Slope Calculator: Equation, Angle and Grade

How to use the calculator

Enter the first point as x₁ and y₁, and the second as x₂ and y₂. Negative numbers and decimals are fine.
For a real-world slope, use the same unit for every coordinate. For a roof, put the run (horizontal) on the x-axis and the rise on the y-axis.
Read the slope and the slope-intercept equation at the top, then the other forms and measurements below.

The two points must be different. When x₁ = x₂ the line is vertical: the slope, grade, y-intercept and point-slope form are not shown because they don't exist, and the equation appears as x = constant. A horizontal line has no x-intercept and no defined perpendicular slope, so those results are hidden.

Formulas used

m = (y₂ − y₁) ÷ (x₂ − x₁) = rise ÷ run

Slope-intercept form: y = mx + b, with b = y₁ − m·x₁.
Point-slope form: y − y₁ = m(x − x₁), built on the first point you enter.
Standard form: Ax + By = C, reduced to the smallest whole numbers with A positive (B positive when A is 0). This works for inputs with up to 6 decimal places.
Angle of inclination: θ = arctan(m), in degrees from the positive x-axis, between −90° and 90°. A negative angle means the line falls from left to right.
Grade: 100 × m, written as a percentage, as on road signs and ramp plans.
Perpendicular slope: −1 ÷ m, the negative reciprocal.
Distance: √(Δx² + Δy²); midpoint: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).

Equations are shown with up to 10 significant digits, so a slope of 1/12 appears as 0.08333333333 in the equation and exactly as 1/12 in the fraction.

Worked examples

1. Homework: through (2, 3) and (6, 5)

Rise 2, run 4, so m = 0.5 = 1/2. The calculator gives y = 0.5x + 2, y − 3 = 0.5(x − 2) and x − 2y = −4. The line crosses the x-axis at −4, rises at 26.565051°, and any line perpendicular to it has slope −2. The points are 4.472136 units apart with midpoint (4, 4).

2. A falling line: (−1, 4) and (3, −2)

Rise −6 over run 4 gives m = −1.5 = −3/2, the equation y = −1.5x + 2.5 or 3x + 2y = 5, and an angle of −56.309932°.

3. Roof pitch and ramp slope

Pitch is rise in inches per 12 inches of run, so enter (0, 0) and (12, rise):

Rise over 12 run	Slope	Grade	Angle
1 (ramp, 1:12)	1/12	8.333333%	4.763642°
4 (4/12 roof)	1/3	33.333333%	18.434949°
6 (6/12 roof)	1/2	50%	26.565051°

The U.S. Access Board's guide to the ADA Standards gives 1:12 as the maximum running slope of a ramp run and limits each run to a 30-inch rise. A 20-inch rise at 1:12 needs 240 inches of run: (0, 0) to (240, 20) gives the same 1/12 slope.

4. Road grade

A road that climbs 264 feet over one mile (5,280 feet) has a slope of 1/20, a 5% grade, and an angle of only 2.862405°.

Reading the results

A positive slope rises from left to right, a negative one falls, 0 is flat and a vertical line has no slope. The larger the absolute value, the steeper the line. Grade and angle are two ways to express the same steepness, but they don't scale together: doubling the grade doesn't double the angle. For building work, follow the plans and the code that applies to your project; the calculator only does the geometry.

Frequently asked questions

How do you find the slope from two points?

Subtract the y-values and divide by the difference of the x-values in the same order: (y₂ − y₁) ÷ (x₂ − x₁). Swapping both points gives the same answer, because the signs of the top and bottom flip together.

Is a 100% grade a vertical line?

No. A 100% grade means the rise equals the run, a slope of 1, which is a 45° angle: the line from (0, 0) to (1, 1) gives exactly 45° and 100%. A vertical line would need an infinite grade.

What is an undefined slope?

When both points have the same x-value, the run is 0 and the division has no answer. The line is vertical, written as x = constant, for example x = 3 through (3, 1) and (3, 7). Its perpendicular is horizontal, with slope 0.

What is the slope of a horizontal line?

Zero. Through (1, 4) and (5, 4) the rise is 0, so m = 0 and the equation is y = 4. It never crosses the x-axis, so there's no x-intercept.

How do I convert roof pitch to degrees?

Enter (0, 0) and (12, rise) and read the angle: a 6/12 pitch is 26.565051°. The grade is the pitch as a percentage of the run, 50% in that case.

Can it find the slope of a line from its equation or from many data points?

Not directly. For an equation, pick two points on the line and enter them. A best-fit slope through scattered data needs linear regression, which this calculator doesn't do.

Sources

Results are computed in floating-point arithmetic and rounded for display. For construction and accessibility work, check the requirements that apply to your project with the local building authority.