Calculating Slope with Degrees

What is Slope?

Slope, often denoted by the letter "m," is a numerical value that describes the steepness and direction of a line. It represents the change in the vertical direction (rise) over the change in the horizontal direction (run) between two points on the line.

Where:

• m is the slope
• Δy is the change in the y-coordinate (rise)
• Δx is the change in the x-coordinate (run)
• (x₁, y₁) and (x₂, y₂) are two points on the line

The slope can be positive, negative, zero, or undefined. A positive slope indicates an upward trend, a negative slope indicates a downward trend, and a zero slope indicates a horizontal line. An undefined slope indicates a vertical line.

Slope with Degrees

When we refer to slope with degrees, we're talking about the angle of inclination of a line. This angle is measured from the positive direction of the x-axis to the line. The relationship between slope and degrees is given by the tangent function from trigonometry.

Where:

• θ is the angle of inclination in degrees
• m is the slope
• arctan is the inverse tangent function
• π is the mathematical constant pi (approximately 3.14159)

The angle of inclination can range from -90° to 90°. A positive angle indicates the line rises to the right, while a negative angle indicates the line falls to the right.

How to Calculate Slope with Degrees

Calculating slope with degrees involves two main steps: first, determine the slope using the slope formula, and then convert that slope to degrees using the arctangent function.

Step 1: Calculate the Slope

1. Identify two points on the line: (x₁, y₁) and (x₂, y₂).
2. Calculate the change in y (Δy) by subtracting y₁ from y₂: Δy = y₂ - y₁.
3. Calculate the change in x (Δx) by subtracting x₁ from x₂: Δx = x₂ - x₁.
4. Divide Δy by Δx to find the slope: m = Δy / Δx.

Step 2: Convert Slope to Degrees

1. Take the arctangent of the slope: θ_radians = arctan(m).
2. Convert the angle from radians to degrees: θ_degrees = θ_radians × (180/π).

Examples

Let's look at some examples to understand how to calculate slope with degrees.

Example 1: Positive Slope

Consider a line passing through the points (2, 3) and (5, 7).

1. Calculate Δy: 7 - 3 = 4
2. Calculate Δx: 5 - 2 = 3
3. Calculate slope: m = 4 / 3 ≈ 1.333
4. Convert to degrees: θ = arctan(1.333) × (180/π) ≈ 53.13°

The angle of inclination is approximately 53.13°, indicating the line rises to the right.

Example 2: Negative Slope

Consider a line passing through the points (1, 8) and (4, 2).

1. Calculate Δy: 2 - 8 = -6
2. Calculate Δx: 4 - 1 = 3
3. Calculate slope: m = -6 / 3 = -2
4. Convert to degrees: θ = arctan(-2) × (180/π) ≈ -63.43°

The angle of inclination is approximately -63.43°, indicating the line falls to the right.