How to use the slope calculator
- Choose whether you know two points or a point and the slope.
- Enter the first point (x₁, y₁).
- Enter the second point (x₂, y₂), or the slope m.
- Read the slope (or the equation) at the top, then the slope-intercept, point-slope and standard forms, the intercepts, the angle with the x-axis and, for two points, the distance and midpoint.
With whole numbers or short decimals, the slope and intercept are shown as exact fractions, such as 1/3, with the decimal beside them.
The slope formula
The slope is the rise (the change in y) divided by the run (the change in x). A positive slope climbs from left to right, a negative slope falls, a slope of 0 is a horizontal line, and a vertical line has no slope at all because its run is 0. The y-intercept b is where the line crosses the y-axis. Standard form uses whole numbers with no common factor and a positive x coefficient, so it is easy to compare two lines.
Worked example
The calculator starts with the points (2, 3) and (6, 11).
- Rise: 11 − 3 = 8.
- Run: 6 − 2 = 4.
- Slope: m = 8 ÷ 4 = 2.
- y-intercept: b = 3 − 2 × 2 = −1, so the line is y = 2x − 1.
- Point-slope form: y − 3 = 2(x − 2). Standard form: 2x − y = 1.
- Angle: arctan(2) ≈ 63.4349°. Distance: √(4² + 8²) = √80 ≈ 8.944272. Midpoint: (4, 7).
Check with the second point: 2 × 6 − 1 = 11. With (2, 3) and (5, 4) instead, the rise is 1 and the run is 3, so the slope is 1/3, about 0.333333.
Slope, angle and grade
The same steepness can be written three ways. Road signs use grade, the slope as a percentage; builders often use rise over run; maps and physics use the angle.
| Slope (rise / run) | Grade | Angle |
|---|---|---|
| 1/12 | 8.33% | 4.76° |
| 0.1 | 10% | 5.71° |
| 1/2 | 50% | 26.57° |
| 1 | 100% | 45° |
| 2 | 200% | 63.43° |
A 100% grade is a 45° slope, not a vertical wall. The 1/12 row is the steepest running slope allowed for a new wheelchair ramp under section 405.2 of the 2010 ADA Standards for Accessible Design: 1 inch of rise for every 12 inches of run.
Parallel and perpendicular lines
Parallel lines have the same slope. Perpendicular lines have slopes whose product is −1, so the perpendicular to a line with slope 2 has slope −1/2. Use the point and slope mode to write that line through any point: through (2, 3) with slope −0.5 the equation is y = −0.5x + 4.
Frequently asked questions
What does an undefined slope mean?
The two points have the same x value, so the run is 0 and the formula would divide by zero. The line is vertical, and its equation is just x = (that value). Its angle with the x-axis is 90°.
Does it matter which point I call the first?
No. Swapping the points flips the sign of both the rise and the run, and the two minus signs cancel. (11 − 3) ÷ (6 − 2) and (3 − 11) ÷ (2 − 6) are both 2.
How do I turn a slope into an angle?
Take the inverse tangent: θ = arctan(m). A slope of 1 is 45°, and a slope of 0.5 is about 26.57°. To go the other way, m = tan θ. The right triangle calculator works with the rise and run as the two legs.
How do I find the distance between the two points?
Use the Pythagorean theorem on the rise and the run: √(Δx² + Δy²). The calculator shows it for two points, and the distance calculator also handles 3D points and distances on the globe.
Related calculators
- Distance Calculator — Straight-line distance between 2D or 3D points, or the great-circle distance between two places on Earth.
- Right Triangle Calculator — Enter any two of the legs, hypotenuse, angles or area to solve a right triangle completely.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0 with exact and decimal roots, the discriminant, vertex and factored form.
- Percentage Calculator — Percent of a number, what percent one number is of another, percentage change and difference.
Results are calculated in your browser. Double-check anything important before you rely on it. See our disclaimer.