Mental math for the cockpit

Nobody computes sines in a cockpit — you use rules of thumb that are close enough to matter. These are the ones that actually get used, grouped by the moment you need them.

The 3:1 rule — descent distance

D_{NM} \approx 3 \times \frac{\Delta h}{1000\ \mathrm{ft}}

$D_{NM}$: track miles needed for the descent
$\Delta h$: altitude to lose, in feet

10,000 ft to lose → ~30 NM out; add 1 NM per 10 kt of speed still to shed

Altitude to loseftSpeed to shedkt ≈ 38 NM

Descent rate on a 3° path

\mathrm{VSI} \approx \mathrm{GS} \times 5

$\mathrm{VSI}$: vertical speed to command, in ft/min
$\mathrm{GS}$: ground speed, in knots

150 kt ground speed → ~750 ft/min down

Ground speedkt ≈ 750 ft/min

Angle ↔ gradient ↔ ft per NM

\mathrm{gradient}\ [\%] \approx \mathrm{angle}\ [^\circ] \times 1.75 \qquad \mathrm{ft/NM} \approx \mathrm{gradient}\ [\%] \times 61

$\mathrm{angle}$: descent or climb angle, in degrees
$\mathrm{gradient}$: the same slope as a percentage

3° → ~5.3% ≈ 320 ft/NM (exact tan: 5.24%, 318 ft/NM) — the same units the climb-gradient rows in a minima table use

Angle° 5.3 % ≈ 320 ft/NM

Glidepath sanity check

h \approx 300\ \mathrm{ft} \times D_{\mathrm{DME}}

$h$: expected height above the runway
$D_{\mathrm{DME}}$: distance to the threshold, in NM

at 5

≈ 1,500 ft above the runway — plus airport elevation

Distance to thresholdNM ≈ 1,500 ft above the runway

Wind components

The clock rule in words: read the wind angle off the nose as minutes past the hour — that fraction of the wind becomes crosswind. 30° off is half, 45° is three quarters, 60° or more is all of it.

Wind components

\mathrm{crosswind} \approx W \times \sin\theta \qquad \mathrm{headwind} \approx W \times \cos\theta

$W$: reported wind speed, in kt
$\theta$: angle between wind direction and runway heading

20 kt at 45° → ~14 kt crosswind + ~14 kt headwind

WindktAngle off runway° 14 kt crosswind · 14 kt headwind

Correction angle to parallel the course

\Delta\theta \approx \frac{d_{\mathrm{off}}}{d_{\mathrm{to\ go}}} \times 60

$\Delta\theta$: heading change needed, in degrees
$d_{\mathrm{off}}$: how far off course you are, in NM
$d_{\mathrm{to\ go}}$: distance remaining, in NM

2 NM off with 60 NM to run → 2° to parallel — double it to rejoin

Distance off courseNMDistance to goNM ≈ 2°

Maximum drift from wind

\mathrm{max\ drift} \approx W \times \frac{60}{\mathrm{TAS}}

$W$: wind speed, in kt
$\mathrm{TAS}$: true airspeed, in kt

24 kt wind at 120 kt

→ up to 12° of drift

WindktTASkt ≈ 12°

Turns and holding

Standard rate is 3° per second — 90° takes 30 seconds, 180° takes a minute. Lead the rollout by about half the bank angle.

Rate-one bank angle (FAA IFH rule)

\phi \approx \frac{\mathrm{TAS}}{10} + 7

$\phi$: bank angle for a standard-rate turn, in degrees
$\mathrm{TAS}$: true airspeed, in kt

180 kt → 18 + 7 ≈ 25° bank

TASkt ≈ 25° bank

Rate-one turn diameter

D_{\mathrm{turn}} \approx \frac{\mathrm{GS}}{100}\ \ [\mathrm{NM}]

$D_{\mathrm{turn}}$: turn diameter, in NM
$\mathrm{GS}$: ground speed, in kt

180 kt → ~1.8 NM across the turn — worth knowing in a valley hold

Ground speedkt ≈ 1.8 NM

QNH error → altitude error

\Delta h \approx 28\ \mathrm{ft} \times \Delta p\ [\mathrm{hPa}]

$\Delta h$: altitude error, in ft
$\Delta p$: QNH setting error, in hPa

990 set instead of 1013 → ~640 ft low

QNH sethPaActual QNHhPa ≈ 644 ft low

Cold-temperature correction ("four per ten")

\Delta h \approx 0.04 \times h_{\mathrm{above}} \times \frac{|\Delta T_{\mathrm{ISA}}|}{10^\circ\mathrm{C}}

$\Delta h$: margin to add, in ft
$h_{\mathrm{above}}$: height above the altimeter source (the airport), in ft
$|\Delta T_{\mathrm{ISA}}|$: temperature below ISA, in °C

20°C below

at 3,000 ft above the field → ~240 ft to add

Height above airportftBelow ISA°C + ≈ 240 ft

Density altitude

\mathrm{DA} \approx \mathrm{PA} + 120\ \mathrm{ft} \times \Delta T_{\mathrm{ISA}}

$\mathrm{DA}$: density altitude
$\mathrm{PA}$: pressure altitude — altimeter at 1013 hPa, in ft
$\Delta T_{\mathrm{ISA}}$: temperature deviation from ISA, in °C

PA 5,000 ft at ISA+10 → ~6,200 ft — the aircraft performs as if it were that high

Pressure altitudeftISA deviation°C ≈ 6,200 ft DA

TAS from CAS

\mathrm{TAS} \approx \mathrm{CAS} \times \left(1 + 0.02 \times \frac{h}{1000\ \mathrm{ft}}\right)

$\mathrm{TAS}$: true airspeed
$\mathrm{CAS}$: calibrated airspeed
$h$: altitude, in ft

150 kt

at 10,000 ft → ~180 kt TAS

CASktAltitudeft ≈ 180 kt TAS

Planning a descent in numbers

FL200 down to a field at 2,000 ft: 18,000 ft to lose → 3 × 18 = 54 NM, plus 8 NM to slow down → top of descent at ~62 NM. At 180 kt ground speed descend at 180 × 5 ≈ 900 ft/min, and cross-check 10 DME at about 3,000 ft above the field.

Tip

In MSFS the FMS does all of this for you — the value of these rules is catching the moments when it is wrong: a descent that starts too late, a crosswind over the limit, a QNH typo. Sanity-check the automation instead of trusting it.

Related terms Climb gradient QNH TAS CAS DME

Sources