PHAK · E6B · study note

Altitude math mental models

This page is about how to think about the numbers—not a substitute for the AFM/POH. Pair it with the conceptual note on density altitude & performance and the interactive Performance Lab.

1. Three different “altitudes”

Do not treat these as interchangeable labels for “how high I am.”

Vertical comparison of field elevation, pressure altitude, and density altitude — **Figure 1.** Geography vs pressure-as-feet vs density-as-feet.

Name	Question it answers	Main inputs
Field elevation	How high is the airport above MSL?	Survey / chart
Pressure altitude (PA)	How high am I if sea-level pressure were 29.92?	Altimeter + elevation (or 29.92 window)
Density altitude (DA)	How high does air density make the airplane feel?	PA + temperature (+ humidity in full physics)

Sticky phrases: elevation = geography · PA = pressure remapped to feet · DA = density remapped to feet. Performance cares about density. Charts often speak PA + OAT because that encodes density under a dry ISA assumption.

2. Pressure altitude math

Cockpit formula (altimeter in inHg, result in feet):

PA ≈ (29.92 − altimeter setting) × 1000 + field elevation

Pressure altitude formula with low and high pressure examples — **Figure 2.** Mental model: about **1000 ft per inHg** between local setting and 29.92.

Low altimeter (e.g. 29.42) → PA above field elevation (thinner pressure column).
High altimeter (e.g. 30.12) → PA below field elevation.
Cockpit method: set Kollsman to 29.92, read altitude → that is PA.

What you are really doing: mapping “local static pressure at the field” onto the standard atmosphere pressure profile. By definition, ambient pressure equals ISA pressure at that PA.

3. The ISA ladder (the ruler)

The International Standard Atmosphere is the reference ruler for both PA and DA.

Quantity	Sea level	Lower troposphere lapse (approx.)
Pressure	29.92 inHg	decreases with height
Temperature	15 °C / 59 °F	≈ −2 °C / 1000 ft
Density	standard ρ₀	decreases with height

Standard temp at a given PA (°C):

T_ISA ≈ 15 − 2 × (PA_ft / 1000)

Example: PA = 5000 ft → T_ISA ≈ 5 °C. On a standard day at that PA, DA ≈ PA. Hotter than ISA → DA > PA; colder → DA < PA.

4. Density altitude math (dry / chart style)

Common training approximation (OAT and T_ISA in °C):

DA ≈ PA + 120 × (OAT − T_ISA)

Worked example

Field elev 5000 ft, altimeter 29.92 → PA = 5000 ft
T_ISA = 15 − 2×5 = 5 °C
OAT = 25 °C → ΔT = +20 °C
DA ≈ 5000 + 120×20 = 7400 ft

E6Bs and many apps implement this dry path (or a close ISA density inversion). It is approximate—good for intuition and planning order-of-magnitude, not a law of nature.

5. Physics under the hood (ideal gas)

Dry air density:

ρ = p / (R × T)

Higher pressure → denser
Higher temperature → less dense
DA = “ISA height whose density matches today’s ρ”

Humidity: water vapor is lighter than dry air, so for the same p and T, moist air is less dense → slightly higher DA. Full models use vapor pressure and virtual temperature. Pilot charts usually ignore humidity on the axes; treat it as a small add-on or margin (see density altitude note).

6. Why performance charts use PA + OAT

Flow from pressure and temperature to density to DA language to AFM charts — **Figure 4.** Physics tracks ρ; the POH speaks altitude language that implies ρ under dry ISA assumptions.

Lift (and related forces) scale with dynamic pressure:

L = ½ ρ V² S C_L

Lower ρ → less lift / thrust / power for a given indicated speed and configuration. Publishing charts vs raw ρ would be awkward; PA and OAT are the pilot-friendly encoding.

7. Altimeter vs performance (don’t mix them)

Number / instrument	Driven mainly by
Altimeter (local setting)	Pressure
True airspeed for a given IAS	Density (and compressibility at high M)
Takeoff / climb performance	Density (+ weight, wind, runway…)

Hot day, same field and altimeter setting: indicated altitude still “looks right” for traffic, but DA is high and the airplane is weak.

8. Humidity “math” (secondary)

Model	Idea
PHAK	Humidity contributes; usually not essential for the calc; no simple cockpit chart
FAASTeam-style	High humidity → consider ~+10% takeoff distance, expect weaker climb (margin)
Dew-point ROT (study)	~20 × dew point (°C) feet added to dry DA — rough, often conservative-ish
Full calculator	Station pressure + T + dew point → humidity-aware DA

9. End-to-end checklist

Field elevation
Altimeter setting → pressure altitude
T_ISA at that PA → compare to OAT
DA ≈ PA + 120×(OAT − T_ISA) (°C)
Optional humidity margin / calculator
Enter AFM/POH (PA+temp or DA as the chart requires)
Weight, wind, surface, slope — then go / no-go

10. Sticky one-liners

Idea	Phrase
PA	“Altitude as if sea-level pressure were 29.92.”
inHg rule	“About 1000 feet per inch of mercury.”
ISA lapse	“Two degrees colder per thousand feet.”
DA dry	“PA plus 120 feet for every °C hotter than standard.”
DA meaning	“The ISA height with today’s density.”
Humidity	“Moist air is lighter; usually hundreds of feet; often margin, not a chart axis.”
Charts	“PA and OAT are how the POH speaks density.”

What not to over-trust

The 120 ft/°C rule is an approximation.
Dew-point rules of thumb are rougher still.
Study-lab numbers are for intuition, not dispatch.
Real decisions use the AFM/POH for that airplane.

Practice

Open Aircraft Performance Lab → Density altitude note

Sources

FAA PHAK FAA-H-8083-25C Ch. 4 & 11 (definitions, humidity, performance framing)
Standard training PA / DA approximations used with E6B-class computers
ISA temperature lapse as used in private pilot ground school

Figures are original educational SVGs, not FAA handbook scans.

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