# How to find Radius of Turn given groundspeed and bank angle? [duplicate]

How to find Radius/Rate of Turn given groundspeed and bank angle? Is GS/bank angle one of the formulas? What are the formulas needed for Turns Around a Point? I'm trying to find a way to explain this maneuver as if I was teaching a student pilot?

• Groundspeed needs wind information, or you must assume still air. Sep 25 '21 at 9:08
– mins
Sep 25 '21 at 18:08

Here are some equations - (pay attention how one changes the other - to visualize)

$$\text{Radius of turn} = \frac{v^2}{g \times \tan(\theta)} \qquad \text{(standard units)}$$

Rate of turn

If you increase speed, rate is smaller since you make a bigger turn

$$\text{Rate of turn} = \frac{g \times \tan(\theta)}{v}$$

Hope this helps

The formula for Turn radius is $$V^2 / G_R$$, where $$V$$ is velocity (ground speed), and $$G_R$$ is radial G, which can be determined by taking the tangent of the bank angle multiplied by what G is in the units you are using for velocity (if velocity is ft/sec, then Gs must be in $$\text{ft}/\text{s}^2$$ ($$\sim 32.2 \text{ft}/\text{s}^2$$), or $$\sim 9.8 \text{m}/\text{s}^2$$). This is to ensure the numbers are all in the same scale.

So turn radius is $$V^2 / \tan(\theta)$$, where $$\theta$$ is the Bank Angle.

To get turn rate, just think about the velocity, and the circumference of a complete 360 degree circle. The circumference of a circle with radius $$R$$ is $$2\pi R$$, and if the velocity is $$V$$, then how long will it take the airplane to travel that distance? Time is distance divided by velocity. So it will take Time $$T = 2\pi R / V$$. And if it takes Time $$2\pi R / V$$ to go 360 degrees, then the turn rate is how many degrees it will change in one second, so that is $$360^\circ V / (2\pi R)$$.

But $$R$$ (from turn radius above) is $$V^2 / \tan(\theta)$$, so turn rate is $$360^\circ V \tan(\theta) / (2\pi V^2)$$, or,

Turn Rate = $$180^\circ \tan(\theta) / (\pi V)$$

NOTE. Again, to use these formulas everything needs to be in the same units: e.g., if V is in $$\text{ft}/\text{s}$$, then radial G must in $$\text{ft}/\text{s}^2$$. So in this case, using feet and seconds, you need to add $$32 \text{ft}/\text{s}^2$$ to the formula for radial G. Instead of $$\tan(\theta)$$, make it $$32\times \tan(\theta)$$. If you are entering velocity in knots, then the value of G must be converted to nautical miles per second squared.

• In your first sentence, I think you mean TAS instead of ground speed, right? Also, I converted the images to MathJax (Basic Tutorial and Quick Reference). Sep 25 '21 at 17:33
• Depends on whether you want to know the Turn rate and radius in the atmospheric frame of reference (Moving with speed = wind velocity), or rate and radius across the ground. Sep 25 '21 at 18:12