13.2 Plumbing Math for the Exam: Offsets, Rolling Offsets, Grade, Areas, Volumes, and Pressure

Calculator and Scratch Paper

The PSI bulletin allows silent, battery-operated, non-programmable calculators without paper tape or an alphabetic keyboard. PSI provides scratch paper and a pencil. Practice these formulas on a basic calculator so exam-day arithmetic is routine.

Offsets

An offset moves a pipe out of line and back to parallel. It forms a right triangle: the offset (also called set, or rise) is one leg, the run is the other leg, and the travel is the hypotenuse, measured center to center of the fittings.

The multipliers are 1 ÷ sine of the angle (travel) and 1 ÷ tangent of the angle (run). To work backward, offset = travel ÷ multiplier. For 45 degrees, offset = travel × 0.707.

Example: A 45° offset must move a drain 18 inches. Travel = 18 × 1.414 = 25.45 inches center to center, and the run is also 18 inches.

Example: The same 18-inch offset with 22-1/2° fittings: travel = 18 × 2.613 = 47.03 inches, and run = 18 × 2.414 = 43.45 inches.

Rolling Offsets

A rolling offset moves the pipe both up (the set) and over (the roll).

1. True offset = square root of (roll² + set²).
2. Travel = true offset × the multiplier for the fitting angle.

Example: roll 12 inches, set 16 inches, 45° fittings. True offset = square root of (144 + 256) = square root of 400 = 20 inches. Travel = 20 × 1.414 = 28.28 inches.

With a calculator, you can also get travel as the square root of (roll² + set² + run²). Here the run equals the true offset for 45 degrees: square root of (144 + 256 + 400) = square root of 800 = 28.28 inches, which matches.

Cut Length

Travel is center to center of the fittings. To get the length of pipe to cut, subtract the fitting allowance (takeoff) of each fitting. The takeoff is the distance from the fitting's center to the point where the pipe end seats.

Cut length = center-to-center length − (takeoff of fitting 1 + takeoff of fitting 2).

For threaded pipe, add back the thread engagement. Takeoffs come from the manufacturer's dimensions for the specific fittings.

Example: 28.28-inch travel between two fittings with 1-1/4-inch takeoffs each: cut length = 28.28 − 2.5 = 25.78 inches (about 25-3/4 inches).

Drainage Grade

• Fall = slope × length. At 1/4 inch per foot, 48 feet falls 12 inches. At 1/8 inch per foot, 48 feet falls 6 inches.
• Percent grade = fall ÷ run × 100. A fall of 1/4 inch per foot is 0.25 ÷ 12 × 100 ≈ 2.08%, which the code calls 2 percent. A fall of 1/8 inch per foot is about 1%.
• Trap arm check: a 1-1/2-inch arm at its 42-inch maximum falls 3.5 × 0.25 = 0.875 inch at 1/4 inch per foot.

Areas (for Aggregate Venting and Combination Waste and Vent)

Area = 0.7854 × d² (d is the inside diameter in inches).

Area grows with the square of the diameter. A 4-inch pipe has four times the area of a 2-inch pipe, not twice.

Volume and Weight

• Pipe capacity: gallons per foot = 0.0408 × d². For a 4-inch pipe: 0.0408 × 16 = 0.65 gallon per foot. A 50-foot, 4-inch drain holds about 32.6 gallons during a water test.
• Cylinder (tank): volume in cubic inches = 0.7854 × d² × h. Divide by 231 for gallons. Alternatively, use cubic feet × 7.48.
• Example: a tank 18 inches in diameter and 48 inches tall holds 0.7854 × 324 × 48 = 12,215 cubic inches, and 12,215 ÷ 231 ≈ 52.9 gallons.
• Weight: water is about 8.33 lb per gallon and 62.4 lb per cubic foot. A 52.9-gallon tank of water weighs about 441 lb, before the tank itself.
• Grease interceptor capacity (Example 1014.2.1): length × width × depth in inches ÷ 231 = gallons, then × 0.75 fill factor.

Pressure and Head

• Pressure (psi) = head (ft) × 0.433. A 10-foot water test head exerts about 4.33 psi at the bottom.
• Head (ft) = psi × 2.31. A 60 psi main can lift water about 139 feet with no flow.
• Table 610.4 method: use 1/2 psi per foot (610.8), not 0.433 (Section 12.3).

Temperature

°F = °C × 1.8 + 32, and °C = (°F − 32) ÷ 1.8. For example, 49°C = 120.2°F (the scald limit), 60°C = 140°F, and 43°C = 109.4°F.