Slope Calculator
Find the slope between two points, with the line equation, angle, distance and midpoint.
Results
| Item | Value |
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Steps
| Step | Working |
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Quick Answer
Slope is rise over run: m = (y₂ − y₁) ÷ (x₂ − x₁). Between the points (1, 2) and (4, 8), the rise is 6 and the run is 3, so the slope is 2. The line equation is y = 2x + 0.
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Table of Contents
How This Slope Calculator Works
Slope measures how steep a line is. Enter two points, and the calculator finds the slope, the line equation in y = mx + b form, the angle, the distance between the points, and the midpoint.
Quick explanation: m = rise ÷ run.
- Positive slope: the line rises from left to right.
- Negative slope: the line falls from left to right.
- Zero: a flat line. A vertical line has an undefined slope.
How to Use the Slope Calculator
- Enter the x and y values of the first point.
- Enter the x and y values of the second point.
- Press Calculate.
- Read the slope, the line equation, the angle, the distance and the midpoint.
- Check the steps table to follow the working.
Core Formulas
- Slope: m = (y₂ − y₁) ÷ (x₂ − x₁)
- Y-intercept: b = y₁ − m × x₁
- Angle: arctan(m)
- Distance: √((x₂ − x₁)² + (y₂ − y₁)²)
- Midpoint: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Key Terms
- Slope (m): how steep a line is: the change in y divided by the change in x.
- Rise: the vertical change between two points.
- Run: the horizontal change between two points.
- Y-intercept (b): the point where the line crosses the y-axis.
- Slope-intercept form: a line written as y = mx + b.
- Midpoint: the point exactly halfway between two points.
- Parallel and perpendicular lines: parallel lines have equal slopes. Perpendicular slopes multiply to −1.
Worked Example
Find the slope between (0, 0) and (2, −1).
- Rise = −1 − 0 = −1
- Run = 2 − 0 = 2
- Slope = −1 ÷ 2 = −0.5
- Equation: y = −0.5x + 0
Real-Life Examples
- Road grade: a road that climbs 30 m over 500 m has a slope of 0.06, a 6% grade, or an angle of about 3.43°.
- Wheelchair ramp: US ADA guidance limits most ramps to a slope of 1:12. That is about 8.33%, or 4.76°.
- Roof pitch: a 6/12 roof rises 6 in for every 12 in across, a slope of 0.5 or about 26.57°.
- Rate of change: if a phone plan costs $40 for 2 GB and $55 for 5 GB, the slope is (55 − 40) ÷ (5 − 2) = $5 per GB.
Slope vs Grade vs Angle
- Slope: rise ÷ run, as a number such as 0.25.
- Grade: slope × 100, as a percentage such as 25%. Used for roads and paths.
- Angle: arctan(slope), in degrees, such as 14.04°.
- Pitch: rise per 12 units of run, such as 3/12. Used for roofs in the US.
Slope, Grade, Angle and Pitch
| Slope | Grade | Angle | Roof pitch |
|---|---|---|---|
| 0.05 | 5.00% | 2.86° | 0.6/12 |
| 0.0833 | 8.33% | 4.76° | 1/12 |
| 0.1 | 10.00% | 5.71° | 1.2/12 |
| 0.25 | 25.00% | 14.04° | 3/12 |
| 0.5 | 50.00% | 26.57° | 6/12 |
| 0.75 | 75.00% | 36.87° | 9/12 |
| 1 | 100.00% | 45.00° | 12/12 |
| 2 | 200.00% | 63.43° | 24/12 |
Common Mistakes
- Mixing the order: subtract x and y in the same point order.
- Dropping a negative: take care when subtracting a negative number.
- Dividing by zero: if both x values match, the line is vertical.
- Swapping rise and run: slope is change in y over change in x, not the other way round.
- Confusing slope with angle: a slope of 1 is 45°, but a slope of 2 is not 90°.
Frequently Asked Questions
Multiply the slope by 100. A slope of 0.06 is a 6% grade.
It rises 6 units for every 12 across, so the slope is 0.5 and the angle is about 26.57°.
It is 0, because the y value stays the same as x changes.
It is the negative reciprocal. If one line has a slope of 2, a line at right angles to it has a slope of −1/2.
Yes. A slope of 2/3 means the line rises 2 units for every 3 units across. It is the same as about 0.667.
Rearrange it into y = mx + b. The number in front of x is the slope.
Subtract the y values, subtract the x values in the same order, then divide: m = (y₂ − y₁) ÷ (x₂ − x₁).
It means the line is flat, because y does not change.
It is the slope of a vertical line. The run is 0, and you cannot divide by 0.
It is the slope-intercept form of a line. m is the slope, and b is where the line crosses the y-axis.
Take the arctangent of the slope. A slope of 1 is an angle of 45°.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Slope | Definition and formula for slope. | https://en.wikipedia.org/wiki/Slope |
| Wikipedia, Linear equation | Slope-intercept form. | https://en.wikipedia.org/wiki/Linear_equation |