The hook
Three ways to write the same slope: percent, ratio, degrees. Civil designers think in ratios, spec sheets quote percent, trig works in degrees. Fluency in all three is muscle memory worth building. Drag the angle slider below until the conversions become automatic.
Percent grade
rise / run × 100
33.32%
Ratio (run : rise)
surveyor shorthand
3.00 : 1
Angle
from horizontal
18.43°
Common rules of thumb:
- 2 % max — ADA-accessible ramp
- 5 % max — comfortable highway grade
- 3:1 — typical highway cut/fill (33%, 18°)
- 2:1 — steepest stable un-vegetated slope (50%, 27°)
- 1:1 — angle of repose for loose dry soil (100%, 45°)
Try this: set the angle to 26.57° — that's exactly 2:1 (50% grade). Now try 14.04° — that's 4:1 (25%). Surveyors and civil designers think in ratios; spec sheets quote percent; trig works in degrees. Fluent translation between all three is muscle memory worth building.
Memorize these
Concepts that show up on the exam
Percent grade
rise / run × 100. Expresses the slope as the percentage of vertical change per unit horizontal.
Ratio (run : rise)
How many feet horizontal for every foot vertical. "3:1" means 3 horizontal, 1 vertical (33% grade, 18.4°).
Angle from horizontal
arctan(percent / 100). What you measure with a clinometer or zenith angle.
Cross slope
Slope perpendicular to the centerline of a road. Typically 2% for drainage on highways, 1% on parking lots.
Stationing along a slope
Stations are measured horizontal, not along the slope. A 100-ft station is 100 ft horizontal — 100.5 ft slope distance on a 10% grade.
Angle of repose
The steepest stable angle for an unsupported pile of loose material. ~33° (1.5:1) for typical soils, ~45° (1:1) for dry sand.
Keep these in muscle memory
Formulas to know cold
Percent ↔ angle
percent = tan(angle) · 100, angle = arctan(percent/100)Percent ↔ ratio
ratio_run_per_rise = 100 / percentSlope distance from horizontal
slope_dist = horiz_dist / cos(angle)| Application | Typical slope | In percent | In ratio | In degrees |
|---|---|---|---|---|
| ADA-accessible ramp (max) | 1:12 | 8.3% | 12 : 1 | 4.8° |
| Comfortable highway grade (max) | 1:20 | 5% | 20 : 1 | 2.9° |
| Highway cut/fill (typical) | 1:3 | 33% | 3 : 1 | 18.4° |
| Steepest stable un-vegetated | 1:2 | 50% | 2 : 1 | 26.6° |
| Angle of repose, dry sand | 1:1 | 100% | 1 : 1 | 45° |
Try it before you peek
Worked example
The problem
A construction stake reads "cut 3.4 ft at +0.3 offset." The design cross-slope is 2:1 (2 horizontal for 1 vertical). At what offset distance does the design surface daylight (intersect existing ground)?
Don't fall for these
What trips people up
Ratio order is run-first by convention
"3:1" in surveying / civil means 3 horizontal : 1 vertical. In some Europe-rooted documents, ratio is written rise-first (1:3 meaning the same thing). When in doubt, do the trig and check against the stated percent.
Percent isn't a fraction
A 100% grade is NOT vertical — it's 45°. Vertical is "infinity %." Above 100%, ratios become fractional (0.5:1 = 200% = 63.4°).
Slope distance vs. horizontal distance
Stationing along a road is HORIZONTAL distance. The actual surface you walk is longer. On a 10% grade the difference is only 0.5%, but on a 50% grade it's 12%.
Test yourself
How well did it stick?
A quick 5-question check on Slopes and Grades. See where you stand and what to review.