4.
Take a long piece of light-colored string and any circular
object. Your object should be at least a few centimeters
across. A diameter of a circle is any line segment stretched
all the way across the circle that passes through the center
of the circle. Use the string to carefully measure out and
cut a diameter of your circle. Fold the diameter in half.
With a dry erase or permanent marker, use the folded
diameter to mark out eight or so radiuses (not diameters)
along your remaining string. (Some people say “radii”
instead of “radiuses.” Either one is fine.) Measure the
number of radiuses that fit around your circle by wrapping
the string around it and counting. Compare your number
of radiuses with the results of your classmates.
If you were very careful in the previous problem, you
probably got something like 6 radiuses, plus a little bit.
If you were to repeat this activity, but had a very smooth
circle, a very accurate ruler, a string that doesn’t stretch,
and a very large supply of patience and luck, you might get
a value somewhere close to 6.28 radiuses. If you tried this
with a bigger circle or a smaller circle, it wouldn’t make a
difference. You’d always get the same number: about 6.28.
For historical reasons, people usually talk about half of
this number. Do you recognize it?

Journeys in Film
: Hidden Figures
8.
Out of all the planets that orbit the sun, Venus has an orbit
that is closest to a perfect circle. On average, Venus is
0.723 AU from the sun. Venus takes about 225 Earth days
to orbit the sun. What is the average speed of Venus in its
orbit, in AU per day? In meters per second?
9.
Draw a diagram that shows how the inside of a circle can
be approximated by a large number of triangles, each with
one corner at the center of the circle. If you increase the
number of triangles, what happens to the area of each
triangle? Does the approximation get better, worse, or stay
the same? Why?
10.
One way to calculate the area of a circle is to use the
formula
. Explain why this formula works.
11.
Another way to calculate the area of a circle is to use the
formula
. Explain why this formula works.
12.
What is the area enclosed by the orbit of Venus, in square
AU? In square meters? (This may seem like a silly thing
to calculate, but, as you’ll find out soon, Johannes Kepler
figured out that it’s not actually all that silly.)
Handout 2
•
p . 2
Lesson
5
(MATHEMATICS)
Problems on Conic Sections
5.
Half the number of radiuses around a circle is called by the
Greek letter
, which has the symbol
. It is pronounced
“pie” by English speakers, as in pecan pie or apple pie,
and is spelled “pi.” Using methods that definitely do not
involve string or scissors, many people have calculated
to much higher accuracy than 3.14. Look up some of
the methods people have used over the years to calculate
approximations of
. (Just for fun, this site has a million
digits of
:
. The world
record for memorizing digits of
is over 70,000 digits
which took over 17 hours to recite.)
6.

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- Winter '17
- Mrs. Manternach