The 1 in 60 Rule — Page 189, Lesson 168

Let’s start with the geometry, because that’s the foundation of the whole 1 in 60 rule. I want you to imagine a circle with a radius of exactly 1 metre. Now, the formula for the circumference of any circle is 2πr, where r is the radius. Taking π as 3.142, which is close enough for our purposes, the circumference of our 1-metre-radius circle is 2 × 3.142 × 1, which equals 6.284 metres. Now, imagine we walk one metre along the circumference — not cutting across the chord, but following the curve itself. That one metre of arc will subtend an angle θ at the centre of the circle. We need to find the value of θ. If you’ve done some engineering or applied maths, you’ll recognise that θ is 1 radian. But let’s derive it from first principles. Starting at any point on the circumference and going all the way round once gives an angle of 360° and a distance of 6.284 metres. Going just one metre along the circumference gives an angle θ. These are in proportion, so we can write the equation: 360 divided by 6.284 equals θ divided by 1. Solving that, θ equals 57.3°. Now here’s the fundamental principle of geometry: as long as we keep the ratios the same, we can scale a diagram up or down and the angles will not change. For instance, if we take a circle of 2 metres radius, then as long as we also take a 2-metre length of the circumference, θ will still be 57.3°. Similarly, θ remains 57.3° if we take a circle of 10 metres radius, as long as we also take 10 metres of circumference. So, if we take a circle with a radius of 57.3 metres and consider a length of the circumference of 57.3 metres, we now have a situation where 57.3 degrees corresponds to 57.3 metres — a perfect one-to-one ratio. But note, this only occurs with a radius of 57.3 metres. Strictly, then, this should be called the “1 in 57.3 rule.” But 57.3 is a difficult number to use in mental arithmetic when flying, so we approximate it to 60. That introduces only about 5% error, which is perfectly acceptable for our purposes. Now let me give you another explanation, a different way of looking at it. Consider a right-angled triangle with an angle z, an opposite side, and an adjacent side. Let’s write a series of values for z between 1 and 20 and compare the angles with their tangents. For z = 1, tan z is 0.017. For z = 2, tan z is 0.035. For z = 5, tan z is 0.087. For z = 10, tan z is 0.176. For z = 15, tan z is 0.268. For z = 20, tan z is 0.364. Nothing leaps out of the page at us. If there is a relationship, it’s not a clearly obvious one. What this tangent relationship is saying is: if the adjacent side is 1 metre, then the length of the opposite side is the number of metres in the “tan z” row — i.e., 0.017 metres for 1°, 0.035 metres for 2°, and so on. Now let’s expand the adjacent side to 60 metres in length. This means multiplying the opposites by 60 as well. And that’s where the magic of the 1 in 60 rule comes from — when the adjacent side is 60 units, the opposite side in those same units becomes almost exactly the angle in degrees. That’s the practical heart of the rule: for small angles, the track error in degrees is approximately equal to the distance off track divided by the distance flown, all multiplied by 60. So, to summarise what we’ve built: the 1 in 60 rule is an approximation of the true 1 in 57.3 relationship, and it works because for small angles, the tangent of the angle is nearly proportional to the angle itself. The 5% error is acceptable in flight because we’re doing mental arithmetic, not precision engineering. That’s the geometry and the tangent explanation. When you’re ready, we can move on to how this rule is actually applied in navigation — things like track error and closing angles.