Slope Calculator: Calculate Slope Between Two Points
Our free slope calculator helps you find the slope, angle, distance, midpoint, and linear equation between any two points on a coordinate plane. Perfect for students, teachers, engineers, and anyone working with linear equations and coordinate geometry.
What is Slope?
Slope (also called gradient) is a measure of the steepness of a line. It represents the rate of change between two points on a coordinate plane. The slope tells you how much the y-value changes for every unit change in the x-value.
The slope formula is: m = (y₂ - y₁) / (x₂ - x₁), where m is the slope and (x₁, y₁) and (x₂, y₂) are two points on the line.
Types of Slope
- Positive Slope: The line rises from left to right (m > 0). As x increases, y increases.
- Negative Slope: The line falls from left to right (m < 0). As x increases, y decreases.
- Zero Slope: The line is horizontal (m = 0). The y-value stays constant.
- Undefined Slope: The line is vertical (division by zero). The x-value stays constant.
How to Use the Slope Calculator
- Enter the x and y coordinates for Point 1 (x₁, y₁)
- Enter the x and y coordinates for Point 2 (x₂, y₂)
- The calculator automatically computes:
- Slope (m)
- Angle of inclination
- Distance between points
- Midpoint coordinates
- Linear equation (y = mx + b)
- Click "Reset" to start over with new values
Important Formulas
Slope Formula
m = (y₂ - y₁) / (x₂ - x₁)
Where m is the slope, and (x₁, y₁) and (x₂, y₂) are two distinct points on the line.
Distance Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Calculates the straight-line distance between two points using the Pythagorean theorem.
Midpoint Formula
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Finds the point exactly halfway between two given points.
Angle Formula
θ = tan⁻¹(m)
Converts slope to angle of inclination in degrees or radians.
Linear Equation (Slope-Intercept Form)
y = mx + b
Where m is the slope and b is the y-intercept (the y-value where the line crosses the y-axis).
Common Applications
- Mathematics: Solving linear equations, graphing lines, coordinate geometry
- Physics: Calculating velocity, acceleration, and rates of change
- Engineering: Designing ramps, roads, and structural inclines
- Construction: Determining roof pitch, drainage slopes, and accessibility ramps
- Economics: Analyzing trends, supply and demand curves
- Geography: Measuring terrain steepness and elevation changes
- Architecture: Planning staircases, wheelchair ramps (ADA compliance requires 1:12 slope)
Step-by-Step Example
Example: Find the slope between points (2, 3) and (6, 11)
Step 1: Identify the coordinates
- Point 1: (x₁, y₁) = (2, 3)
- Point 2: (x₂, y₂) = (6, 11)
Step 2: Apply the slope formula
m = (y₂ - y₁) / (x₂ - x₁)
m = (11 - 3) / (6 - 2)
m = 8 / 4
m = 2
Step 3: Interpret the result
The slope is 2, which means for every 1 unit increase in x, y increases by 2 units. This is a positive slope, so the line rises from left to right.
Tips for Accurate Calculations
- Always subtract coordinates in the same order (y₂ - y₁ and x₂ - x₁)
- Check for vertical lines (x₁ = x₂) which have undefined slope
- Remember that horizontal lines (y₁ = y₂) have zero slope
- Use decimal values for more precise results
- Verify your answer by plotting the points on a graph
Common Mistakes to Avoid
- Reversing the order: Make sure to subtract in the correct order (y₂ - y₁) / (x₂ - x₁)
- Confusing rise and run: Rise is the vertical change (Δy), run is the horizontal change (Δx)
- Dividing by zero: When x₁ = x₂, the slope is undefined, not zero
- Sign errors: Pay attention to negative coordinates and subtraction
- Mixing up points: Keep track of which coordinates belong to which point
Frequently Asked Questions
What does a slope of 1 mean?
A slope of 1 means the line rises at a 45-degree angle. For every 1 unit you move right, you move 1 unit up.
Can slope be negative?
Yes! A negative slope means the line falls from left to right. As x increases, y decreases.
What is the difference between slope and gradient?
Slope and gradient are the same thing - they both measure the steepness of a line. "Slope" is more common in American English, while "gradient" is preferred in British English.
How do you find slope from a graph?
Pick two points on the line, count the vertical change (rise) and horizontal change (run) between them, then divide rise by run.
What is slope-intercept form?
Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. It's the most common way to write linear equations.
Why is vertical line slope undefined?
For a vertical line, x₁ = x₂, so the denominator (x₂ - x₁) equals zero. Division by zero is undefined in mathematics.