DGCA Air Navigation Formulas & Key Concepts

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Earth, Convergence & Departure4 Chart Scale3 Map Projections3 Compass & Direction1 Units & Conversions2 Wind Triangle & Drift5 Simple Navigation Maths4 1 in 60 Rule & Bearings5 Solar System & Time2 Descent Planning4 Critical Point & PNR4 Payload, Fuel & Weight and Balance3

Earth, Convergence & Departure

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Define convergence and give the formula.

The angle between two meridians at a given latitude: how much a great-circle track changes direction relative to the meridians.

\text{Convergence}=\Delta\lambda\times\sin(\text{mean latitude})

Zero at the equator, equal to the change of longitude at the pole
Example: Δλ = 20° at 45°N gives 20 × sin 45° ≈ 14°

What is departure, and how do you work it out?

The distance in nautical miles along a parallel between two meridians.

\text{Departure}=\Delta\lambda\ (\text{minutes of arc})\times\cos(\text{latitude})

1° of longitude is 60 NM at the equator, shrinking to 0 at the poles
Example: 2° at 60°N is 120′ × cos 60° = 60 NM

What is the conversion angle?

The angle between the rhumb-line track and the great-circle track at the start or end of a leg. It is half the convergence.

\text{Conversion angle}=\tfrac12\,\Delta\lambda\,\sin(\text{mean latitude})

Tip: Use it to convert a great-circle bearing into a rhumb-line bearing for plotting on a Mercator chart.

How long is one degree of longitude at 40°N?

About 46 NM.

60\ \text{NM}\times\cos 40^\circ\approx 46\ \text{NM}

Cosine of latitude shrinks east–west distance as you go north

Chart Scale

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How do you calculate chart scale?

Scale (representative fraction) is chart length divided by the real distance, both in the same unit.

\text{Scale}=\frac{\text{chart length}}{\text{earth distance}}

1 : 1 000 000 means 1 cm on the chart is 10 km on the ground (5.4 NM)
Larger denominator = smaller scale = less detail

How does scale vary on a Mercator chart?

Scale increases with latitude. At latitude φ it equals the equatorial scale multiplied by sec φ.

\text{Scale at }\phi=\frac{\text{scale at equator}}{\cos\phi}

Measure distance against the latitude scale at the same latitude as the leg

How do you find chart scale from a measurement?

Scale = chart distance / earth distance (same units).

\text{Scale}=\dfrac{\text{chart distance}}{\text{earth distance}}

1:1 000 000 means 1 cm = 10 km
1 NM = 1 852 m
Chart length (cm) = distance (NM) × 1 852 × 100 / scale denominator

Map Projections

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Why is the Lambert conformal conic popular for aeronautical charts?

Great circles are nearly straight lines and scale is almost constant across the chart.

n=\sin\phi_0

Two standard parallels where scale is exact
Scale is contracted between them and expanded outside
Meridians are straight lines that converge; parallels are arcs of circles
Convergence factor (constant of the cone) ≈ sin of the parallel of origin

What are the properties of a Mercator chart?

A cylindrical, conformal projection. Rhumb lines are straight lines.

\text{Scale}_\phi=\dfrac{\text{Scale}_0}{\cos\phi}

Meridians parallel, parallels parallel and unequal spacing
Scale expands as 1/cos(latitude)
Great circles curve towards the equator
Not usable near the poles

What are the properties of a Lambert conformal conic chart?

A conic, conformal projection with two standard parallels.

\text{Convergence}=\Delta\lambda\times n,\quad n=\sin\phi_0

Great circles are almost straight lines
Convergence factor (n) = sin of the parallel of origin
Scale is correct at the standard parallels and slightly small between them
Used for aeronautical charts of mid-latitudes

Compass & Direction

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How do you convert a relative bearing to a true bearing?

Add the relative bearing to the true heading.

\text{True bearing}=\text{True heading}+\text{relative bearing}

Subtract 360° if the answer is above 360°
Reciprocal = ±180°

Units & Conversions

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Mass, volume and fuel conversions?

Quick table:

\text{mass}=\text{volume}\times\text{density}

1 kg = 2.205 lb
1 US gallon = 3.785 L · 1 Imp gallon = 4.546 L
Avgas density ≈ 0.72 kg/L
Jet A-1 density ≈ 0.80 kg/L

Tip: Always check the unit of fuel on the question: litres, US gallons or kg.

Speed, distance, time: the formulas?

Keep minutes and hours straight.

D=S\times T

T(\text{min})=\frac{D(\text{NM})}{GS(\text{kt})}\times 60

GS 120 kt means 2 NM per minute
GS 180 kt means 3 NM per minute

Wind Triangle & Drift

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How do you calculate the wind correction angle?

Use the sine rule on the wind triangle.

\sin(\text{WCA})=\frac{W}{TAS}\,\sin(\text{angle between track and wind})

Small angles: WCA ≈ 60 × crosswind ÷ TAS (degrees)
Wind from the right means crab right, so heading is to the right of track

How do you find headwind and crosswind components?

Resolve the wind against the runway or track.

\text{Crosswind}=V\sin\theta

\text{Headwind}=V\cos\theta

θ is the angle between wind direction and runway heading (or track)
Example: 30 kt at 40° off: crosswind 19 kt, headwind 23 kt

How do you find drift and groundspeed from a wind triangle?

Draw heading/TAS from the start, add the wind vector, and the closing line is the track/groundspeed.

\sin(\text{WCA})=\dfrac{W}{V}\sin(\text{wind angle})

Drift = angle between heading and track
Drift right: track is to the right of heading
Correct for wind: heading = track − drift (drift right → steer left of track)

Tip: Wind from the right blows you to the left of heading (drift left); the correction is to steer into the wind.

What is the maximum drift angle for a given wind and TAS?

Maximum drift occurs when the wind is perpendicular to the track (a pure crosswind).

\text{Drift}_{max}=\sin^{-1}\left(\dfrac{W}{TAS}\right)

Maximum drift ≈ arcsin(W / TAS)
Example: W = 30 kt, TAS = 120 kt → about 14.5°

How do you use the 1-in-60 rule for a quick wind correction?

Drift (°) ≈ crosswind component × 60 / TAS.

\text{Drift}\approx\dfrac{60\times\text{xwind}}{TAS}

Crosswind = wind speed × sin(wind angle)
Example: 40 kt wind at 30° to track, TAS 120 kt: xw = 20; drift = 20 × 60/120 = 10°

Simple Navigation Maths

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Convert a climb or descent gradient into a rate of climb.

Rate of climb in ft/min depends on gradient and ground speed.

ROC(\text{ft/min})\approx GS(\text{kt})\times\text{gradient}(\%)\times 1.013

5 % gradient at 120 kt ≈ 600 ft/min
ft per NM = ROC ÷ GS × 60

Gradient in percent from the angle?

Percent gradient is 100 × tan of the angle.

\text{gradient}\ (\%)=100\tan\theta

3° ≈ 5.2 % ≈ 318 ft per NM
1° ≈ 1.75 % ≈ 106 ft per NM

Fuel, time and distance: how do you solve “how much fuel for this leg”?

Time first, then fuel.

\text{fuel}=\text{fuel flow}\times\frac{\text{distance}}{GS}

Use the leg’s ground speed, not TAS
Add reserve according to the rules in the question

How do you find time, distance and speed quickly?

Use the triangle: Distance = Speed × Time.

D=S\times T

Minutes = distance × 60 / groundspeed
Mach × 10 ≈ NM per minute at cruise (M 0.8 ≈ 8 NM per min)
Fuel required = flow × time

1 in 60 Rule & Bearings

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State the 1 in 60 rule.

At 60 NM from the point of departure, 1° of track error gives 1 NM of lateral displacement. Distances scale in proportion.

\text{Track error}^\circ=\frac{\text{off-track distance}}{\text{distance flown}}\times 60

How do you regain track using 1 in 60?

Add the track error and the closing angle.

\text{Closing angle}=\frac{\text{off-track}}{\text{distance to go}}\times 60

To rejoin track in the same distance you have already flown, alter heading by twice the track error
Turn the correct way: towards the track

Time to a station from a bearing change?

Fly at a steady speed on a track passing abeam, note the time for a given bearing change.

\text{Time to station (min)}=\frac{60\times\text{time between bearings (min)}}{\text{change of bearing (}^\circ)}

Valid when the bearing change is measured at the wingtip or on the beam
Distance = ground speed × time

State the 1-in-60 rule.

At 60 NM from the start, an error of 1 NM equals 1° of track error.

\text{Error}^\circ=\dfrac{\text{off-track NM}\times60}{\text{distance gone NM}}

Track error (°) = off-track distance × 60 / distance gone
Closing angle (°) = off-track distance × 60 / distance to go
Heading change to regain track = track error + closing angle

How do you work out the descent angle and rate of descent with the 1-in-60 rule?

Descent angle (°) = height to lose / (distance in NM × 100). Rate of descent ≈ groundspeed × 5 for 3°.

\text{ROD}\approx GS\times 5\ (\text{ft/min, for }3^\circ)

3° path: 300 ft per NM
Rate of descent (ft/min) ≈ GS (kt) × 5 (more precisely × 5.3 for 3°)
To lose 9 000 ft at 3°: 30 NM

Solar System & Time

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How do longitude and time relate?

The Earth turns 360° in 24 h, so 15° = 1 hour and 1° = 4 minutes.

\text{LMT}=\text{UTC}+\frac{\text{longitude E}}{15}

East is later (add), west is earlier (subtract)
IST = UTC + 5 h 30 min, based on 82.5° E

How are standard time and UTC related?

Standard time = UTC + longitude/15 (rounded to the nearest hour, east positive).

\text{LMT}=\text{UTC}+\dfrac{\lambda_E}{15}

India: UTC + 5:30
Local mean time: 4 minutes per degree of longitude
Crossing the date line westward: add a day

Descent Planning

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How far before the destination should you start a descent?

Use the 3-to-1 rule for a 3° path.

D(\text{NM})\approx\frac{\Delta h(\text{ft})}{1000}\times 3

From FL 330 down to 3000 ft: 30 000 ft ÷ 1000 × 3 = 90 NM
Add distance to slow down, and allow for wind

What rate of descent holds a 3° path?

About 5 times the ground speed (more precisely 5.3 ×).

ROD(\text{ft/min})\approx GS(\text{kt})\times 5

GS 140 kt → ROD ≈ 700 ft/min
A tailwind raises GS and needs a higher rate

Descent gradient from rate and speed?

ft per NM = rate of descent ÷ ground speed × 60.

\text{ft/NM}=\frac{ROD}{GS}\times 60

1000 ft/min at 120 kt = 500 ft/NM (too steep for a 3° path)

How do you calculate the top of descent distance?

Distance (NM) = height to lose (ft) / 300 for a 3° path.

\text{Dist (NM)}\approx\dfrac{\Delta h\ (\text{ft})}{300}

FL350 to 3 000 ft: 32 000 / 300 ≈ 107 NM
Common cockpit rule: about 3 NM per 1 000 ft to lose (3 × 32 = 96 NM)
Allow extra distance to decelerate

Critical Point & PNR

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What is the critical point (CP) or equal time point (ETP)?

The point on a route from which the flight time to continue to the destination equals the time to return to the departure (or alternate) airfield.

\text{Distance to CP}=\frac{D\times GS_{back}}{GS_{out}+GS_{back}}

\text{Time to CP}=\frac{D}{GS_{out}+GS_{back}}

D is the whole distance between the two airfields
The CP moves into the wind

What is the point of no return (PNR)?

The last point from which you can still return to the departure aerodrome using the fuel carried (safe endurance E).

\text{Time to PNR}=\frac{E\times GS_{back}}{GS_{out}+GS_{back}}

\text{Distance to PNR}=GS_{out}\times\text{time to PNR}

Based on fuel, whereas CP is based on time
A tailwind outbound brings the PNR further out

What is the point of no return (PNR) and its formula?

The last point from which you can return to the departure airfield with the fuel on board.

T_{PNR}=\dfrac{E\times H}{O+H}

Endurance E = usable fuel / fuel flow
Time to PNR T = E × H / (O + H) where O = outbound GS, H = homebound GS
Distance to PNR = T × O

Tip: The wind is the key: a tailwind out means a headwind home, so the PNR moves closer.

What is the critical point (CP) or equal time point?

The point from which the time to continue to the destination equals the time to return to the departure.

\text{Dist to CP}=\dfrac{D\times H}{O+H}

Time to CP = D × H / (O + H) where D = total distance
Distance to CP = D × H / (O + H)
With no wind CP is the midpoint

Payload, Fuel & Weight and Balance

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How do you find the centre of gravity?

Take moments about a datum and divide by total mass.

\text{CG}=\frac{\sum(\text{mass}\times\text{arm})}{\sum\text{mass}}

Moment = mass × arm
Arm is measured from the datum, positive aft
Check the CG lies inside the envelope for every loading

How far does the CG move when you shift cargo?

The shift equals mass moved times distance moved, over total mass.

\text{CG shift}=\frac{m_{moved}\times d}{M_{total}}

Moving mass aft moves the CG aft; forward moves it forward
The shift is always in the direction the mass moved

How do you find the centre of gravity of a loaded aeroplane?

Take moments about the datum: CG = total moment / total mass.

CG=\dfrac{\sum(m\times\text{arm})}{\sum m}

Moment = mass × arm
Adding mass behind the CG moves it aft
Shifting mass forward moves the CG forward by (mass × distance)/total mass

Make the Air Navigation formulas stick

In class every formula is built from a diagram first, then practised until it is automatic.

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