Slope Calculator

Slope Formula

The slope of a line through two points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ - y₁) / (x₂ - x₁) = Δy / Δx

Slope is also equal to the tangent of the angle of incline: m = tan(θ)

Types of Slope

• Positive slope: Line rises from left to right (m > 0)
• Negative slope: Line falls from left to right (m < 0)
• Zero slope: Horizontal line (m = 0)
• Undefined slope: Vertical line (Δx = 0)

Slope-Intercept Form

The equation of a line in slope-intercept form is y = mx + b, where:

• m = slope of the line
• b = y-intercept (where the line crosses the y-axis)

Given slope m and a point (x₁, y₁), find b: b = y₁ - m · x₁

Point-Slope Form

An alternative form using a known point and slope:

y - y₁ = m(x - x₁)

Distance Between Two Points

The distance between (x₁, y₁) and (x₂, y₂) uses the Pythagorean theorem:

d = √(Δx² + Δy²)

Percent Grade

Grade is slope expressed as a percentage, commonly used for roads, ramps, and construction:

Grade (%) = (rise / run) × 100 = slope × 100

• A 5% grade rises 5 feet for every 100 feet horizontally
• ADA-compliant ramps require a maximum 8.33% grade (1:12 ratio)
• Steep roads are typically 6-10% grade

Example

Find the slope between (3, 2) and (7, 6):

• Δx = 7 - 3 = 4
• Δy = 6 - 2 = 4
• m = 4 / 4 = 1
• θ = arctan(1) = 45°
• d = √(16 + 16) = 5.6569
• y-intercept: b = 2 - 1(3) = -1
• Equation: y = x - 1