Pi Whats?
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Lab 41: π Whats? Discussion: What is π? Is it possible to cut a piece of string with exactly π length? Materials:
Circular object Metric ruler Marker String Strip of Paper Tape Scissors
1. Measure the diameter of your circular object. Calculate the radius to the nearest 10th of a centimeter and record: r = _______ cm 2. Cut a strip of paper at least 8 times longer than your radius, and tape it to a flat surface. 3. With a marker, make a mark on the edge of your circular object. Make a corresponding mark near the end of your strip of paper. Line up the two marks, then roll your circular object one complete revolution. Make a second mark on the paper at exactly one complete revolution. What does the distance between the two marks on the paper represent? Verify this value by using the radius to calculate it.
4. Cut a length of string the same length as the distance between the marks. Set it aside. 5. Using one of the marks on your strip of paper as “zero”, use your ruler and a pencil to mark off intervals along the paper of the same length as your radius. About how many radii before you get to the onerevolution mark? This value should be the same for all of the various circles. Why? If you need a hint, divide it by 2.
6. Take your length of string, fold it in half, and cut it exactly in half. This string is precisely πlong. You and your partner can walk around all day with πin your pocket! The question is….
π whats? (spoiler alert: please don’t turn over the paper until you have figured this out.)
7. You hopefully determined that the string is π radii of your circle. For convenience sake, we frequently use the number of radii along the circle to refer to measures of central angles of the circle. Except we call the measure “radians”. = 1 radius of the circle
∠ measures 1 radian
8. Armed with this information and thinking about the lab you just completed… In terms of π, how many radians in an angle of… 180° 90° 270°
write a general formula
360°
for converting x° to
30°
radians:
60° 210° 18° 20°
How many degrees in an angle of… Write a general formula 2π radians π
−
π 4
for converting θ radians to °:
3π 4 4π 3 1 radian
10π 9
Circular object Metric ruler Marker String Strip of Paper Tape Scissors
1. Measure the diameter of your circular object. Calculate the radius to the nearest 10th of a centimeter and record: r = _______ cm 2. Cut a strip of paper at least 8 times longer than your radius, and tape it to a flat surface. 3. With a marker, make a mark on the edge of your circular object. Make a corresponding mark near the end of your strip of paper. Line up the two marks, then roll your circular object one complete revolution. Make a second mark on the paper at exactly one complete revolution. What does the distance between the two marks on the paper represent? Verify this value by using the radius to calculate it.
4. Cut a length of string the same length as the distance between the marks. Set it aside. 5. Using one of the marks on your strip of paper as “zero”, use your ruler and a pencil to mark off intervals along the paper of the same length as your radius. About how many radii before you get to the onerevolution mark? This value should be the same for all of the various circles. Why? If you need a hint, divide it by 2.
6. Take your length of string, fold it in half, and cut it exactly in half. This string is precisely πlong. You and your partner can walk around all day with πin your pocket! The question is….
π whats? (spoiler alert: please don’t turn over the paper until you have figured this out.)
7. You hopefully determined that the string is π radii of your circle. For convenience sake, we frequently use the number of radii along the circle to refer to measures of central angles of the circle. Except we call the measure “radians”. = 1 radius of the circle
∠ measures 1 radian
8. Armed with this information and thinking about the lab you just completed… In terms of π, how many radians in an angle of… 180° 90° 270°
write a general formula
360°
for converting x° to
30°
radians:
60° 210° 18° 20°
How many degrees in an angle of… Write a general formula 2π radians π
−
π 4
for converting θ radians to °:
3π 4 4π 3 1 radian
10π 9
