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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