Slope Calculator
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About Slope
Slope is algebra's rate of change — the single number that says how steeply a line climbs or falls. It's the m in y = mx + b, the grade sign on mountain highways, the pitch of a roof, and the derivative's whole ancestry. Two points determine it completely.
Enter both points (negatives welcome) and get the slope with the rise-over-run worked, the complete line equation, the angle in degrees, and the distance between the points — the full geometry package most homework questions want in one place.
Curved rather than straight? Parabolas live in the Quadratic Equation Solver
Rise Over Run, and Everything After
One subtraction stacked on another, then two bonuses:
m = (y₂ − y₁) ÷ (x₂ − x₁) b = y₁ − m·x₁ → line: y = mx + b Angle = arctan(m) · Distance = √((Δx)² + (Δy)²)
Worked example: through (1, 2) and (4, 8): m = 6 ÷ 3 = 2, intercept b = 0, so y = 2x, climbing at 63.4° with the points 6.71 units apart. Order doesn't matter — swap the points and both differences flip sign, leaving m unchanged.
Slopes, Angles & Grades
How slope, angle, and percent-grade language line up:
| Slope m | Angle | As a grade | Feels like |
|---|---|---|---|
| 0 | 0° | 0% | Flat road |
| 0.05 | 2.9° | 5% | Steep highway warning signs |
| 0.1 | 5.7° | 10% | Hard cycling climb |
| 0.5 | 26.6° | 50% | Steep staircase ramp territory |
| 1 | 45° | 100% | 45° — rise equals run |
| 2 | 63.4° | 200% | Steeper than most roofs |
| Undefined (vertical) | 90° | — | A wall |
Note the vocabulary trap: a “100% grade” is only 45°, not vertical — percent grade is slope × 100, rise over run, not a fraction of upright.
Reading Slopes in the Wild
Sign tells direction: positive slopes rise left-to-right, negative fall, zero is horizontal — and vertical is its own category with no slope value at all (division by zero, honestly undefined). Parallel lines share slopes; perpendicular ones multiply to −1 (slope 2 meets slope −½ at right angles) — the two facts that unlock most coordinate-geometry proofs.
Outside class, slope wears different uniforms: road grade (rise/run × 100 — the same m as a percentage), roof pitch (rise per 12 of run — a 6/12 pitch is m = 0.5), wheelchair-ramp standards (1:12 maximum, m ≈ 0.083), and in data, the regression line's slope is the “per-unit effect” everyone actually cares about. Same number, many costumes.
Frequently Asked Questions
How do I find slope from two points?
Subtract y-coordinates, subtract x-coordinates (same order!), divide: through (1,2) and (4,8), m = (8−2)/(4−1) = 2. The classic error is mixing subtraction orders between top and bottom — which silently flips the sign.
What is the slope of a vertical line?
Undefined — the run is zero and division by zero has no value. Not “infinity,” not zero (that's horizontal lines). Vertical lines are written x = constant because y = mx + b can't express them.
How do I get the line equation from the slope?
Solve b from either point: b = y₁ − m·x₁, then write y = mx + b. Through (1,2) with m = 2: b = 0, so y = 2x. The calculator does both steps and shows the finished equation.
How does slope convert to an angle?
Angle = arctan(m): slope 1 is 45°, slope 2 is 63.4°, slope 0.1 is 5.7°. It's not proportional — doubling a steep slope adds few degrees — which is why grades and angles feel so different as numbers.
What slope do perpendicular lines have?
Negative reciprocals: perpendicular to m = 2 is m = −½; their product is −1. Exception: horizontal (m = 0) is perpendicular to vertical (undefined) — the pair the rule can't express but geometry still honors.
What does a negative slope mean?
The line falls left-to-right: y drops as x grows. Through (−2, 4) and (2, −4), m = −2 — each unit rightward loses two units of height. In data terms, a negative relationship; on a road sign, downhill.
Methodology. This calculator uses standard, peer-reviewed mathematical formulas. It is reviewed and maintained by the Vast Calculators editorial team.
Last updated · July 11, 2026
Results are estimates for general use; verify critical figures independently.