Slope from Two Points
Enter coordinates of Point A and Point B
Point A (x₁, y₁)
Point B (x₂, y₂)
Examples:
📐 Step-by-Step Solution
📈 Coordinate Graph
Slope Formula & How to Calculate It
The slope of a line (denoted m) measures its steepness and direction. It is defined as rise over run — how much the line moves vertically for each unit of horizontal movement.
m = (y₂ - y₁) / (x₂ - x₁) = Rise / Run = Δy / Δx
Angle = arctan(m) in degrees
Grade (%) = m × 100
Distance = √((x₂-x₁)² + (y₂-y₁)²)
Angle = arctan(m) in degrees
Grade (%) = m × 100
Distance = √((x₂-x₁)² + (y₂-y₁)²)
Types of Slope
- Positive slope (m > 0): Line goes up from left to right — increasing
- Negative slope (m < 0): Line goes down from left to right — decreasing
- Zero slope (m = 0): Horizontal line — no rise, constant y
- Undefined slope: Vertical line — x₁ = x₂, run = 0, division by zero
Line Equation Forms
Slope-Intercept: y = mx + b
Point-Slope: y - y₁ = m(x - x₁)
Standard Form: Ax + By = C
Point-Slope: y - y₁ = m(x - x₁)
Standard Form: Ax + By = C
Grade Classification (Civil Engineering)
- Flat (0–3%): Wheelchair accessible, parking lots
- Gentle (3–8%): ADA ramp max = 8.33% (1:12)
- Moderate (8–15%): Residential driveways
- Steep (15–30%): Steep driveways, mountain roads
- Very Steep (30–50%): Ski slopes, extreme terrain
- Extreme (>50%): Cliff-like, not typical for roads
Real-World Applications
- Construction: Roof pitch (4:12 = 18.4°), ramp grades, stair slope
- Roads: Highway grades (5–8%), mountain roads (up to 12%)
- Accessibility: ADA ramp max = 1:12 = 8.33% grade
- Math/Physics: Rate of change, velocity vs time graphs
Slope Examples — Common Values
m = 0 → Horizontal line (y = b)
m = 1 → 45° angle, 100% grade
m = undefined → Vertical line (x = a)
m = 1/12 ≈ 0.083 → ADA ramp (8.33% grade)
m = 4/12 ≈ 0.333 → Roof 4:12 pitch (33.33% grade)
m = 1 → 45° angle, 100% grade
m = undefined → Vertical line (x = a)
m = 1/12 ≈ 0.083 → ADA ramp (8.33% grade)
m = 4/12 ≈ 0.333 → Roof 4:12 pitch (33.33% grade)
Slope vs Angle vs Grade — Conversion
Slope (m) = Rise ÷ Run = tan(angle)
Angle = arctan(m) in degrees
Grade (%) = m × 100
Example: m=0.05 → angle=2.86° → grade=5%
Angle = arctan(m) in degrees
Grade (%) = m × 100
Example: m=0.05 → angle=2.86° → grade=5%
Parallel and Perpendicular Lines
Parallel lines have the same slope: m₁ = m₂. Example: y=2x+1 and y=2x-3 are parallel (both slope = 2).
Perpendicular lines have slopes that are negative reciprocals: m₁ × m₂ = -1. Example: slope 2 is perpendicular to slope -1/2.
How to Find Y-Intercept from Slope and Point
b = y₁ - m × x₁
Example: Point (3, 5), slope m=2
b = 5 - 2×3 = 5 - 6 = -1 → y = 2x - 1
Example: Point (3, 5), slope m=2
b = 5 - 2×3 = 5 - 6 = -1 → y = 2x - 1
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