Dive theory formulas: the cheat sheet
In short
The main dive theory formulas: absolute pressure = depth ÷ 10 + 1 bar (÷ 33 + 1 ata); Boyle: P1 × V1 = P2 × V2; Dalton: partial pressure = fraction × absolute pressure; MOD = (1.4 ÷ oxygen fraction − 1) × 10 m (× 33 ft), rounded down; gas use = SAC × absolute pressure.
An exam question gives you a lift bag at 30 m (99 ft) and a volume to find. You know there’s a formula, but not whether to multiply or divide, or where the 1 goes. This sheet puts every dive theory formula on one page, each worked once in metric and once in imperial, with the trap that catches people. The theory page linked in each section explains why the formula works.
Which conventions do these formulas use?
They use the rounded conventions of Octo’s Divemaster lessons. Metric and imperial examples are separate problems, not conversions of each other.
| Quantity | Metric | Imperial |
|---|---|---|
| Seawater | 1 bar per 10 m | 1 atm per 33 ft |
| Fresh water | about 1 bar per 10.3 m | about 1 atm per 34 ft |
| Air | 21% oxygen, 79% nitrogen | same |
| Weight of water | 1.03 kg per liter (sea), 1.00 (fresh) | 64 lb per cu ft (sea), 62.4 (fresh) |
| Absolute temperature | K = °C + 273 | °R = °F + 460 |
| Example cylinder | 12 L at 200 bar = 2,400 L | 80 cu ft at 3,000 psi: 37.5 psi per cu ft |
Freshwater figures vary slightly between texts: use the one your course uses.
How do you calculate absolute pressure?
Absolute pressure is the water plus the atmosphere: depth ÷ 10 + 1 bar, or depth ÷ 33 + 1 ata. Gauge pressure is the water alone, and every gas law uses absolute pressure. Back to depth: (pressure − 1) × 10 m, or × 33 ft.
Worked example
Metric: at 25 m, 25 ÷ 10 + 1 = 3.5 bar absolute (2.5 bar gauge).
Imperial: at 66 ft, 66 ÷ 33 + 1 = 3 ata (2 atm gauge).
How do you use Boyle’s law?
At a constant temperature, a gas’s volume varies inversely with absolute pressure: P1 × V1 = P2 × V2. Its density varies the other way, which is why you use more gas at depth. Full lesson: Boyle’s law.
Worked example
Metric: a sealed bag holding 12 L at the surface (1 bar) holds 12 × 1 ÷ 4 = 3 L at 30 m (4 bar).
Imperial: a sealed bag holding 2 cu ft at the surface (1 ata) holds 2 × 1 ÷ 4 = 0.5 cu ft at 99 ft (4 ata).
Drag the depth: Boyle’s law in action
How does temperature change cylinder pressure?
In a rigid cylinder, pressure follows absolute temperature: P2 = P1 × T2 ÷ T1, in kelvin or degrees Rankine. Dive texts often call this Charles’s law; others credit Gay-Lussac. With Boyle’s law it forms the general gas law, P1 × V1 ÷ T1 = P2 × V2 ÷ T2. See Charles’s law.
Worked example
Metric: filled to 200 bar at 40 °C (313 K), cooled to 20 °C (293 K): 200 × 293 ÷ 313 ≈ 187 bar.
Imperial: filled to 3,000 psi at 110 °F (570 °R), cooled to 80 °F (540 °R): 3,000 × 540 ÷ 570 ≈ 2,840 psi.
How do you calculate a partial pressure?
Dalton’s law: partial pressure = fraction × absolute pressure, and the partial pressures add up to the total. Use fractions (0.32), not percentages. Rearranged, it gives the richest mix for a depth (partial pressure ÷ absolute pressure). See Dalton’s law.
Worked example
Metric: air at 30 m (4 bar): ppO2 = 0.21 × 4 = 0.84 bar; ppN2 = 0.79 × 4 = 3.16 bar.
Imperial: air at 99 ft (4 ata): ppO2 = 0.21 × 4 = 0.84 ata; ppN2 = 0.79 × 4 = 3.16 ata.
How do you calculate MOD and best mix?
MOD = (ppO2 limit ÷ oxygen fraction − 1) × 10 m, or × 33 ft, rounded down. The usual limits are 1.4 for the dive and 1.6 as a contingency, never a target; some agencies and operators set lower ones. DAN calls 1.4 or less the “green light” zone for open-circuit scuba and says recreational divers should not exceed 1.6.
Worked example
Metric: EAN32 at 1.4: 1.4 ÷ 0.32 = 4.375 bar; (4.375 − 1) × 10 = 33.75, so 33 m.
Imperial: EAN32 at 1.4: 1.4 ÷ 0.32 = 4.375 ata; (4.375 − 1) × 33 = 111.4, so 111 ft.
Best mix: 1.4 ÷ absolute pressure, rounded down. For 30 m (99 ft), 1.4 ÷ 4 = 0.35: EAN35 at most. More in oxygen toxicity and nitrox.
Try a question from the app
MOD of EAN32
What is the maximum operating depth of EAN32 for a ppO2 of 1.4?
Show the answer
Answer D. 1.4 ÷ 0.32 = 4.375 bar; (4.375 − 1) × 10 = 33.75 m, rounded down to 33 m. 43 m forgets to subtract the atmosphere; 40 m is the limit at 1.6; 34 m rounds up and crosses 1.4.
What does Henry’s law say, in words?
At a given temperature, the amount of gas dissolved in a liquid is proportional to the partial pressure of that gas. Double the nitrogen partial pressure and, given enough time, your tissues hold twice as much nitrogen. Each half-time covers half the remaining gap: 50% after one, 75% after two, 87.5% after three, about 98% after six. See Henry’s law.
Worked example
Metric: on air at 30 m, ppN2 = 0.79 × 4 = 3.16 bar, four times the surface value: once saturated, a tissue holds four times as much nitrogen.
Imperial: on air at 99 ft, ppN2 = 3.16 ata, also four times. In both systems, a 20-minute compartment is 75% of the way after 40 minutes.
How do you calculate buoyancy and lift?
Archimedes’ principle: water pushes an object up with the weight of the water it displaces. Apparent weight = weight − volume displaced × weight of water per unit volume. For a lift bag, air needed = apparent weight ÷ weight of water per unit volume, ignoring the bag’s own weight. See buoyancy.
Worked example
Metric: 30 kg displacing 20 L in the sea: 30 − 20 × 1.03 = 9.4 kg; air needed: 9.4 ÷ 1.03 ≈ 9.1 L.
Imperial: 60 lb displacing 0.5 cu ft in the sea: 60 − 0.5 × 64 = 28 lb; air needed: 28 ÷ 64 ≈ 0.44 cu ft.
How do you calculate SAC rate and gas time?
Surface air consumption (SAC, also called RMV, as in Octo’s lessons) is your breathing rate brought back to the surface. DAN keeps “SAC” for a pressure rate in bar or psi per minute, which only fits one size of cylinder; here it’s a volume rate: SAC = gas used ÷ minutes ÷ absolute pressure. Then gas use at depth = SAC × absolute pressure, and gas time = usable gas ÷ gas use at depth, before the ascent and safety stop. See SAC rate.
Worked example
Metric: 125 bar from a 12 L cylinder = 1,500 L in 25 minutes at 20 m (3 bar): 1,500 ÷ 25 ÷ 3 = 20 L/min. With 150 bar usable (1,800 L) at 30 m: 1,800 ÷ (20 × 4) = 22.5 minutes.
Imperial: 1,800 psi of an 80 cu ft, 3,000 psi cylinder = 48 cu ft in 20 minutes at 66 ft (3 ata): 48 ÷ 20 ÷ 3 = 0.8 cu ft/min. With 2,500 psi usable (about 67 cu ft) at 99 ft: 67 ÷ (0.8 × 4) ≈ 21 minutes.
Try a question from the app
consumption at depth
You breathe 18 L/min at the surface. How much surface-equivalent air do you breathe per minute at 30 m?
Show the answer
Answer B. Each breath is four times as dense at 4 bar: 18 × 4 = 72 L/min. 54 L/min uses the gauge pressure (3 bar); 18 L/min forgets the density; 4.5 L/min applies Boyle’s law backward.
Frequently asked questions
Do dive formulas use gauge or absolute pressure?
Gas laws use absolute pressure: gauge pressure plus 1 bar (1 atm). Gauge pressure is behind many wrong answers, such as 2.5 instead of 3.5 bar at 25 m. One exception in practice: when a cylinder heats or cools, using the gauge reading barely changes the result.
Why is the MOD rounded down?
Because rounding up takes you past your ppO2 limit. EAN32 at 1.4 gives 33.75 m; at 34 m the ppO2 would be slightly above 1.4, so the MOD is 33 m (111 ft). Each diver analyzes their own cylinder and marks the mix and MOD on it.
Are 10 m and 33 ft the same thing?
Not exactly. NOAA gives one atmosphere per 33 ft, or 10.06 m, of seawater; metric texts round to 1 bar per 10 m. 30 m and 99 ft are close, but work each problem in one system and don’t convert halfway. More in metric vs imperial dive math.
How do you turn liters per minute into bar per minute?
Divide by the cylinder’s volume in liters: 80 L/min from a 12 L cylinder is 80 ÷ 12 ≈ 6.7 bar per minute. In imperial, multiply by the psi per cubic foot: with an 80 cu ft, 3,000 psi cylinder, 2.8 cu ft/min × 37.5 = 105 psi per minute.
