Slope Four Ways: Percent, Ratio, Inches Per Foot and Degrees

Why this matters

The same slope gets written four different ways depending on who wrote the spec. A plumbing inspector says 1/4 inch per foot. A civil or landscape plan says 2 percent. An accessibility drawing says 1:12. A digital level reads out in degrees unless someone changed the setting. None of these is wrong, and none of them is the same number, so the tech holding the level has to convert in their head, on site, often between the two that are most easily confused: percent and degrees.

Get that conversion wrong and a drain line that should fall is flat, a patio sheds water toward the house, or a ramp is built steeper than the drawing intended. This card gives the conversions with one table to keep, the formulas behind it, and the cases where the conversion is easy to botch. The required slopes themselves live elsewhere: see 'Drain Pipe Slope + Venting Quick Reference' for drainage and 'Ramp Slope, Rise and Landings' for ramps. This card only translates between the ways of writing them.

The four expressions

All four describe the same thing: how much the surface drops (or rises) for a given horizontal distance.

• Inches per foot. Inches of fall per 12 in of horizontal run. Standard in plumbing and drainage.
• Ratio (1:N). One unit of rise for N units of run, in any unit as long as both are the same. 1:12 means 1 in of rise per 12 in of run. Standard in accessibility and some roofing contexts.
• Percent. Rise divided by run, times 100. 2 percent means 2 ft of fall per 100 ft of run, or 0.02 of anything per unit of run. Standard on site, civil and landscape plans.
• Degrees. The angle between the surface and horizontal. Standard on digital levels, angle finders and some equipment specs.

The first three are all the same kind of number, a rise-over-run ratio written differently. Degrees is a different kind of number, an angle, and it is the one that does not convert by simple multiplication.

The table

SLOPE, FOUR WAYS   (rise over horizontal run)
----------------------------------------------------
 in per ft   ratio    percent    degrees
 ---------   -----    -------    -------
   1/16      1:192     0.52       0.30
   1/8       1:96      1.04       0.60
   1/4       1:48      2.08       1.19
   3/8       1:32      3.13       1.79
   1/2       1:24      4.17       2.39
   0.24      1:50      2.00       1.15
   1         1:12      8.33       4.76
   0.6       1:20      5.00       2.86
----------------------------------------------------
 percent = in per ft x 8.333
 N in 1:N = 12 / in per ft
 degrees  = arctan(percent / 100)

Two anchors are worth memorizing: 1/4 in per ft is just over 2 percent, and 1 in per ft (1:12) is 8.33 percent, a little under 5 degrees. Almost every slope you meet in drainage, flatwork and accessibility falls in or near that range.

The conversion formulas

Inches per foot to percent. Divide by 12 to get rise over run, then multiply by 100. That is the same as multiplying by 8.333. 1/4 in per ft = 0.25 / 12 x 100 = 2.083 percent.

Percent to inches per foot. Divide by 8.333, or multiply the decimal fraction by 12. 2 percent = 0.02 x 12 = 0.24 in per ft, which is just under 1/4 (0.25).

Inches per foot to ratio. N = 12 / (inches per foot). 1/4 in per ft gives N = 12 / 0.25 = 48, so 1:48. 1/8 in per ft gives 1:96.

Any of those to degrees. Degrees = arctan(rise / run), using the inverse tangent key on a calculator or phone. 1:12 is arctan(1 / 12) = arctan(0.0833) = 4.76 degrees.

Degrees to rise over run. Rise / run = tan(degrees). 2 degrees = tan(2) = 0.0349, which is 3.49 percent, or 0.0349 x 12 = 0.42 in per ft.

The shortcut for small slopes. Below about 10 percent, degrees is close to percent times 0.573. 2.08 percent x 0.573 = 1.19 degrees, which matches the exact value. At 8.33 percent the shortcut gives 4.77 against an exact 4.76, still within a hair. It drifts as the slope steepens: at 33 percent (a 4/12 roof) it gives 19.1 degrees against an exact 18.4. For roof pitches, use the arctangent or the pitch tables in 'Roof Slope, Pitch, + Measurement Reference'.

The worked example: a digital level on a drain line

A tech is checking a horizontal drain line that is supposed to fall at 1/4 in per ft, and the digital level laid on the pipe reads in degrees. On one section it reads 1.3, on the next 1.1. Do both pass?

Step 1, convert the target into the level's unit. 1/4 in per ft is 2.083 percent, which is arctan(0.02083) = 1.19 degrees. So the target on the display is 1.19 degrees.

Step 2, convert each reading into the spec's unit.

• 1.3 degrees: tan(1.3) = 0.0227, times 12 = 0.272 in per ft. That is just over 1/4. It passes the 1/4 in per ft minimum.
• 1.1 degrees: tan(1.1) = 0.0192, times 12 = 0.230 in per ft. That is under 1/4. It does not meet a 1/4 in per ft minimum.

Step 3, consider what the display can resolve. A level that shows one decimal place of a degree moves in steps of 0.1 degree, and tan(0.1) x 12 = 0.021 in per ft. So each step on the display is about 8 percent of a 1/4 in per ft slope. Readings of 1.2 and 1.1 are one display step apart and fall on opposite sides of the target. Before failing a section on a 1.1 reading, take it at more than one spot along the section, check the level reads 0.0 on a known level surface and reverses correctly (flip it end for end on the same spot and confirm the reading's magnitude holds), and if the reading sits right at the line, measure the drop over a known length with a tape instead. See related: Measurement Resolution and Honest Rounding.

Step 4, what the tech writes. "Section A: 1.3 deg, 0.27 in/ft, passes. Section B: 1.1 deg, 0.23 in/ft, under 1/4 in/ft; rechecked at three points, all 1.1; to be regraded." Both units on the record means the office or the inspector can read it without converting again.

The mistake this prevents. A tech who reads "1.1" and thinks of 1/4 in per ft as "about 2" has compared degrees to percent and passed a line that does not meet the spec. 2 percent and 2 degrees are not close: 2 degrees is 3.49 percent, about 1.7 times as steep.

The level-and-shim method

Without a digital level, a plain level and a shim give a slope directly in inches per foot. The idea: raise the low end of the level until the bubble centers, and the thickness of the shim is the fall over the level's length.

• On a 24 in level, 1/4 in per ft of slope means 2 ft x 1/4 in = 1/2 in under the low end.
• On a 48 in level, it is 4 ft x 1/4 in = 1 in.
• 1/8 in per ft on a 48 in level is 1/2 in.

Set the shim under the downhill end, lay the level on the work, and the bubble centers when the slope matches. This is often more trustworthy than a digital reading near a threshold, because it has no display resolution and no calibration offset, only the vial. The vial does need to be accurate, which you check by reading the level on a surface, flipping it end for end, and confirming the bubble sits the same.

Write a slope with its direction and two units

A slope number with no direction is half a spec. "2 percent" on a patio does not say whether it falls away from the house or toward it, and a crew that sets it backward has built the exact problem the slope was meant to prevent. Every slope written on a ticket, a sketch or a punch list gets three things:

• The number in the unit of the spec, because that is what an inspector or the drawing will be checked against.
• The same number in the unit of the tool, because that is what the person holding the level will read. "2 percent (1.15 deg, about 1/4 in per ft)" costs a few characters and removes the conversion from the job site.
• The direction, as a fall toward something: "falls to the floor drain", "falls away from the foundation", "falls toward the street". An arrow on a sketch should point downhill and say so.

This matters most on handoffs, when the person who read the plan is not the person who sets the forms or hangs the pipe.

Where the conversions go wrong

Percent read as degrees, or degrees as percent. The most common error, covered above. On a slope of a few percent they differ by nearly a factor of two. Write the unit every time you write a slope.

A ratio read as a percent. 1:12 is 8.33 percent, not 12 percent and not 1.2 percent. 1:20 is 5 percent. The ratio's second number is the run per unit of rise, so a bigger number is a gentler slope, the opposite of percent and degrees.

Rise over run versus rise over slope length. Every expression on this card is rise over horizontal run. Some field measurements are taken along the sloped surface (a tape laid on a ramp, a string along a pipe). For gentle slopes the difference is negligible (at 8.33 percent the slope length is only 0.35 percent longer than the run), but on steep surfaces like roofs it is not. See related: Roof Pitch Multipliers for Any Sloped Surface.

Inches per foot over a long run. A slope converts cleanly, but turning it into a total drop over a long distance means multiplying by the run in feet. That is its own procedure, with its own traps (measuring from one datum, not chaining); see related: How to Convert a Slope Spec Into a Drop Over a Run.

The minimum is not the target. A required slope in a code or a drawing is usually a minimum, a maximum or a range, and which one it is changes what passes. This card converts the number; whether a reading above or below it passes is set by the requirement, in the reference card that owns it.

References

• Drain Pipe Slope + Venting Quick Reference (library card, for required drainage slopes)
• Ramp Slope, Rise and Landings (library card, for required ramp slopes and landings)
• Roof Slope, Pitch, + Measurement Reference (library card, for roof pitch in degrees)
• See related: How to Convert a Slope Spec Into a Drop Over a Run
• See related: Measurement Resolution and Honest Rounding