Slope Calculator
What is Slope? (The Concept)
In algebra and geometry, the slope describes the steepness and direction of a line. You encounter slope every day in the real world without realizing it. The steepness of a mountain road, the pitch of a roof, and the ramp for a wheelchair are all real-world examples of slope.
In mathematics, we define slope (often represented by the letter m) as the ratio of the vertical change to the horizontal change between any two distinct points on a line.
The Slope Formula
If you know the coordinates of two points, (x1, y1) and (x2, y2), calculating the slope involves simple subtraction and division.
The top part (y2 - y1) represents the Change in Y (The Rise).
The bottom part (x2 - x1) represents the Change in X (The Run).
Step-by-Step Calculation Example
Let's find the slope between Point 1 (1, 2) and Point 2 (4, 8).
- Identify your coordinates:
x1=1, y1=2 and x2=4, y2=8. - Find the Rise (change in Y):
8 - 2 = 6. - Find the Run (change in X):
4 - 1 = 3. - Divide Rise by Run:
6 / 3 = 2.
The slope is 2. This means for every 1 step you move to the right, the line goes up 2 steps.
Interpreting the Result: The 4 Types of Slope
The value of m tells you exactly what the line looks like before you even graph it.
| Slope Value (m) | Line Direction | Example |
|---|---|---|
| Positive (+) | Lines goes UP from left to right. | Uphill road. |
| Negative (-) | Lines goes DOWN from left to right. | Downhill ski slope. |
| Zero (0) | A perfectly HORIZONTAL line. | A flat floor. |
| Undefined | A perfectly VERTICAL line. | A wall. (You cannot divide by zero). |
Slope Intercept Form (y = mx + b)
Once you have the slope (m), you can write the equation of the line. The most common format is the Slope-Intercept Form:
y = mx + b
- m is the slope you just calculated.
- b is the y-intercept (where the line crosses the vertical Y-axis).
To find b, you can rearrange the formula: b = y - mx. Simply plug in your slope and one of your points, and our calculator handles the rest automatically.
Relation to Angle and Distance
Our calculator also provides two other useful metrics derived from the coordinates:
1. Euclidean Distance
How far apart are the two points "as the crow flies"? This is calculated using the Pythagorean Theorem:
d = √(x2-x1)2 + (y2-y1)2
2. The Angle (θ)
This is the angle the line makes with the horizontal X-axis. It is found using trigonometry (inverse tangent).
θ = tan-1(m)
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