fixes the whole line: substitute to find c, the value of y where the line crosses the vertical axis. The x-intercept comes from setting y to zero, giving x = −c/m. Through (2, 3) and (8, 15), the slope is 2, c is −1, and the line is y = 2x − 1, crossing the x-axis at 0.5.

Two special cases break the formula. A horizontal line has a slope of zero, which is fine. A vertical line has a run of zero, so the slope is undefined rather than infinite, and the line cannot be written as y = mx + c at all — it is written x = k. Any calculator that reports a vertical line's slope as a huge number is misleading you.

Perpendicular lines have slopes whose product is −1, so the perpendicular to a slope of 2 is −0.5. Parallel lines share the same slope. These two facts solve most geometry problems that look like they need trigonometry.

Distance and midpoint come free

Distance = √[(x₂−x₁)² + (y₂−y₁)²]  ·  Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)

The distance formula is Pythagoras applied to the rise and the run, which is why the triangle calculator solves the same problem from a different angle. For the points above, the distance is √180, about 13.4164, and the midpoint is (5, 9).

Where slope shows up outside a textbook

Road signs use gradient as a percentage, and so do railway grades, usually expressed as 1 in 50 rather than 2%. Roofs use rise per 12 units of run — a 6:12 roof is a slope of 0.5 — which is exactly what the roofing calculator converts into a pitch multiplier. Drainage pipes are specified as a fall, commonly 1 in 40 for waste. Accessibility standards cap ramps at 1:12, about 8.3%, with the ADA and most building codes using that figure.

In data rather than geometry, slope is the rate of change: the coefficient in a linear regression is a slope, measured in units of y per unit of x. The interpretation is identical, which is why a line fitted through a scatter of points answers “how much does y change when x goes up by one”.

A worked example

A drainage run must fall 1 in 80 over 24 metres. The slope is 0.0125, the gradient is 1.25%, the angle is 0.72 degrees, and the total fall is 0.3 metres — so the outlet sits 300 mm below the inlet. If the available drop is only 200 mm, the gradient achievable is 1 in 120, flatter than the specification, and the pipe diameter or the route has to change. The arithmetic is trivial; getting the direction of the inequality right is what matters on site.

Two neighbours worth knowing: the triangle calculator, for the question this one leads into, and the quadratic formula calculator, which answers it from the other direction.

Frequently Asked Questions (FAQs)