Today, a simple, practical guide for estimating the height of a tree from Ian Stewart's recent "Professor Stewart's Casebook of Mathematical Mysteries."
Stewart describes this as "an old forester's trick (the trick is old, not the forester) for estimating the height of a tree without climbing it or using surveying equipment."
"Stand at a reasonable distance from the tree, with your back towards it. Bend over and look back at it through your legs. If you can't see the top, move away until you can. If you can see it easily, move closer until it's just visible. At that point, your distance from the base of the tree will be roughly equal to its height."
The angle your line-of-sight is forming with the ground is now roughly 45 degrees, and thus the line-of-sight itself is the hypotenuse of an isosceles right triangle with the base-distance and tree height equal side values (thus, measure or walk off the base, and you have the height).
...As for those of us of an age where bending over and looking through our legs isn't such a practical affair, I guess we're out-of-luck :-(
Anyway, Stewart's volume is his usual mix of older and fresher, easier and more technical, math entertainment. Give it a gander.
One of my old favorites that shows up in Stewart's book is what he calls the "Square Peg Problem" (it goes by some different names) -- another one of those seemingly easy, yet exquisitely difficult-to-prove (100+ year-old) conjectures. It asks whether one or more squares can always be fitted upon every closed planar curve by selecting four points on that curve, or stated more succinctly as "Does every simple closed curve have an inscribed square?" For a fuller treatment of it see here:
and here's a 2014 article on it (pdf) from AMS: