What the Slope Calculator does
This slope calculator finds the slope of the line through two points and gives the equation of that line in slope-intercept, point-slope and standard form, with the intercepts, the midpoint and the distance between the points. It also expresses the same gradient the way builders and road engineers do - as rise over run, a percent grade, an angle in degrees and a ratio such as 1 : 12 - and converts between them.
Fractions and decimals are handled exactly, so the points (1, 2) and (4, 6.5) give a slope of 3/2 and the line y = (3/2)x + 1/2 rather than a rounded decimal. For ramps it compares the result with the 1:12 maximum running slope in the US ADA Standards, as a reference point. Everything is calculated in your browser.
How to use it
- Choose your input: two points on a line, a rise and a run (for a ramp, path, roof or pipe), or a grade or angle you want converted.
- For two points, type x₁, y₁, x₂ and y₂. Negative numbers, decimals and fractions such as 3/4 are all accepted.
- For rise and run, measure both in the same unit: the vertical height gained and the horizontal distance it is gained over.
- Read the slope at the top, the grade, angle and ratio underneath, the equations of the line and a quick plot. The steps show the subtraction and division so you can copy the method into homework.
Reading the results
A positive slope rises from left to right, a negative one falls, zero is a horizontal line and an undefined slope is a vertical line, which has the equation x = constant.
Percent grade and angle are not the same number: a 100% grade is 45 degrees, not 90, because 100% means the rise equals the run. Road signs, railways and ramps use percent or ratio; roofs in the US use rise per 12 inches of run; maths uses the plain slope m.
The ramp check only compares the steepness with 1:12. Accessibility codes also set limits on rise per run, landings, width, handrails and cross slope, and local regulations vary.
Worked example: a line through two points and a wheelchair ramp
Points (1, 2) and (4, 6.5): the rise is 6.5 - 2 = 4.5 and the run is 4 - 1 = 3, so m = 4.5 / 3 = 3/2. The intercept is b = 2 - (3/2)(1) = 1/2, so y = (3/2)x + 1/2. Multiplying through by 2 gives the standard form 3x - 2y = -1.
The same slope is a grade of 150% and an angle of arctan(1.5) = 56.31 degrees. The distance between the points is √(3² + 4.5²) = 5.41.
A ramp rising 24 inches over a run of 288 inches has a slope of 24 / 288 = 1/12: a grade of 8.33% and an angle of 4.76 degrees. That is exactly the 1:12 maximum, so the calculator marks it as not steeper than the reference.
Formulas and scoring rules
- Slope
m = (y₂ − y₁) / (x₂ − x₁) = rise / runUndefined when x₂ = x₁ (a vertical line).- Slope-intercept form
y = mx + b, b = y₁ − m·x₁- Point-slope form
y − y₁ = m(x − x₁)- Standard form
Ax + By = C, A = y₂ − y₁, B = x₁ − x₂, C = A·x₁ + B·y₁Scaled to whole numbers with no common factor and A positive.- Grade, angle and ratio
grade % = 100·m; angle = arctan(m); ratio = 1 : (1/m)Fractions are exact; decimals are rounded to 4 places (2 for percent and degrees) only for display.- Distance and midpoint
d = √((x₂ − x₁)² + (y₂ − y₁)²); M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The 1:12 ramp rule as a reference
Section 405.2 of the 2010 ADA Standards for Accessible Design limits the running slope of a ramp to 1:12 - one unit of rise for every twelve units of run, or 8.33% - and section 405.3 limits the cross slope to 1:48. Many national codes elsewhere use the same or similar figures, and some allow steeper slopes over short rises or require gentler ones.
The calculator shows how your gradient compares with 1:12 because it is the number most people are looking for when they measure a ramp. It is not a compliance check: treat it as a quick sense-check and confirm the design against the code that applies where you are, with a qualified professional where it matters.
Limitations: what the result does not prove
- It describes straight lines and constant gradients. A path or road whose steepness changes needs measuring section by section.
- Angles are measured from the horizontal in degrees. Some trades measure slope from the vertical or quote roof pitch as rise in inches per 12 inches of run; convert those before entering them.
- The ramp comparison is one dimension of accessibility. It says nothing about landings, length of run between landings, width, handrails, surfaces or local law.
- Very long decimals (more than 30 digits) are refused rather than rounded silently.
Privacy: where your data goes
Everything you paste, type or drop is processed in this browser tab. It is not uploaded, logged, stored or sent to analytics. Session recording and tag-manager scripts are switched off on this page.
Standards and sources
- US Access Board - 2010 ADA Standards for Accessible Design, 405 Ramps
- OpenStax Algebra and Trigonometry 2e - Linear functions and slope
- US Department of Justice - 2010 ADA Standards for Accessible Design
Frequently asked questions
How do you find the slope between two points?
Subtract the y-values and divide by the difference of the x-values in the same order: m = (y₂ − y₁) / (x₂ − x₁). For (1, 2) and (4, 6.5) that is 4.5 / 3 = 1.5. The calculator shows the subtraction and division step by step.
How do I convert slope to percent grade?
Multiply the slope by 100. A slope of 0.05, a rise of 5 over a run of 100, is a 5% grade. Percent grade is not an angle: 5% is about 2.86 degrees and 100% is 45 degrees.
What is the slope of a 1:12 ramp in percent and degrees?
A 1:12 ramp rises one unit for every twelve of horizontal run, which is 8.33% or about 4.76 degrees. That is the maximum running slope for ramps in the US ADA Standards.
How do I write the equation of a line from two points?
Find the slope m, then the intercept b = y₁ − m·x₁, and write y = mx + b. The calculator gives that form, the point-slope form y − y₁ = m(x − x₁) and the standard form Ax + By = C with whole numbers.
What does an undefined slope mean?
The two points have the same x-value, so the line is vertical and the run is zero. You cannot divide by zero, so the slope has no value; the equation of the line is simply x = that value.
How do I convert a slope angle in degrees to a ratio?
Take the tangent of the angle to get the slope, then divide 1 by it for the 1 : n ratio. Ten degrees has a tangent of 0.1763, which is about 1 : 5.67, or a 17.63% grade. Choose the grade conversion mode to do it instantly.
Last reviewed by the A2Z.Tools team against the sources listed above.
