How big is the circle you are flying?
Turn radius is V² / (g tan φ), and almost every calculator that prints it feeds in the number off your airspeed indicator. That is the wrong number. The formula wants true airspeed, and radius goes as the square of speed, so at 10,000 ft the circle you are really flying is about a third larger than the ASI implies and sweeps nearly twice the area. If you have ever wondered why a thermal that worked beautifully at 3,000 ft keeps sliding out from under you at 10,000 ft, this is a large part of the answer.
The second half of the page is the more useful one: given your own stall speed and the margin you choose to keep over it, what is the smallest circle you can fly at all — and is it small enough to stay inside the core?
Your circle right now
What the ASI in front of you actually reads. The conversion to true airspeed happens below.
Both ends of the slider are real answers and both are degenerate. Zero is straight flight; 90 is not a level turn at all.
Your altimeter with 1013 hPa (29.92 inHg) set. Close enough to your indicated altitude on most days for this purpose.
Radius of turn
True airspeed
Diameter
Rate of turn
Time for a full circle
Load factor
What the ASI reading alone would have told you
Radius using indicated airspeed
Radius using true airspeed
Area swept, compared with sea level
At a density altitude of the density ratio σ is , so true airspeed runs above indicated and your circle is bigger than the ASI-only figure. You have not changed anything you can feel: the attitude is the same, the ASI reads the same, the stick forces are the same. The circle is simply larger.
At this bank you are inside the margin you chose below. Stall speed at ° is indicated, so is only times the stall rather than the you asked for. Rolling into a steeper turn without adding speed closes the gap from both directions at once — the stall speed rises while your airspeed decays.
The smallest circle you can actually fly
You do not get to pick radius and bank independently. Steeper bank means more load factor, more load factor means a higher stall speed, and a higher stall speed means you must fly faster — which pushes the radius back up. Put in your own numbers and the whole trade becomes visible.
Indicated, from your own flight manual, at the weight you are flying today. Water ballast raises it as the square root of mass.
Against the stall speed at the bank flown, not the wings-level figure. 1.2 is a conventional thermalling margin; it is a choice, not a rule.
Your judgement, not a limit from anywhere else. The search stops here.
Tightest circle in that range
radius, at ° and indicated
The floor, at any bank
what 1/sin φ approaches as bank goes to 90°
Smallest core you could work
across, i.e. twice the floor
Notice where that minimum landed. It is sitting on the steepest bank you said you would use, and it always will, because the radius falls monotonically with bank right up to 90°. That number is the edge of your own input box, not an optimum this page discovered. The real limit is the floor beside it, and the floor depends on your stall speed and your margin — not on your willingness to bank.
| Bank | Radius | Speed needed | g | Circle in | |
|---|---|---|---|---|---|
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Bars are radius, to scale. The teal mark is half the core width you set below — bars ending left of it fit inside the core. Rows past ° are greyed because you said you would not use them, not because anything forbids them. Read the speed column and ask whether that is a speed you would genuinely circle at.
Will you fit in the core?
The useful lift, not the whole rising bubble. You need to stay inside a circle of radius.
Why true airspeed is the only speed this formula accepts
The airspeed indicator is not a speedometer. It is a pressure gauge, calibrated so that it reads correctly in sea-level air, and what it actually measures is dynamic pressure — half the air density times the square of your speed through the air. As you climb, the density falls. To keep producing the same dynamic pressure, and therefore the same reading on the dial and the same lift from the wing, you have to move through the air faster. That is why the ASI is such a good instrument for flying the aeroplane: it tells you about lift and stall directly, and those are what keep you alive.
It is a terrible instrument for geometry. Turn radius is a distance in the real world, swept out by an aircraft physically moving through the air at true airspeed. Feed the formula an indicated reading and the whole answer is wrong by the density ratio.
Work the standard-day numbers through and the size of the error is unpleasant. At 10,000 ft the density ratio σ is 0.74, so true airspeed is 16% above indicated. Radius goes as the square of speed, so the circle is 35% larger. Area goes as the square of radius, so you are sweeping 1.83 times the ground — nearly double. Nothing in the cockpit tells you this. Same attitude, same ASI reading, same stick position, same feel. The circle is simply bigger, and if the core is narrow you are now spending part of every turn outside it.
The effect compounds with heat, because it is really density altitude that matters rather than height. Ten thousand feet on a day 10 °C above standard is a density altitude closer to 11,200 ft, and the circle grows again. The density altitude calculator works that half through on its own.
One consequence people miss: your rate of turn falls too. Rate is g tan φ / V, so the same bank at the same ASI reading takes about 16% longer to get round at 10,000 ft than at sea level. If you centre a thermal by counting seconds through the good part of the circle, your timing is quietly wrong at height.
Steeper really is tighter — but the returns collapse
You cannot choose your radius and your bank separately, because the speed you can fly depends on the bank. In a level turn the load factor is n = 1/cos φ, and stall speed rises with the square root of load factor, so the slowest speed you will accept at bank φ is m × Vs1g × √n, where m is whatever margin you keep. That is an indicated speed, because Vs1g is the indicated figure out of your flight manual, so it has to be divided by √σ before it goes anywhere near a radius — the same correction the first half of this page is about, and the reason σ appears below. Substitute, and the n cancels against the cos φ hiding inside the tangent:
Rmin(φ) = m² Vs1g² n / (σ g tan φ) = m² Vs1g² / (σ g cos φ tan φ) = m² Vs1g² / (σ g sin φ)
That is 1/sin φ, and 1/sin φ falls all the way to 90°. There is no interior optimum. It is sometimes claimed that the tightest circle sits at some middling bank because the required speed rises too — that result comes from mistakenly using n = 1/cos²φ, which gives 1/(sin φ cos φ) and a spurious minimum at 45°. The load factor in a level turn is 1/cos φ. Steeper is always tighter, and this page will not tell you otherwise to make a nicer-looking graph.
The genuinely interesting result is what steepening buys. Because 1/sin φ flattens out, the radius does not head for zero — it heads for a floor of m² Vs1g²/(σ g), which is exactly the radius you would fly at 90° of bank if 90° of bank were possible. Each step of steepening returns less than the last, while the load factor and the speed you must hold both climb without limit:
- cuts the radius by .
Those percentages are pure geometry — stall speed, margin and air density all cancel out of the ratio, so they are the same for every glider on every day. Going from 30° to 45° is worth having. Going from 60° to 75° buys you a tenth of your radius in exchange for going from 2.0g to 3.9g and needing 39% more airspeed — because the speed you must hold goes as √n, and √(3.86/2) is 1.39. That is the trade, stated plainly. Whether it is worth taking is a judgement about your aircraft, your currency and how accurately you can hold a steep turn — and none of those are things a web page knows.
The other reason to be sceptical of very steep circling is one this page cannot model: sink rate. Circling sink grows roughly as n3/2, so the tightest circle is often not the best climb even when it is comfortably flyable. That trade — radius against sink, in a core of a given width — is what the thermalling and ballast calculator is for.
Core width, and the circle that fits inside it
A thermal core is not a point. If the useful lift is 150 m across, staying in it means keeping your circle inside a 75 m radius, and that single requirement ties your speed and your bank together. Take a glider with a 35 kt wings-level stall speed and a 1.2 margin: at sea level it needs about 39° of bank at 48 kt indicated to fit, and its floor — the tightest circle it could fly at any bank at all — is 48 m radius. There is room to spare, so the pilot has choices.
Take the same glider and the same core to 10,000 ft and the arithmetic turns unpleasant. The floor rises from 48 m to 64 m, because the true speed at every bank is 16% higher. Fitting the same 75 m circle now takes about 59° of bank rather than 39°. Twenty degrees more bank, for the same glider in the same core, with nothing on any instrument to warn you. And the smallest core that glider can work at all has gone from 95 m across to 129 m.
This is why narrow high-altitude thermals feel so much harder than narrow low ones, and why the answer so often is not "bank more" but "get lighter". Dumping water reduces Vs1g, and the floor goes as Vs1g² — so a 10% reduction in stall speed buys a 19% reduction in the tightest circle you can fly. No amount of bank does that, because the floor already assumes infinite bank.
Two honest caveats about the core-fitting number above. First, it assumes you are perfectly centred; a circle displaced from the core by even 20 m needs a much tighter circle to stay inside the lift, and nobody is perfectly centred. Second, a glider is not a point. A 15 m span sweeps 7.5 m either side of the path you are flying, so part of the wing is outside a marginal core even when the fuselage is inside it. Both of those push in the same direction: the circle you need is smaller than the one this page computes.
What this assumes, and what it can't know
The turn geometry is exact, not modelled. R = V²/(g tan φ) and rate = g tan φ/V follow from resolving the lift vector in a steady, balanced, level turn. There is no aerodynamic model and no aircraft-specific assumption anywhere in them. If the turn is descending — and every glider turn is — the radius is very slightly smaller than shown, by a factor of the cosine of the glide angle. At a 40:1 glide that is under a tenth of a percent, which is far below the precision of anything else on this page.
Every limit here is one you typed in. Your stall speed, your stall margin, and the steepest bank you are willing to use. This site ships no maximum bank angle, no manoeuvring speed and no stall speed for any type, because all three are placarded per airframe and vary with weight, water and modification state. Read yours out of your own flight manual, at today's weight.
Stall speed rises with weight, and this page does not know your weight. Vs1g goes as the square root of mass, so a glider carrying 100 kg of water at 350 kg dry stalls about 13% faster and its floor radius is about 29% larger. If you are ballasted, use the ballasted stall speed — the difference is the whole reason people dump water in weak, narrow conditions.
The smallest-circle figure is a floor, not a target. It is the circle you fly at exactly your chosen margin over the banked stall, perfectly coordinated, in still smooth air, holding the bank precisely. Thermals are none of those things. Turbulence produces momentary gust loads that raise the stall speed further, and a thermal edge produces a rolling moment you have to fight. Keep more margin than the arithmetic demands.
Nothing here says anything about your climb rate. The tightest circle is frequently not the best one, because circling sink grows faster than radius shrinks. This page answers a geometry question only.
The atmosphere is the standard one, and humidity is ignored. True airspeed comes from pressure altitude and temperature via the ISA density ratio. Moist air is slightly less dense than dry air, so a very humid day is marginally worse than shown — a small effect next to temperature, and conventionally left out.
Not yet checked by anyone but me. If you instruct, or one of these assumptions is wrong, I would genuinely rather hear it than not — tell me and I will credit you here. This is ground school, not a flight computer, and not an authority on your aircraft. Fly the numbers in your own flight manual and the instruments in front of you.