Slope Calculator
Find the slope of a line from two points, from rise and run, from a point and an angle or grade, or from its equation. Get the slope as a fraction, decimal, angle and percent grade, plus the distance, midpoint, the line's equation in every form, its intercepts and a graph, with every step shown.
Whole numbers, decimals and fractions all work: 3, −2.5, 3/4 or 1 1/2.
Changing the unit converts the four numbers above. The slope stays the same; the distance gets a unit.
Mix units if you like, such as a rise in cm over a run in m. Example: a ramp that rises 1 inch for every 12 inches.
Any form: y = 2x + 3, 2x + 3y = 6, y − 3 = 2(x − 1), x = 4 or 3x − 2y + 6 = 0.
How your slope is worked out
Your points: rise over run, then the line through them.
- 1
Find the rise
y₂ − y₁8 − 2= 6
- 2
Find the run
x₂ − x₁4 − 1= 3
- 3
Divide
m = rise ÷ run6 ÷ 3m = 2
- 4
Angle and line
63.43° · 200%y = 2x
Read more: the formulas, worked examples and typical slopes
The slope (or gradient) of a line says how steep it is: how far it goes up for every 1 step across. It is the same everywhere along a straight line, so any two points on the line give the same answer. Slopes are written as a number m (maths), an angle in degrees, a percent grade (roads and ramps), a ratio 1 : x (ramps and railways) or a pitch x/12 (roofs).
The formula
Slope m = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁)
Angle = tan⁻¹(m) · Percent grade = m × 100 · Ratio = 1 : (1 ÷ m) · Pitch = 12 × m in 12
Distance = √((x₂ − x₁)² + (y₂ − y₁)²) · Midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Slope-intercept form: y = mx + b, where b = y₁ − m × x₁
Point-slope form: y − y₁ = m(x − x₁) · Standard form: Ax + By = C, slope = −A ÷ B
Parallel: same m · Perpendicular: −1 ÷ m · Distance from (p, q) to Ax + By = C: |Ap + Bq − C| ÷ √(A² + B²)
Worked example
For the default points (1, 2) and (4, 8): the rise is 8 − 2 = 6 and the run is 4 − 1 = 3, so the slope is m = 6 ÷ 3 = 2. The angle is tan⁻¹(2) ≈ 63.43° and the grade is 2 × 100 = 200%. Putting (1, 2) into y = mx + b: 2 = 2 × 1 + b, so b = 0 and the line is y = 2x (point-slope y − 2 = 2(x − 1), standard 2x − y = 0). It crosses both axes at (0, 0). The distance between the points is √(3² + 6²) = √45 = 3√5 ≈ 6.7082, and the midpoint is ((1 + 4) ÷ 2, (2 + 8) ÷ 2) = (2.5, 5).
Rise and run: a ramp that rises 1 inch for every 12 inches has a slope of 1/12 ≈ 0.0833, a grade of 8.33% and an angle of tan⁻¹(1/12) ≈ 4.76°. Rising 50 cm over 2 m is 0.5 ÷ 2 = 1/4, a 25% grade.
From an equation: 2x + 3y = 6 → 3y = −2x + 6 → y = −(2/3)x + 2. The slope is −2/3 (falling), the y-intercept is (0, 2) and the x-intercept is (3, 0).
How steep is that? (typical slopes)
| Example (typical) | Grade | Angle | Slope |
|---|---|---|---|
| Main-line railway, steep section | 2% | 1.15° | 1/50 |
| Aircraft approach glide path (3°) | 5.24% | 3° | 0.0524 |
| Steep motorway or highway hill | 6% | 3.43° | 3/50 |
| Wheelchair ramp, steepest usually allowed (1 : 12) | 8.33% | 4.76° | 1/12 |
| Steep city street | 15% | 8.53° | 3/20 |
| Baldwin Street, New Zealand, among the steepest streets | about 35% | 19.3° | 0.35 |
| Roof at 6/12 pitch | 50% | 26.57° | 1/2 |
| House stairs (18 cm rise, 28 cm going) | 64% | 32.7° | 9/14 |
| 45° slope | 100% | 45° | 1 |
Notes
- The typical figures are rough guides only. Ramp, road and stair limits are set by each country's own rules (for example the US ADA and the UK Building Regulations both use 1 : 12 for many ramps), so check the rules where you are before you build.
- Grade is not the same as angle: a 100% grade is 45°, not 90°. A vertical wall has no grade at all (the slope is undefined).
- Answers are exact fractions where the inputs are exact. A slope worked out from an angle uses tan, so it is a rounded decimal.
- Road signs give grades along the road surface or the map distance; over short distances the difference is tiny for normal road slopes.
Common questions
How do you find the slope from two points?
Take the change in y (the rise) and divide it by the change in x (the run): m = (y₂ − y₁) ÷ (x₂ − x₁). For the points (1, 2) and (4, 8) the rise is 8 − 2 = 6 and the run is 4 − 1 = 3, so the slope is 6 ÷ 3 = 2. The line goes up 2 for every 1 across.
What does a negative, zero or undefined slope mean?
A positive slope rises from left to right and a negative slope falls. A slope of 0 is a flat, horizontal line such as y = 1. When the two points have the same x, the run is 0 and you would divide by 0, so a vertical line such as x = 3 has an undefined slope.
How do you turn a slope into an angle or a percent grade?
The angle is the inverse tangent of the slope: a slope of 2 is tan⁻¹(2) ≈ 63.43°. The percent grade is the slope times 100, so a slope of 1/12 is 8.33%. A 45° angle is a slope of 1, or a 100% grade, where the rise equals the run.
What is the equation of a line through two points?
Work out the slope m, then put one point into y = mx + b to find b. For (1, 2) and (4, 8), m = 2, and 2 = 2 × 1 + b gives b = 0, so the line is y = 2x. In point-slope form it is y − 2 = 2(x − 1), and in standard form 2x − y = 0.
What are the slopes of parallel and perpendicular lines?
Parallel lines have the same slope. A perpendicular line has the negative reciprocal slope: flip the fraction and change the sign. So a line with slope −2/3 has perpendicular lines with slope 3/2. Multiplying the two slopes always gives −1, except for a flat line, whose perpendicular is vertical.
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