Slope Calculator

Formula

m = (y₂ - y₁) / (x₂ - x₁)

Slope is the ratio of the vertical change (rise) to the horizontal change (run).

Worked Examples

Example 1: Points (0,0) and (4,4)

Problem:(0,0) to (4,4)

Solution:Slope = (4-0)/(4-0) = 1. Distance = √(4²+4²) ≈ 5.657

Result:m=1

Example 2: Negative Slope

Problem:(0,5) to (5,0)

Solution:Slope = (0-5)/(5-0) = -1.

Result:m=-1

Frequently Asked Questions

What does "Rise over Run" mean?

It is a simple way to remember the slope formula. "Rise" is the vertical change (Δy), and "Run" is the horizontal change (Δx).

What is the slope of a horizontal line?

The slope is 0. There is no vertical change (rise = 0), so 0 divided by any run is 0.

What is the slope of a vertical line?

The slope is Undefined. There is no horizontal change (run = 0), and division by zero is mathematically undefined.

What does a negative slope mean?

A negative slope means the line is going downhill from left to right. As x increases, y decreases.

How do I find the angle of the slope?

You can calculate the angle using trigonometry: Angle = arctan(slope). A slope of 1 corresponds to 45°.

What is the slope of perpendicular lines?

Perpendicular lines have negative reciprocal slopes. If line A has slope m, line B has slope -1/m.

How do I convert slope to grade percentage?

Multiply the slope by 100. A slope of 0.05 is a 5% grade (common in roads).

What is Slope-Intercept Form?

It is the equation y = mx + b, where m is the slope and b is the y-intercept.

How is slope used in real life?

It describes the steepness of roads, roofs, and wheelchair ramps, as well as rates of change in economics and physics.

Background & Theory

**Slope (m):** Measures the steepness and direction of a line. * **Formula:** m = (y₂ - y₁) / (x₂ - x₁) * **Rise:** Change in y (vertical) * **Run:** Change in x (horizontal) **Distance:** The length of the line segment connecting two points. * **Formula:** d = √((x₂ - x₁)² + (y₂ - y₁)²) * Derived from the Pythagorean theorem. **Types of Slope:** * **Positive:** Line goes up from left to right. * **Negative:** Line goes down from left to right. * **Zero:** Horizontal line. * **Undefined:** Vertical line.

History

The concept of slope (or gradient) has origins in algebra and geometry. René Descartes (17th century) introduced the coordinate system, allowing geometric shapes to be described by algebraic equations. This paved the way for calculus, where slope (derivative) is a central concept developed by Newton and Leibniz.