Slope Calculator

Math

Find the slope of a line, rise over run, angle of inclination, slope-intercept form (y = mx + b), point-slope form, standard form (Ax + By = C), midpoint, distance, and perpendicular slope.

Presets:

Slope: m = 2

Line Equations & Geometric Properties

Slope-Intercept Form y = 2x

Standard Form (Ax + By = C) 2x - y = 0

Point-Slope Form y - 2 = 2(x - 1)

Angle of Inclination (θ) 63.43° (1.1071 rad)

Midpoint (2.50, 5.00)

Distance 3√5 (6.7082)

Parallel Slope m = 2

Perpendicular Slope m⊥ = -0.5

Algebraic Derivations

1 1. Delta (Rise and Run)

Δx (Run) = x₂ - x₁ = 4 - 1 = 3 | Δy (Rise) = y₂ - y₁ = 8 - 2 = 6

The horizontal change (run) and vertical change (rise) between the two coordinates.

2 2. Slope Formula (m = Rise / Run)

m = (y₂ - y₁) / (x₂ - x₁) = 6 / 3 = 2 ≈ 2.000000

Ratio of vertical change to horizontal change.

3 3. Angle of Inclination (θ)

θ = arctan(m) = arctan(2.0000) = 63.43° (1.1071 rad)

The angle between the positive x-axis and the line.

4 4. Slope-Intercept Form (y = mx + b)

b = y₁ - m·x₁ = 2 - (2.0000 × 1) = 0.0000 ⟹ y = 2x

b is the y-intercept value where the line crosses the y-axis at (0, b).

5 5. Midpoint & Distance

Midpoint = ((1 + 4)/2, (2 + 8)/2) = (2.50, 5.00) | Distance = √(3² + 6²) = 3√5 ≈ 6.708204

Geometric midpoint and Euclidean distance between the two coordinates.

How This Calculator Works

The slope of a line describes its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points on the line.

📐 Formulas for Slope and Line Equations

m = (y₂ - y₁) / (x₂ - x₁) = Δy / Δx | y = mx + b | Ax + By = C

m (Slope): Rate of change; positive slopes rise from left to right, negative slopes fall

b (y-intercept): The y-coordinate of the point where the line crosses the y-axis: (0, b)

Angle of Inclination (θ): θ = arctan(m), the angle made with the positive x-axis (0° ≤ θ < 180°)

Parallel Lines: Parallel lines share identical slopes: m₁ = m₂

Perpendicular Lines: Perpendicular lines have negative reciprocal slopes: m₁ · m₂ = -1 ⟹ m₂ = -1/m₁

Key Mathematical Rules & Properties

Horizontal Lines

When y₁ = y₂, rise is 0, so slope m = 0. The equation simplifies to y = b.

Vertical Lines

When x₁ = x₂, run is 0, so slope is undefined. The equation simplifies to x = constant.

Understanding your calculation

Method, example and limitations

How the calculation works

Slope equals vertical change divided by horizontal change: (y₂ − y₁)/(x₂ − x₁). A percent slope is 100 times that ratio.

Example to check

From (1, 2) to (5, 4), slope is 2/4 = 0.5, or 50%.

Before using the result

A vertical line has undefined finite slope. Percent slope and an angle in degrees are different quantities.

Calculation methodology · Report a result that needs review