Angle Cards - A "business card" for measuring angle

This page has "business cards", which held at arm's length, help you measure angle, and thus to estimate distance and size.

Here is a sample.

Example usage: Hold the card at arm's length, looking past it at a distant door. Tall people, and doors, are about 2 meters (yards) tall. If a distant door comes up to the x30 mark on the card, then then it is 2 times 30 meters away, so 60 meters. If it only comes up to the x50 mark, then it would be 2 x 50 = 100 meters away. Same with tall people. A short 1 meter (3 ft) person at x50 would be 1 x 50 = 50 meters away. A 3 or 4 meter high building floor at x20 would be (3 or 4) x 20 = (60 or 80) meters away.

An x30
door

A 2 yard high door,
which appears to be "x30" high,
is 2 x 30 = 60 yards away.

Creating your card: Hold a piece of paper out at arm's length. Measure the distance from your eye to the paper. Print out the page closest to your distance. Chop it up into cards.

Mix.pdf (.ps)
30 cm (11.8 in) (.ps)
35 cm (13.8 in) (.ps)
40 cm (15.7 in) (.ps)
45 cm (17.7 in) (.ps)
50 cm (19.7 in) (.ps)
55 cm (21.7 in) (.ps)
60 cm (23.6 in) (.ps)
65 cm (25.6 in) (.ps)
70 cm (27.6 in) (.ps)
75 cm (29.5 in) (.ps)

Getting a feel for distance Go for a walk. Estimate the distance to houses, using their doors as above. Or instead of houses, you can use people, or posts, or cars, or anything else you know the size of. First just look at the house, and guess its distance. Then check your guess using the angle card. Then if you know the size of your stride, you can pace the distance to get another check. First start out by just trying to do "is it more like 1 meter, or 10 meters, or 100 meters?". Then as your accuracy improves with practice, you can try to get within 50%, 25%, etc. A good habit to practice is giving bounds on your estimate - "Well, I am sure it is more than xxx, and I am sure it is less than xxx,... and, well, my estimate is xxx.". Even if all you are sure about to start with is that it is more than nothing, and less than a zillion.

Measuring Angle and Distance with your Thumb has a few more examples.

Misc... You can carry the card around in your wallet, backpack, or even pocket. Printing on card-stock, rather than on paper, helps them last longer. Or you can just print out more, and treat them as disposable. A paper chopper helps save time when cutting out the cards. Or you can fold the paper, and run a scissor edge up inside. If a door is 2 meters tall, how tall is that house? How wide?

Links

A chart for measuring angle and distance from arm's length
Measuring Angle and Distance with your Thumb
At Arm's Length a NOVA Teacher's Guide.
Up, Up, and How Far Away?

Notes

You can estimate Earth's diameter by viewing the horizon from a tall building. [Caveat - I've not yet tried this myself.] First, determine what is horizontal (using railings, tables, pairs of doorknobs, whatever). Then measure the angle between horizontal and the horizon. (It can be helpful to choose a visual reference for horizontal, like some distant cloud.) The horizon will be about 0.01 radians down. Finally, determine how high you are (4 to 4.5 meters per floor is reasonable). (If your ground level is higher than the ground at the horizon, a city on a hill for instance, you should include it in your height.) The distance to the horizon is the height, times two (the earth is dropping away too), divided by the angle (well, sin(angle), but for small angles, they are the same (in radians, but not in degrees - that's one reason radians are nice)). Say 50 km. But the distance to the horizon is also an angle's-worth of the Earth's circumference! So, (2 * 3.14 / angle), about 700, times the distance, is the circumference. (We use 2 * 3.14 because 2 pi radians is 360 degrees.) The diameter is just the circumference / 3.14. Something like 13000 km (13 Megameters). Well, 12740 km +/- about 20 km.
Example: On a 45 story, 200 meter building, one might estimate the angle at between 0.007 and 0.01 radians, for a distance of 200 * 2 / 0.007 = 57 km to 40 km. Now, 2 * pi = 6.28. 6.28 / 0.007 = 900, and 6.28 / 0.01 = 628, suggesting the distance to the horizon is between 1/900th and 1/628th of the Earth's circumference. 900 * 57 km / 3.14 = 16000 km (16 Megameters). And 628 * 40 / 3.14 = 8000 km (8 Mm). Which nicely bracket the actual diameter of 13 Megameters (13000 km). :)
[Thanks to a reader for this idea.]

Thanks to a reader, Lamar Fussell, for motivating this page.

Comments encouraged - Mitchell Charity <mcharity@lcs.mit.edu>

Notes:
  The postscript is the hand-coded source.  Feel free to fiddle.

Doables:
  Next card draft:
    Add url.
    Drop ".5" from Mix - centering not worth silly precision.
    Fine lines for x300+.  Add rads to x side to show connection?
    Use sin rather than small-angle-approx to place deg and rad ticks.
    Do cleanup pass on source.
    Would color be interesting?  Allowing x, deg, and rad on same
    side, showing connections?  Or just have rules go in same direction?
    Use clipping rather than conditional for degree inclusion.
  So much more could be done with this page...
    Discuss pacing back and forth to give a parallax baseline.
  Gen card with ruler, (deg and rad) protractor.
    What else could be done with this form factor?  Reference data?
    Having someone else hold the card at two-arms length, for greater accuracy.
  Autogen cards with distance scale for known size targets (archery).
  Search engine fodder: range finder, astrolabe, protractor.

History:
  2002-Mar-05  Added `estimating Earth's diameter' note.  Thanks to a reader.
  2001-Nov-07  Added name to credit.
  2001-Aug-14  Fleshed out.  Online.
  2001-Aug-12  Card postscript.  Initial draft.