The 1 in 60 Rule — Page 189, Lesson 168

I want to walk you through the 1 in 60 Rule — one of the most practical mental shortcuts you'll use in navigation as a professional pilot. Let's start with the geometry behind it, because understanding where it comes from helps you trust it in the cockpit. Imagine a circle with a radius of exactly 1 metre. The formula for circumference is 2πr. Taking π as 3.142 — close enough for our purposes — the circumference works out to 2 × 3.142 × 1, which is 6.284 metres. Now picture this: you travel exactly 1 metre along the circumference — not cutting across the chord, but following the curve itself. That 1-metre arc will subtend an angle θ at the centre of the circle. We need to find the value of θ. If you've studied engineering or applied maths, you might recognise that θ equals 1 radian. But let's calculate it from first principles. Starting at any point on the circumference and going all the way around once gives you an angle of 360° and a distance of 6.284 metres. Going just 1 metre along the circumference gives you the angle θ. These are in proportion, so we can write: 360° divided by 6.284 metres equals θ divided by 1 metre. Solving that, θ equals 57.3°. Now here's a fundamental principle of geometry: as long as we keep the ratios the same, we can scale a diagram up or down and the angles won't change. For example, if we take a circle of 2 metres radius and also take a 2-metre length of circumference, θ is still 57.3°. Same for a 10-metre radius with a 10-metre arc — θ stays at 57.3°. So here's the key step: 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 this only works with a radius of 57.3 metres. Strictly speaking, then, this should be called the "1 in 57.3 rule." But 57.3 is a difficult number to use in mental arithmetic when you're flying. So we approximate it to 60. That introduces only about a 5% error, which is perfectly acceptable for our purposes in navigation. Let me give you another way to look at it. Consider a right-angled triangle where the adjacent side is some length, the opposite side is some length, and the angle at the vertex is z degrees. If we write a series of values for z between 1 and 20, and compare the angles with their tangents, here's what we get: For z = 1°, tan z is 0.017. For 2°, it's 0.035. For 5°, 0.087. For 10°, 0.176. For 15°, 0.268. For 20°, 0.364. Nothing immediately obvious jumps out. 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 — 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. That means we multiply the opposite sides by 60 as well. And that's where the 1 in 60 rule starts to reveal itself — but we'll pick up from there next time. For now, the core takeaway is this: the 1 in 60 rule is an approximation of a precise geometric relationship. One degree at the centre of a circle subtends an arc of about 1 unit for every 60 units of radius. The exact value is 57.3, but we round it to 60 for easy mental arithmetic, accepting a small 5% error.