The one ratio behind all three notations
Every roof pitch expression starts from the same two numbers: rise (vertical height gained) and run (horizontal distance covered). "X-in-12" notation states how many units of rise occur for every 12 units of run — a "4 in 12" roof rises 4 units for every 12 units horizontal. The angle in degrees is simply the arctangent of rise ÷ run. Slope percent is rise ÷ run × 100. All three are different ways of presenting the exact same underlying ratio.
A worked example — two common pitches
A "4 in 12" roof (rise 4, run 12) works out to an angle of 18.43° and a slope of 33.33%. A steeper "6 in 12" roof (rise 6, run 12) works out to 26.57° and a 50% slope. Notice the angle doesn't scale linearly with the "X" in "X-in-12" — going from 4-in-12 to 6-in-12 is a 50% increase in the rise ratio, but the angle only increases by about 8 degrees, since angle is related to the ratio through arctangent, not a straight multiplication.
Why construction uses "X-in-12" instead of degrees
"X-in-12" notation ties directly to how roofs are actually framed and measured on site — a carpenter's speed square and framing square are marked in this notation specifically so pitch can be set and checked directly with common tools, without needing a separate angle-measuring instrument. Degrees are more common in engineering and architectural drawings, while slope percent shows up in drainage and civil-engineering contexts (where percent grade is the standard way to describe any sloped surface, roofs included).
Why pitch matters beyond just appearance
Pitch affects water and snow shedding (steeper roofs shed both faster, which matters more in heavy-rainfall or snow-load regions), the choice of roofing material (some materials have a minimum pitch requirement to shed water properly), attic usability (steeper pitches create more usable headroom inside the roof structure), and — as covered in a related article — how much extra roofing material is needed versus the flat footprint area, since steeper roofs have proportionally more actual surface area to cover.
Converting between the three notations yourself
Given any one form, the other two follow directly: from "X in 12," slope percent = X ÷ 12 × 100, and angle = arctan(X ÷ 12). From an angle, rise ÷ run = tan(angle), which converts directly into either other notation. From a slope percentage, rise ÷ run = percent ÷ 100. Because they're all expressions of the same ratio, none of them requires knowing the roof's actual physical dimensions — only the relationship between rise and run.