Slope of a line through two points
The slope (or gradient) measures how steeply a line rises or falls between two points. This tool returns the slope, the line equation and the line’s angle — for students, surveyors and anyone working with linear graphs or gradients.
How it works
The slope is the change in y divided by the change in x, often called “rise over run”:
m = (y₂ − y₁) ÷ (x₂ − x₁)
From the slope and one point, the y-intercept is b = y₁ − m × x₁, giving the
line equation y = mx + b. The angle of the line is atan2(Δy, Δx) converted
to degrees. A positive slope rises left to right, a negative slope falls, a slope
of 0 is horizontal, and a vertical line has an undefined slope because the run
is zero — the tool reports this rather than dividing by zero.
Worked example
For the points (1, 2) and (4, 8):
| Result | Value |
|---|---|
| Δy / Δx | 6 ÷ 3 |
| Slope (m) | 2 |
| Equation | y = 2x + 0 |
| Angle | ≈ 63.43° |
Step by step: the rise is 8 − 2 = 6 and the run is 4 − 1 = 3, so the slope is 6/3 = 2. The y-intercept is 2 − (2 × 1) = 0, giving the equation y = 2x. The angle in degrees is arctan(2) ≈ 63.43°.
Reading the results
Positive slope: the line rises from left to right. A slope of 1 means equal rise and run — a 45° angle.
Negative slope: the line falls from left to right. The steeper the fall, the more negative the value.
Slope of 0: the line is perfectly horizontal — y stays constant no matter what x does. The equation simplifies to y = b.
Undefined slope (vertical line): when both x values are equal, the run is zero and division by zero is undefined. A vertical line has no slope, only an x-intercept of the form x = a.
Converting slope to angle: the arctan function gives the angle in radians; multiply by 180/π to convert to degrees. Note that angles returned by arctan live in the range −90° to 90°, matching the natural range of a non-vertical line’s inclination.
Common uses
- Algebra coursework: finding line equations from two points is a staple of coordinate geometry.
- Data analysis: the slope of a best-fit line is the rate of change — how much y changes per unit increase in x.
- Surveying and engineering: gradient between two elevation points on a profile.
- Physics: velocity from a displacement-time graph, or force from a potential-energy graph, both read directly as slopes.
All calculations run in your browser, with nothing uploaded.