Calibrate Yourself

While flying low on the approach to an airport¹, I started wondering how high we were. A common enough thought. This presented a problem - I felt like I had few means to satisfy this curiosity. The use of phones was prohibited during the approach - so there was no GPS or barometer on hand that I could use.

In hindsight I suppose I could have asked the attendant, or tried gaining intimate knowledge of typical approach parameters for a variety of common passenger aircraft and airports… Yawn.

I wanted to know how high we were somehow, so… What do I know?

• I suppose I am best at estimating lengths of 1 mm to 10 m by sight². Outside this range my estimates get real hairy, and I couldn’t tell 1 km from 5 km³.
• s = rθ
• I know the approximate size of many objects that are visible from an airplane window; houses, cars, roads.

I looked at my fingers and mulled a little. They could help me. If I knew the size of a smallish object and held it in front of me at a known distance then I can define a sector of my vision, which is a known angle. A smallish object that I will likely always carry with me is my thumbnail. This looked a little shy of 20mm so I went with 18mm.

Thanks to an old friend’s junior science project⁴, I recall that people (at least the little ones that we were at the time) have an arm-span that is approximately equal to their height. Knowing I am ≈1.8m tall, I could then say that my outstretched arm would rest somewhere a little less than 0.9m from my eyes (accounting for triangles and such). Make that 0.75m from the eye on the same side as my outstretched arm.

Cranking it out

The size of the angle I make by the width of my thumbnail when my arm is outstretched is:

θ = s/r

θ = 18mm/750mm

θ = 0.024

To make this useful for getting distances:

distance_object = length_object/θ

distance_object = length_object × 1/θ

And:

1/θ = 1/0.024

1/θ = 41.67

I rather happily rounded the above to 42.

Which makes:

distance_object = length_object × 42

Units for distance and length don’t matter as long as the same unit is used for both.

All super simple stuff that could be done a myriad of ways (similar triangles would save a step). Still, it managed to give the old mind something to chew on for a bit.

Post flight check

How off were my estimates?

width_thumbnail = 16.5mm

distance_{eyeToThumbnail} = 0.79m

Which would make the magic number 48, not 42. [cramps arm, sticks with 42].

Some days after …

Update - Military Estimation

‘Military Estimation’ is the field that covers much of what I’m grasping for in this post.