MEASURING PI WITH STRING
An easy alternative to find the value of pi is to use something round (like a
soup can). Carefully measure the diameter of the can by stretching a string
across the top (Figure 7-8). To do this, hold one side of the string tight on one
side while holding the other side more loosely. Swing the looser side across
the can, letting the string slip through your fingers as you go around the cir-
FIGURE 78: Finding the maximum distance across the can (the diameter)
FIGURE 79: Holding the maximum value to measure it
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cumference and your hands get further apart. The
farthest apart your two hands get is the diameter.
Once you pass that point, the string will no longer
be pulled farther through your fingers (Figure 7-9),
and the length between your fingers will be the
diameter. Measure it and write it down.
Then wrap a piece of string around the can.
Measure the length of the string and divide the
value you get by the diameter you measured. You
should get close to pi, since the circumference
around the can equals pi times the diameter.
LATITUDE AND
LONGITUDE
We’ve now learned enough math to figure out where
we are on the earth’s surface. In the next few sections of this chapter, we are
going to build your ability to use observations to be able to estimate where
you are on earth based on the positions of the sun and other objects in the
sky, plus a few common tools like a protractor and ruler.
About 2100 years ago, the Greek astronomer Hipparchus is credited with
figuring out the basics of measuring what we now call latitude and longitude.
He is also credited with early work on what became trigonometry. The 1989
European Space Agency’s Hipparcos astrometry mission was named in his
honor (astrometry being the precise measurement of the locations of the
stars in the sky) as well as craters on the moon and Mars.
On a globe, longitude lines (called meridians) run from pole to pole, marking
distance east and west. These are the green lines on Figure 7-10. You could
start counting anywhere since the earth is round, but by convention a line
running north and south through Greenwich, England, is considered to be
the zero meridian. Distances to the west of Greenwich are treated as neg-
ative numbers, or “Longitude West”. Los Angeles, California is about 118°
west, or -118°. 118° east runs through the middle of Nanjing, China.
The red lines (circles) on Figure 7-10 are latitude, marking distance north
or south of the equator (the heavy red line) which is 0°. The poles are plus
(north) and minus (south) 90° latitude. Typically north is at the top of this
imaginary system, and south at the bottom.
FIGURE 710: Latitude and longitude lines.
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Let’s walk through figuring out your latitude and longitude just from simple
observations of the sun at local noon. We will take it one step at a time,
using the information about sine and cosine we just learned in Chapter 6. Of
course, you can look up your address in an online map program, but what’s
the fun of that? You can check your answer that way, though.
FINDING YOUR LATITUDE
The two numbers you need to figure out your latitude are the solar decli-
nation and the elevation angle. Elevation angle is the angle of the sun above
the horizon at its highest point for the day. The solar declination is the angle
between a line connecting the centers of the earth and sun, and the earth’s
equator, and it adds or subtracts as much as 23.44° to the elevation angle
over the course of the year. (We discuss how to compute it for a particular
day of the year in a later section in this chapter.) Both of these vary over the
year (and longer). Let’s see how to get those numbers in a sunny south-fac-
ing window, or your backyard.
Your latitude measures how far north or south you are from the equator. It is
measured as an angle between two imaginary lines: one that goes through
the center of the earth at the equator, and one from the center of the earth to
your location (marked by the red dot in Figure 7-11).
We talk about being located at so many
degrees north or south latitude. Los
Angeles, for example, is at about 34°
north. Latitudes south of the equator
are negative numbers; the South Pole is
at 90° south, or -90°. Most online maps
have a way to show latitude and longi-
tude. Google Maps will show latitude and
longitude when you click on a particular
spot.
As it turns out, the earth’s north pole
doesn’t poke straight up at right angles
to the earth’s orbit. Instead, it is tilted at
about 23.44° to that right angle. If you
imagine the solar system with the sun in
the middle, the earth’s axis always points
off to one point in space all year long.
FIGURE 711: Latitude of the point denoted
by the red spot (north pole shown pointing
directly upwards).
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Figure 7-12 shows how that works,
noting the seasons for the northern
hemisphere.
We will talk through this in diagrams
here, but you might want to make
yourself a ball out of something like
PlayDoh or modeling clay and stick a
toothpick through it (Figure 7-13). The
toothpick is the earth’s axis. Draw a
line around the “equator” of your mod-
eling clay earth. This line should be around the center, and shouldn’t appear
to wobble when you twist the toothpick between your fingers. Then, make a
little mark on the earth to show where you are, or use a straight sewing pin
poked in where you are down toward the center of the earth. The angle of the
pin can help you think about your latitude.
Put a lamp on a table in the middle of a room so you can walk around it.
Then go ahead and circle it once (you’re the earth orbiting the sun). Keep the
toothpick pointed to one corner of the room. Think about what happens at the
equator, and where you are, at local noon on the first days of fall, winter, and
spring.
FIGURE 712: How the earth’s axis is tilted relative to its orbit.
FIGURE 713: A modeling clay earth with axis
and northern hemisphere latitude noted.
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Let’s go into a bit more detail for the northern hemisphere example. On
the shortest day of the year (the first day of winter, which is the winter sol-
stice) the sun will appear to be 90° minus your latitude minus 23.44° off the
horizon. In Los Angeles, that would be 90° - 34.1° - 23.44° = 32.44°. So on
December 21st the sun’s highest point is only about a third of the way up
the sky, even in sunny Los Angeles. The winter solstice is often used as the
first “day of the year” when we calculate sun elevation angles, since it is the
lowest “high” point of the sun (Figure 7-14).
We call the longest day of the year the first day of summer, or the summer
solstice. If you look to the south at noon on that day, the sun will appear to
be 90° overhead, minus the difference between your latitude and 23.44°, as
we can see in Figure 7-15. In Los Angeles, this would be 90° - (34.1° - 23.44°)
= 79.34°, as shown in Figure 7-15. This usually happens on June 21, about
182 days after the solstice (183 in leap years, but it’s really about 182 plus a
quarter day).
On the first day of spring or fall, the earth’s tilt is not toward or away from the
sun, and the highest point of the sun in the sky is just 90° minus your latitude.
FIGURE 714: The sun and the earth alignment on the winter solstice (day 1 of the solar year).
FIGURE 715: The sun and the earth alignment on the summer solstice (day 183 of the solar
year).
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