Geometry
Slope Calculator
Enter two points — or a point and a slope — and instantly get the slope, the y-intercept, and the equation of the line in slope-intercept form (y = mx + b), plus the angle, percent grade, and a graph. Every step is shown as you type.
Enter two points
Point 1 — (x1, y1)
Point 2 — (x2, y2)
m =2
Steep, rising line
y = 2x
rise (Δy)
4
run (Δx)
2
y-intercept b
0
angle θ
63.43°
grade %
200%
distance
4.4721
Step by step
m = (y₂ − y₁) / (x₂ − x₁)
m = (6 − 2) / (3 − 1)
m = 4 / 2 = 2
Try an example
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The basics
What is slope?
The slope of a line measures its steepness — how much it climbs or falls for each step sideways. It is the single number that defines a straight line's direction, written m. Two equivalent pictures help: a ratio you compute, and a triangle you can see.
m = (y₂ − y₁) / (x₂ − x₁)
Rise over run
The vertical change (the rise) divided by the horizontal change (the run). This is the form you actually calculate with from two points.
(6 − 2) / (3 − 1) = 4 / 2 = 2
m = tan θ
A direction you can see
Slope is the tangent of the angle the line makes with the horizontal. A larger slope is a steeper angle; a slope of 0 is flat and a vertical line has no defined slope at all.
m = 1 → θ = 45°
Method
How to find slope by hand
Three moves take you from two points to the slope every time.
- 1
Find the rise (change in y)
Subtract the first point's y from the second's. This is the vertical distance between the points.
rise = y₂ − y₁ = 6 − 2 = 4 - 2
Find the run (change in x)
Subtract the x-values in the same order. This is the horizontal distance — keep the order consistent with the rise.
run = x₂ − x₁ = 3 − 1 = 2 - 3
Divide rise by run
The quotient is the slope. If the run is zero, the line is vertical and the slope is undefined.
m = rise / run = 4 / 2 = 2
Slope-intercept form
Slope-intercept form: y = mx + b
Slope on its own describes steepness. Slope-intercept form describes the whole line: m is the slope you just found and b is the y-intercept, the height where the line crosses the y-axis. The calculator writes the equation for you in both modes — enter two points, or a point and a slope, and read the finished equation straight off the result.
y = mx + b
What each letter does
The coefficient of x sets the steepness and direction, while the constant term slides the whole line up or down without tilting it. Change one and the other is untouched.
b = y₁ − m·x₁
Getting the intercept from a point
Once the slope is known, substitute either point back into the equation and solve for the constant. Both points give the same answer, so use whichever has the friendlier numbers.
(1, 2), m = 2 → b = 2 − 2·1 = 0 → y = 2x
y − y₁ = m(x − x₁)
Point-slope form converts to it
Point-slope is the natural first write-up when you have a point and a slope. Expanding the bracket and collecting the constants turns it into slope-intercept form.
y − 2 = 2(x − 1) → y = 2x
Ax + By = C
Standard form converts too
Solve a standard-form equation for y and the coefficient of x becomes the slope while the leftover constant becomes the intercept.
4x − 2y = 0 → y = 2x
Worked examples
See every kind of slope
The same rise-over-run procedure handles positive, negative, zero, undefined, and fractional slopes — switch between them to watch the arithmetic change.
Given (1, 2) and (3, 6)
Subtract y's (rise): 6 − 2 = 4
Subtract x's (run): 3 − 1 = 2
Divide: m = 4 / 2
m = 2
Both points climb — the slope is positive and the line rises left to right.
Reference
The four kinds of slope
Every straight line falls into one of four cases. The sign and size of the slope tell you the line's direction and steepness before you even plot it.
m > 0
Positive — rises
The line climbs from left to right. As x grows, y grows too.
(1, 2) → (3, 6): m = 2
m < 0
Negative — falls
The line drops from left to right. As x grows, y shrinks.
(−2, 5) → (4, 1): m = −⅔
m = 0
Zero — horizontal
A flat line. The rise is zero, so there is no climb at all — written y = constant.
(0, 3) → (5, 3): m = 0
m = undefined
Vertical
The run is zero, so the formula divides by zero. The line is x = constant.
(3, 1) → (3, 7): undefined
Where it shows up
Why slope matters
Slope is a rate of change, so it appears anywhere one quantity grows or shrinks against another — far beyond the math classroom.
grade = rise / run × 100
Roads & ramps
Road signs and accessibility codes state steepness as a percent grade — the slope of the surface written as a percentage.
rate = Δy / Δx
Data & economics
On a graph, slope is the rate: speed on a distance–time plot, cost per unit on a price line, or a trend's direction over time.
pitch = rise / run
Building & roofing
Roof pitch and stair rise are slopes. A 6:12 roof rises 6 units for every 12 it runs — a slope of one half.
Worth knowing
Slope rules worth remembering
A handful of facts make slope problems faster and connect them to the equation of a line.
y = mx + b
Slope-intercept form
Once you know the slope m and the y-intercept b, the whole line is determined. The calculator gives you both.
b = y₁ − m·x₁
Finding the intercept
Plug the slope and one point into this to get where the line crosses the y-axis — no graphing needed.
m₁ = m₂
Parallel lines
Two lines are parallel exactly when their slopes are equal. They climb at the same rate and never meet.
m₁ · m₂ = −1
Perpendicular lines
Perpendicular lines have slopes that are negative reciprocals. Flip the fraction and change the sign.
order is free
Point order doesn't matter
Swapping which point is first flips the sign of both the rise and the run, so the slope is unchanged — just stay consistent.
θ = arctan(m)
Slope to angle
The angle a line makes with the horizontal is the inverse tangent of its slope. A slope of one is a clean 45 degrees.
Quick chart
Slope, angle & grade converter
Slope, the angle of incline, and percent grade are three names for the same steepness. Type any one to convert to the others, or scan the table for common road, ramp, and roof values.
I know the
Slope m
0.1
Angle θ
5.71°
Grade %
10%
| Slope | Ratio | Angle | Grade | Where you see it |
|---|---|---|---|---|
| 0 | 0 | 0° | 0% | Flat / level |
| 0.05 | 1/20 | 2.86° | 5% | ADA ramp max |
| 0.0833 | 1/12 | 4.76° | 8.33% | Wheelchair ramp |
| 0.1 | 1/10 | 5.71° | 10% | Steep road |
| 0.125 | 1/8 | 7.13° | 12.5% | — |
| 0.1667 | 1/6 | 9.46° | 16.67% | — |
| 0.25 | 1/4 | 14.04° | 25% | — |
| 0.3333 | 1/3 | 18.43° | 33.33% | — |
| 0.5 | 1/2 | 26.57° | 50% | 6:12 roof pitch |
| 1 | 1/1 | 45° | 100% | 45° — 12:12 roof |
| 2 | 2/1 | 63.43° | 200% | — |
| 3 | 3/1 | 71.57° | 300% | — |
Pitfalls
Common slope mistakes
Almost every wrong answer comes from one of these slips. Knowing them is half of avoiding them.
rise / run
Flipping rise and run
Slope is rise over run, not run over rise. Putting the x-difference on top inverts your answer.
keep the order
Mixing up the order
If y₂ is on top, x₂ must lead on the bottom. Subtracting the points in different orders flips the sign.
0 vs undefined
Zero vs. undefined
A horizontal line has a slope of zero; a vertical line is undefined. They are opposite cases, not the same thing.
watch the signs
Dropping a minus sign
Subtracting a negative coordinate adds. Missing the double-negative is the most common arithmetic error in slope problems.
FAQ
Common questions
Subtract the y-coordinates to get the rise, subtract the x-coordinates to get the run, then divide: m = (y₂ − y₁) / (x₂ − x₁). For (1, 2) and (3, 6) that is (6 − 2) / (3 − 1) = 4 / 2 = 2. The order of the points doesn't matter as long as you keep it consistent on top and bottom.
The slope formula is m = (y₂ − y₁) / (x₂ − x₁) — rise over run. It measures how steeply a line climbs or falls: the change in y for every one-unit change in x. A slope of 3 means the line rises 3 units for every 1 it moves right.
A vertical line has an undefined slope. Both points share the same x-value, so the run (x₂ − x₁) is zero, and dividing by zero is undefined. Its equation is written x = constant. A horizontal line, by contrast, has a slope of exactly zero.
A negative slope means the line falls from left to right: as x increases, y decreases. A positive slope rises, a zero slope is flat, and an undefined slope is vertical. The larger the absolute value, the steeper the line.
Once you have the slope m, substitute one known point into b = y₁ − m·x₁. The y-intercept b is where the line crosses the y-axis. Together they give the slope-intercept equation y = mx + b.
The angle of incline is θ = arctan(m), and the percent grade is simply the slope times 100. A slope of 1 is a 45° angle and a 100% grade. A 1/12 wheelchair ramp is about 4.76° and an 8.33% grade. Use the converter and chart above to look up any value.
Yes. Every pair of points on the same straight line produces the identical slope — that constant rate of change is exactly what makes the line straight. If two pairs give different slopes, the four points are not collinear.
They describe the same steepness in different units. Slope is a plain ratio (rise ÷ run); grade is that ratio written as a percentage (rise ÷ run × 100). A road with a slope of 0.06 is signed as a 6% grade.
Keep going
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Further reading
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The Cylinder Volume Formula (V = πr²h): How It Works, With Examples
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The Distance Formula: How to Find the Distance Between Two Points
The distance between two points is d = √((x₂−x₁)² + (y₂−y₁)²). It comes straight from the Pythagorean theorem: the horizontal and vertical gaps between the points are the two legs of a right triangle, and the distance is the hypotenuse.
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Written by
UC
Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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