VectorEFB
Menu
Request access

Learning hub · General Navigation · Lesson 8

Radius of action, point of no return and point of equal time

Before a long leg over the sea, a mountain range or empty country, two questions matter more than the arrival time. How far can I go and still get home with my reserve? And where is the point from which continuing and turning back take the same time? Both come from a few lines of arithmetic.

About 15 minutes, plus the quiz. Part of General Navigation.

1. Endurance and safe endurance

Endurance is the time the fuel in the tanks will keep the engine running at the power you plan to use. Fuel on board divided by the fuel flow gives it: 60 litres at 20 litres an hour is 3 hours.

You never plan to use all of it. The reserve stays in the tanks when you land. It is the final reserve fuel that the rules require, plus whatever contingency your school or your own rule adds. The fuel lesson shows how the pieces are built up. What you may actually use is the safe endurance:

safe endurance = endurance − reserve

A tank that lasts 4 hours with a 1 hour reserve gives a safe endurance of 3 hours. Every number in this lesson uses safe endurance and not the full tank.

2. Radius of action

You take off from A, fly out along a track for some time, turn round and fly back along the same track to land at A, all inside the safe endurance. The farthest point you can turn at is the radius of action. In calm air it is easy: half the range. With wind it is not, because you go out and come home at different groundspeeds.

Call the safe endurance E, the groundspeed out Go and the groundspeed home Gh. If you turn back after a distance d, the time out is d ÷ Go and the time home is d ÷ Gh. Together they must equal E:

d ÷ Go + d ÷ Gh = E

Take d out as a common factor, bring the fractions together and solve for d:

d × (Go + Gh) ÷ (Go × Gh) = E radius of action = E × Go × Gh ÷ (Go + Gh)

With a wind that blows straight along the track, Go is the TAS minus the headwind and Gh is the TAS plus it (or the other way round for a tailwind out). In calm air both equal the TAS, and the formula gives E × TAS ÷ 2, which is half the still-air range, as it should.

3. The point of no return

The point of no return (PNR) is the same place seen from a different side: the last point from which you can still turn round and reach the departure aerodrome with the reserve intact. Its distance from A is the radius of action. The time to reach it is the safe endurance share spent going out:

time to PNR = E × Gh ÷ (Go + Gh)

You can check it: the time to the PNR multiplied by Go gives the same distance as the radius formula. Notice that Gh sits on top. A strong headwind on the way out makes Go small and Gh large, so the PNR comes late in time but early in distance.

Beyond the PNR you cannot go home on the fuel you planned to use. That is not an emergency by itself. It means you are committed to a landing somewhere ahead, so that place must be open, reachable and suitable. In flying schools the PNR is mostly an idea for planning, and a reason to check destination weather and alternates before you cross it, not after.

4. The point of equal time

The point of equal time (PET, also called the critical point) is a different question. Two aerodromes, A and B, are a distance D apart on one track. The PET is the point on that track from which flying on to B and flying back to A take the same time. It matters when something goes wrong and you must pick one of the two: the engine runs rough, a window of weather closes, a passenger feels ill.

Let the PET be a distance x from A. Going on takes (D − x) ÷ Go, because you keep flying A to B. Going back takes x ÷ Gh. Set them equal:

(D − x) ÷ Go = x ÷ Gh x = D × Gh ÷ (Go + Gh)

In calm air Go equals Gh and x is D ÷ 2, the halfway point. With a headwind on the way out, Gh is larger, so x is greater than half: the PET moves towards B, because getting home is quick and going on is slow. With a tailwind out, it moves towards A. The time to reach it from A is x ÷ Go.

The PET only compares times. It says nothing about fuel, weather or the state of the two aerodromes. It is a decision aid and not a rule.

Try it: set the fuel, the reserve, the airspeed, the wind and the distance between A and B. The orange marker is the point of no return, the blue one is the point of equal time, and the aeroplane flies out to the PNR and back in proportion to the times. The bar at the front is the tank: green is the time out, blue the time back, amber the reserve. Move the wind from the nose to behind and watch what happens to the radius.

This diagram needs JavaScript to move. The explanation above and the exercises below cover the same ideas.

5. Why a student pilot needs this

  • Water. On a coastal route, a leg across a bay or to an island gives you no landing place for part of the way. Know the width of the gap and the time you are over it.
  • Mountains and empty country. A ridge with cloud on the far side is a place where you may want to turn back. Knowing the PET beforehand lets you make that decision early and calmly.
  • Fuel at the far end. If B has no fuel, or you are not sure it is open, the return leg must fit in the tanks. The radius of action tells you how far out you can plan to land.
  • The wind changes the answer. The forecast wind aloft at your cruising level is the number to use, and it can be different at the destination. Redo the sum if the wind is stronger than forecast.

The numbers here assume the track is a straight line and the wind lies along it. A crosswind lowers the groundspeed both ways, so real figures are a little worse. Use the wind triangle for the exact groundspeed on each leg.

6. A worked example

These numbers are invented. Tanks give 4 hours, you keep a 1 hour reserve, TAS is 100 kt, and the wind gives a 20 kt headwind on the way out from A. Aerodrome B is 200 NM from A.

  1. Safe endurance: 4 − 1 = 3 hours. Groundspeed out: 100 − 20 = 80 kt. Groundspeed home: 100 + 20 = 120 kt.
  2. Radius of action: 3 × 80 × 120 ÷ (80 + 120) = 28,800 ÷ 200 = 144 NM. In calm air it would be 150 NM, so the wind has cost 6 NM.
  3. Time to the PNR: 3 × 120 ÷ 200 = 1.8 hours. Check: 1.8 h at 80 kt is 144 NM, and the way home is 144 ÷ 120 = 1.2 h. Together 3.0 hours.
  4. Point of equal time to B: 200 × 120 ÷ 200 = 120 NM from A. Going on is 80 NM at 80 kt, 1 hour. Going back is 120 NM at 120 kt, 1 hour.
  5. Can you land at B and come back? Out takes 200 ÷ 80 = 2.5 h, back 200 ÷ 120 = 1.67 h, together 4.17 h, more than 3 h. B is beyond the radius. You can still plan to land at B with the reserve intact (2.5 h is under 3 h), but you cannot plan to return on the same fuel.

Open the diagram, choose Strong headwind out and compare with the figures above.

7. Try it yourself

Work each one out on paper first, then open the answer to compare. The numbers are invented.

  1. Tanks give 3 h 30 min and you keep a reserve of 45 min. TAS is 120 kt and the wind gives a 15 kt tailwind on the way out. Find the safe endurance, the radius of action and the time to the point of no return.
    Show the answer

    Safe endurance: 3.5 − 0.75 = 2.75 hours. Groundspeed out is 120 + 15 = 135 kt and home is 120 − 15 = 105 kt.

    Radius: 2.75 × 135 × 105 ÷ 240 = 38,981 ÷ 240 ≈ 162 NM. Time to the PNR: 2.75 × 105 ÷ 240 ≈ 1.2 h, about 72 minutes. Check: 1.2 h at 135 kt is 162 NM.

    In calm air the radius would be 165 NM (2.75 × 120 ÷ 2). The wind cost about 3 NM, even though it was a tailwind on the way out.

  2. Aerodromes A and B are 180 NM apart on one track. TAS is 110 kt and there is a 30 kt headwind on the way from A to B. Where is the point of equal time, and how long does either choice take from there?
    Show the answer

    Groundspeed A to B is 110 − 30 = 80 kt. B to A it is 110 + 30 = 140 kt.

    x = 180 × 140 ÷ (80 + 140) = 25,200 ÷ 220 ≈ 114.5 NM from A. In calm air it would be 90 NM, so the PET has moved about 24 NM towards B.

    Going on: 180 − 114.5 = 65.5 NM at 80 kt, about 0.82 h, 49 minutes. Going back: 114.5 NM at 140 kt, also 0.82 h. The two times match, as they must.

  3. Your radius of action works out at 144 NM, and your destination is 150 NM away. A friend says you must turn back at 144 NM. Is that right?
    Show the answer

    Not necessarily. The radius says how far you can go and still come back to the start. If you plan to land at the destination and not return, what matters is whether the whole trip to it fits in the safe endurance, and whether the destination is open and has fuel. In the worked example, 150 NM at 80 kt is 1.9 hours, well inside 3 hours.

    But once you pass 144 NM you have no home to return to on your planned fuel, so the destination weather and an alternative must be checked before you get there.

8. Check yourself

Five original practice questions. Answer them, press Check answers and read each explanation.

1. Your tanks give 4 hours and you keep a reserve of 1 hour. What is the safe endurance?

Safe endurance is the endurance minus the reserve: 4 − 1 = 3 hours. The reserve stays in the tanks.

2. TAS is 100 kt and there is a 20 kt headwind on the way out along the track. What is the groundspeed home?

On the way home the same wind is behind you, so the groundspeed is TAS plus the wind: 100 + 20 = 120 kt. Out it was 80 kt.

3. What is the point of no return?

The point of no return is the limit of the radius of action. Past it, your planned fuel no longer takes you home with the reserve, so you are committed to a landing ahead.

4. Compared with calm air, a headwind on the way from A to B moves the point of equal time:

With the headwind ahead, going on is slow and going back is quick. You have to be closer to B before going on takes as little time as turning back, so the point of equal time moves towards B.

5. Safe endurance is 2 hours, groundspeed out is 90 kt and groundspeed home is 110 kt. What is the radius of action?

Radius = E × Go × Gh ÷ (Go + Gh) = 2 × 90 × 110 ÷ 200 = 99 NM. Calm air with a TAS of 100 kt would give 100 NM, so any wind makes the radius a little smaller.

Back to the learning hub.

For training and situational awareness only. Not certified for navigation. Always use official charts, the AIP, NOTAMs and a proper weather briefing for real flights.