Calculate Slope in Degrees

Calculating slope in degrees is essential in geometry, physics, and engineering. This guide explains how to find the angle of inclination of a line using two points, the formula, and practical applications.

What is Slope in Degrees?

The slope of a line in degrees represents the angle of inclination relative to the positive x-axis. It measures how steep a line is and in which direction it rises or falls.

Key points about slope in degrees:

Measures the steepness and direction of a line
Ranges from -90° to +90°
Positive slope indicates upward direction
Negative slope indicates downward direction
0° means a horizontal line
90° means a vertical line (undefined slope)

How to Calculate Slope in Degrees

To find the slope in degrees, you need two points on the line: (x₁, y₁) and (x₂, y₂). The process involves:

Calculate the change in y (Δy) and change in x (Δx)
Compute the slope in decimal form (m = Δy/Δx)
Convert the decimal slope to degrees using the arctangent function

Note: The arctangent function (atan) gives the angle in radians, which must be converted to degrees.

Slope Formula

Slope in degrees formula:

θ = atan(Δy/Δx) × (180/π)

Where:

θ = slope in degrees
Δy = y₂ - y₁ (change in y)
Δx = x₂ - x₁ (change in x)
atan = arctangent function
π ≈ 3.14159

The formula converts the decimal slope to degrees by multiplying the arctangent result by 180/π.

Worked Example

Let's calculate the slope in degrees for a line passing through points (2, 3) and (5, 7).

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

The slope of the line is approximately 53.13°.

Interpreting the Result

The slope in degrees tells you:

The steepness of the line
The direction of the line's rise or fall
Whether the line is increasing or decreasing

For example, a 45° slope means the line rises at a 45° angle to the x-axis. A negative slope indicates the line falls as it moves right.

FAQ

What is the difference between slope in decimal and degrees?: The decimal slope (m) represents the ratio of vertical to horizontal change, while the degree slope represents the angle of inclination. The degree slope is calculated from the decimal slope using the arctangent function.
Can a line have a slope of 90°?: Yes, a vertical line has an undefined decimal slope but a 90° degree slope. This occurs when Δx = 0 in the formula.
How do I calculate slope in degrees from a graph?: Identify two points on the line, calculate Δy and Δx, then use the formula θ = atan(Δy/Δx) × (180/π).
What does a negative slope in degrees mean?: A negative slope in degrees indicates the line falls as it moves right. The absolute value represents the steepness, while the negative sign indicates direction.
Is the degree slope the same as the angle of elevation?: Yes, the degree slope is equivalent to the angle of elevation or inclination of the line relative to the positive x-axis.