How to Find Rise Over Run (Math, Roofs, Stairs, Roads, Ramps)

The rise-over-run concept

Slope is a measure of how much something climbs versus how far it travels horizontally. If a road climbs 10 feet over 100 feet of horizontal distance, its slope is 10/100 — usually expressed as 10%, or as 1 in 10, or as 5.71° from horizontal. Same number, three different formats. The phrase rise over run means exactly that: the rise (vertical change) divided by the run (horizontal change), expressed as a ratio. Rise and run math is the same whether you are working with grid coordinates on graph paper or measurements on a real roof — the formula does not care about the units.

In some scientific and European contexts, the same idea is called gradient. The term "gradient" describes a slope as a ratio of vertical change to horizontal change — identical to slope, just different vocabulary. In U.S. K-12 math, the term is "slope" and the symbol is m. In civil engineering and physics, "gradient" is more common. The number is the same.

For roofs, U.S. building convention normalises run to 12 inches. So a roof that climbs 4 inches per foot of horizontal travel is "4/12" or "4 over 12". The 12 is implied — when a builder says "4 pitch", they mean 4/12. For stairs, run is the depth of one tread (typically 10 to 11 inches) and rise is the height of one riser (typically 7 to 7.75 inches). The IRC limits maximum rise to 7-3/4 inches and minimum run to 10 inches for residential stairs (R311.7.5). For ramps, slope is usually expressed as a percentage. ADA-compliant ramps are limited to 8.33% (1:12) — meaning 1 inch of rise per 12 inches of run.

Finding the slope of a line on a graph

In algebra, the slope of a line on a graph is found using two points and the slope formula. Pick any two points on the line — call them (x₁, y₁) and (x₂, y₂). The formula is m = (y₂ − y₁) / (x₂ − x₁), which is the standard rise over run equation in coordinate-geometry form. The numerator (y₂ − y₁) is the rise — how much the y-value changes between the two points. The denominator (x₂ − x₁) is the run — how much the x-value changes. Divide one by the other and you have the slope. This is the answer to "how do u find slope" and "how to find the slope of a line graph" — the same five-step recipe works for every line on a coordinate plane.

How to find the slope of a graph step by step: first, pick two points on the line where it passes through clear grid intersections (so you can read the coordinates without estimating). Second, label the leftmost point (x₁, y₁) and the rightmost (x₂, y₂). Third, count how many units up or down the line goes from the first point to the second — that is the rise (positive if up, negative if down). Fourth, count how many units to the right the line goes — that is the run. Fifth, write the slope as the ratio rise/run and simplify the fraction. The same procedure tells you how to determine rise over run from any two points on any straight line.

Worked example for graph slope: a line passes through (1, 2) and (4, 8). Rise = 8 − 2 = 6. Run = 4 − 1 = 3. Slope = 6/3 = 2. The line rises 2 units for every 1 unit of horizontal travel. Equivalently, it has slope m = 2, which would write as y = 2x + b in slope-intercept form. The line is going up (positive slope); a line going down would have a negative slope. A horizontal line has zero slope. A vertical line has undefined slope (you would be dividing by zero).

How to determine the slope of a graph when the line does not cross clean grid intersections: pick two points where the line is closest to grid intersections, accept that the result is approximate, and round to a sensible fraction. Or, if the graph shows the equation of the line directly (like y = 3x + 5), the slope is the coefficient of x — in that example, slope = 3. This is how to find the slope of a line without points listed explicitly: read the equation if available, count rise and run from any visible point, or measure if the graph has a scale.

Rise over run example — step by step

Concrete worked example using a real coordinate graph. Two points are visible on a line: (2, 3) and (5, 9). Find the slope.

Step 1: identify the rise. Subtract the y-coordinate of the first point from the y-coordinate of the second: 9 − 3 = 6. The line rises 6 units between these two points.

Step 2: identify the run. Subtract the x-coordinate of the first point from the x-coordinate of the second: 5 − 2 = 3. The line runs 3 units to the right.

Step 3: write the slope as the ratio rise/run. Slope = 6 / 3 = 2. The line has a slope of 2, meaning it climbs 2 units for every 1 unit of horizontal travel.

Step 4: check the sign. Both rise and run came out positive, so the slope is positive — the line goes up from left to right. If the rise had come out negative (the line going down), the slope would be negative. The sign convention is built into the formula automatically when you subtract in the same order both times.

Step 5: simplify if needed. 6/3 simplifies to 2/1, usually written as just 2. If the fraction did not simplify cleanly, you would express it as a fraction in lowest terms (e.g., 4/6 = 2/3) or as a decimal. Either is correct; fractions are more common in textbook problems and decimals are more common in applied work.

The same five-step process works for any two points on any line. The slope you compute is the same number regardless of which two points you pick — the line is straight, so it has the same slope everywhere along its length. If you pick different points and get different slopes, you are not on a line; you are on a curve.

Finding rise over run for a roof

For a roof, run is normalised to 12 and rise is the variable. Place a level horizontally against the roof or rafter, mark 12 inches along the level, measure vertically from the 12-inch mark to the surface above (or below) it. That vertical distance in inches is the rise. The pitch is rise/12. A 5-inch rise is 5/12 pitch. A 7-inch rise is 7/12 pitch. The math is identical to the graph case — you are still computing rise ÷ run — but the run is fixed at 12 by U.S. building convention.

For different field methods (level + tape on the roof, level + tape in the attic, digital angle finder, speed square), see the dedicated guide on how to measure roof pitch. The attic method is the safest and is what most roofers default to on existing homes. Once you have the pitch, the home page calculator converts it to angle, slope percent, and slope factor for material ordering and rafter math.

Slope for stairs (rise per step over tread depth)

For stairs, the rise is the vertical height of one step (the riser) and the run is the horizontal depth of one step (the tread, measured nose-to-nose). The IRC R311.7.5 sets the residential limits: maximum riser height is 7-3/4 inches, minimum tread depth is 10 inches. So a code-compliant residential stair has rise/run between 7.75/10 (steepest allowed) and lower values (shallower stairs). The strictest interpretation of the formula makes a residential stair somewhere between 0.625 and 0.775 in slope ratio — much steeper than any roof or ramp.

How to find rise over run for an existing stair: measure the total height from one floor to the next (total rise), measure the total horizontal distance the stair covers (total run), and divide both by the number of steps. For a stair climbing 96 inches over 144 inches of run with 13 steps: rise per step = 96 ÷ 13 = 7.38 inches; run per step = 144 ÷ 13 = 11.08 inches. The slope of one step = 7.38/11.08 = 0.666. Both individual values are within IRC limits.

Why the limits exist: stairs steeper than 7.75/10 become dangerous to descend — the foot does not have enough horizontal landing room and the tendency to fall forward increases sharply. Stairs shallower than the minimum eat floor space and become awkwardly long. The IRC band is the result of decades of accident-data-driven refinement; staying within it is what makes a stair feel intuitive to walk.

Slope for a wheelchair ramp

For a ramp, you typically know the rise (the height you need to overcome — say, 30 inches up to a porch) and you want to find the minimum run that meets the slope limit. ADA requires no steeper than 1:12, so a 30-inch rise needs at least 30 × 12 = 360 inches of run, or 30 feet. That is the absolute minimum; many municipalities require gentler slopes for outdoor ramps in icy climates.

For a non-ADA private use ramp, you have more flexibility but should not exceed about 2:12 (16.67%) for safety. Steeper than 2:12 becomes hard to push a wheelchair up and risky to walk down with anything in your hands.

Slope for a road or driveway

Roads and driveways are typically slope-limited by drainage and traction. Most municipal codes cap residential driveway slopes at 12% to 15% — beyond that, cars bottom out at the transition and ice becomes dangerous. Public roads are usually held to 6% to 8% maximum on long grades.

To find your driveway's slope, measure the rise (the elevation change from street to garage) and the run (the horizontal distance along the driveway, not the angled length). Slope percent = (rise ÷ run) × 100. For a 4-foot rise over a 50-foot run, slope is 8%.

Different ways to find slope

Slope can be found in several different ways depending on the context. Each method gives the same answer when applied correctly to the same line or surface — but the right method depends on what you have to work with.

In the field, the level-and-tape method is fastest and most accurate when you have physical access. In the classroom, the slope formula with two points is the standard. For a graph with clean grid intersections, the visual count is fastest. For a digital field measurement, an angle finder reads the angle directly and you convert. For an equation, the slope is encoded in the form. For a curve rather than a line, calculus gives the instantaneous slope at any point.

Converting between rise/run, percent, and degrees

The three formats are mathematically equivalent. To convert from rise/run to percent: divide rise by run and multiply by 100. To convert from percent to degrees: take the arctangent of (percent ÷ 100) and multiply by 180/π. To convert from degrees to rise/run: take the tangent of the angle and that is the slope ratio (or slope ÷ 1 in the m form).

Worked examples. A 4/12 roof: 4 ÷ 12 × 100 = 33.3% slope. arctan(0.333) × 57.296 = 18.43°. A 6% road: arctan(0.06) × 57.296 = 3.43°. An 8.33% ADA ramp: arctan(0.0833) × 57.296 = 4.76° — the same angle as a 1/12 roof, which is a useful reference if you ever need to picture the limit. A graph-line with slope m = 2: tan⁻¹(2) × 57.296 = 63.43°, far steeper than anything legal in residential construction; the line would be a very steep ramp or stair. Slope m = 0.5: 26.57°, which is exactly a 6/12 roof.

How we sourced these numbers

The slope formula and the rise-over-run identity are accepted mathematical conventions — not sourced to a single citation. The IRC limits for residential stairs (max 7-3/4 inch rise, min 10 inch tread) come from the 2024 International Residential Code, Section R311.7.5. The ADA ramp slope limit (1:12, 8.33%) comes from the 2010 ADA Standards for Accessible Design, Section 405.2. Roof pitch material thresholds (2/12 minimum for asphalt with double underlayment, 4/12 for single underlayment) come from IRC R905.1.1. Driveway slope limits vary by jurisdiction — the ranges given here are typical but not universal; check your local code for exact requirements.