Measuring Angle and Distance with your Thumb
(Now part of A
View from the Back of the Envelope.)
The idea: By holding out your thumb to measure
angles, you can tell your distance to things (people, cars, buildings,
planes, clouds, etc).
This is a
I hold out my arm, look at my thumb, and see a distant car half as
high. Cars are about 5 feet (1.5 meters) high. So my thumb appears
10 feet (3 meters) wide. And since I know (see below) my thumb is x30
times as far as it seems tall... I know the car is something like 300
feet (90 meters) away!
Hold your arm straight out in front of you. Make a fist. Point it at
a window. The window will appear a couple of fists high. The further
away the window, the fewer the fists. Windows in nearby buildings
won't even be a fist tall, they will perhaps be a couple of
thumb-widths high. Or less. Stick your thumb out sideways. Point
your arm at a nearby building. Count how many thumbs high its windows
Ok, but how do you tell how far away it is?
You need to know two things:
very roughly how big windows are (I don't know, say something like 5 feet),
and that your thumb covers things 30 times bigger.
So, a 5 foot window, which seems 2 thumbs high, makes your thumb look
3 feet wide (5 feet / 2 thumbs), and thus something
like 90 feet (3 x 30) away.
Thumbs and Fists and ROUGH Distance
In general, people's thumbs and fists are about:
Or you can just remember that your thumb is x30, and each time you
halve your thumb, its twice as big.
|size ||distance is ||(angle)|
|fist ||x 6 ||10°|
|thumb ||x 30 ||2°|
|½ thumb||x 60 ||1°|
|¼ thumb||x 120||½°|
- Cars are about 4 meters (12 feet) long.
A parked car across the street which is about 3 thumb-widths long
is... 40 meters (120 feet) long. (12 / 3 x 30)
- Cars are about 1.5 meters (5 feet) high.
The same parked car is about 1 thumb high, so...
45 meters (150 feet). (1.5 m (or 5 ft) x 30)
Notice this is only roughly the same as the previous estimate.
How roughly is discussed in the next section.
- Cars are about 2 meters (6 feet) wide.
So the car, ahead of you on the road, which is 1/2 thumb wide,
is roughly 120 meters (360 feet) away.
- People are about 1-2 meters (3-6 feet) tall (kids/adults).
- Office building floors are about 4 meters (10 feet) high.
So if a 30 story building is 6° (two fingers) high, then
its about 30 x 4 (or 10) x 20 = 2400 meters (6000 feet)...
so something like a mile away.
- Airline jets are about 40 meters (150 feet) long.
So a jet flying by and 1/4 thumb wide is about 150 x 120 = 18000 feet away (3+ miles).
If its say 3 fists up (30°) [described below], then its
1/2 x 18000 = at about 9000 feet altitude, and about the
the same 3'ish miles away on the ground.
- How far away is that cloud?
Clouds can come at different altitudes, so you need to find out
the current cloud ceiling (how high the bottom of
the clouds are). If the clouds are at 1500 feet, and the cloud you
are interested in is say one fist up from the ground, then its
approx 10 x 1500 = 15000 feet (3ish miles) away.
How to find out cloud ceilings?
Well, if anyone knows an easy to use web resource, I'd like to know it.
There are a couple of interfaces to the
but not all of them decode the ceiling information,
and they seem a bit experimental/unstable (as of 1997.Mar.25).
I expect things will improve.
for instance under
at the bottom (most recent) say something like
KBOS 251756Z 10014KT 10SM FEW110 BKN250 04/M05 A3040 RMK AO2 SLP292 ACSL SE BKN LYR MSTLY THN T00441050 10050 20006
BroKeN clouds at 25,0oo feet.
UIUC gopher[link broken]
(gopher?! Yes. At the moment (1997.Mar.25), its the best way I've found to get ceiling info.)
Surface Reports For Stations In Massachusetts,US
station ceiling sky tmp dwpt wind alt. vis weather
location (ft) cover kts in. mi
BOS BOSTON/LOGA 25000 broken 40 23 E 14 30.40 10
but says "There are a number of problems here", and that they have stopped
work on their gopher stuff.
The University of Wyoming[link broken],
for instance has a line in the MAssachusets report
ID TIME T TD RH DIR SPD GST ALT SLP VIS CIL COV
KBOS 1556 39 19 44 100 10 048 319 10 250 BKN
where CeILing is 250oo feet.
- You can call a recorded weather for pilots number. In the
back of the phonebook White Pages, there are government blue pages,
where it should be under United States Government, T, Transportation
Department, Federal Aviation Admin, Pilot Automatic Terminal
How "roughly, approximately, something like, about" is
You will often be within a factor of 2 or 3, and by using
greater care, can get it under 20%.
Sources of error:
- Measuring the angle. Error in the angle measured
shows up linearly as error in the distance estimate. So if your angle
measurement is off by 30%, so will your distance estimate be. And
similarly, if you mismeasure something as 1 thumb rather than 2, or
as 1/2 rather than as 1/4, you'll add a factor of 2 error. The
smaller the angle, the harder it is to get close. Sources of error
- Did a quick and dirty measurement. To improve, take
some time and do a more careful one.
- Angle too small to be convenient. Like 1/10 of a thumb. To
improve, find something else to measure with. Like the thickness of a
key, or a pencil lead, or ... See calibration below.
- Calibration error. If your thumb is really 10% narrower than
our imaginary 2° thumb, then your distance estimates with be 10%
high. To improve, fudge in a correction (ie, nudge your distance
estimates up or down to compensate), or calibrate part of your thumb
more precisely. See calibration below.
- Holding your arm out. Small changes in the distance from
thumb to eye don't cause much error.
- Drifting calibration. Parts of your thumb are bigger
than others. Once you start trying for close calibration, you need to
choose one part of your thumb and stick with it.
- Estimating the object size. Error in object size
shows up linearly as error in the distance estimate. So if your
object size estimate is 20% high, your distance estimate will be too.
If you use an object a lot, but dont have a good estimate, you might
want to improve it. (Like finding out how tall a skyscraper really
is, rather than estimating from the number of floors.)
- Doing the math. You can simplify your math by
being approximate (5 / 3 = 2 , rather than 2.5). This can add error.
So if you want a closer estimate, you can avoid such simplifications.
But, there is no point in trying to make your math more precise than
your measurement and estimates. For instance, the 5/3=2 adds a 20%
error, but if your object size estimate is only good to within a
factor of 3 (ie, it could be 3 times bigger, or 1/3 smaller), the 20%
is not important.
- The angle multiplyer.I've said a 2° degree
thumb gives you a x30 distance. Its really more like 28.7
(1/sin(2°)). The error is consistently about 5% high, which
seems well worth the simplification.
The idea is to see how big various things are
(fingers, different parts of fingers, combinations of fingers, fist,
and perhaps some things you generally have with you, like keys). Then
you can find some specific part of your body (like the middle of your
thumb nail, or the end of one of your fingers), which is just under 2
degrees. This can then be your angle measurer.
For instance, I personally use the outermost joint of my pointer
finger. Not my thumb, which is slightly too big.
There are lots of ways to set up for calibration.
The easiest is to just put things against a rule that has already
been calibrated, like this chart
or these cards.
Another approach is to draw some degree lines on a wall, and
then step back the right distance.
The key is that one stand back ~57 times the degree-line separation.
Here are some example(s):
- Draw lines on pieces of paper, dividing them into 4 parts ( ie,
a line in the middle, and 2 in the middle of them, and maybe one on
the very edge, just so you can see the edges when the papers are put
together). Tape the papers to the wall in a tower, and step back 14
papers-worth of distance. (14 is 1/sin(4°). We
could have done 6° per paper, but at only 10 papers distance,
shifting your body confuses things.)
Things up in the air
When you have measured the distance to something up in the air, like
an airplane, it can be nice to separate its altitude from its ground
Here is one simplification, admittedly not very simple.
should work out one with just 3 or 4 numbers to remember, as this
one's 5% error is excessively precise for most stuff.
which hovers around 5% error.
|0° (its on the ground)||0||1|
|10° (1 fist up)||1/6||1|
|20° (2 fists up)||1/3||1|
|30° (3 fists up)||1/2||5/6|
|40° (4 fists up)||2/3||2/3|
|50° (5 fists up)||2/3||2/3|
|60° (3 fists down)||5/6||1/2|
|70° (2 fists down)||1||1/3|
|80° (1 fist down)||1||1/6|
[** ... **]
[** ... **]
Blinking your eyes instead
A reader writes
When I was about 13 years old (1947) I remember reading somewhere
(maybe a Boy Scout handbook) that you could estimate distance by
holding your thumb out in front of you and sight an object with one
eye closed, then without moving open that eye and sight the object
with the other eye closed; then estimate how far your thumb
appeared to move and multiply that number by eight.
Which seems about right. Instead of defining an angle using one eye
and two thumb edges, it uses two eyes and one thumb edge. The angle
is a bit wider. The distance between one's eyes is about 4 thumbs
wide, so 8x sounds reasonable (30x/4).
I wonder if people vary less in their eye "widths", then in
their thumb widths. If so, then this approach would have less need
for personal calibration.
At Arm's Length a NOVA Teacher's Guide.
Up, Up, and How Far Away?
I welcome questions, comments, and thoughts. -
This page hasn't drawn much comment over the years,
and thus hasn't been pursued.
Try describing from an similar triangles / LvH ratio perspecive.
2003-Jul-16 Added a reader's note on blinking. Thanks!
2003-Feb-04 Fixed 1 link, flagged 4. Moved WeatherNet ahead of broken services.
2001-Aug-14 Ripped out `hold rule at arms length' approach to here.
2001-Aug-09 Added `hold rule approach' (thanks to a reader's question
about measuring distance for archery).
1998.Dec.01 Added links to NOVA Balloon science pages. (Thanks go to a reader.)
1997.May.05 Added link to A View from the Back of the Envelope.
1997.Mar.25 Draft fleshed out.
I will work it further if people are interested.
1997.Mar.24 Draft begun.