Measurement in the Round

“Measurement in the Round”

Adapted from “Navigating Through Mathematical Connections”, NCTM Navigations Series and “Core-Plus Mathematics: Contemporary Mathematics in Context, Course 3”

Radian Measure and the Unit Circle

Many real-life situations occur in cycles or are repetitive in motion, such as the ocean’s tide, the wheels of a moving car, or the seats on a Ferris wheel. These periodic or repetitive situations can be mathematically modeled using trigonometric functions, which is why the unit circle is the foundation of this investigation.

I. Circular Motion and Degree Measure

Consider the angles that can be modeled when two radii of a circle are rotated as shown in the diagrams below. Notice one radius is fixed, called the initial side, and the rotating side

(the terminal side) rotates forming angles. As the terminal side rotates counterclockwise around the center of the circle, the angles formed have measures from 0 to 360 degrees. The degree is the unit of angle measure used in some practical applications, such as surveying, navigation, and industrial design.

1. Imagine the rotating side moves through the angles indicated below. Describe the location of the terminal side at the end of each rotation.

a. 1 revolution: ______

b. : ______

c. of a revolution: ______

d. : ______

e. revolutions: ______

f. : ______

2. Think about the relationship between circular revolutions and degrees.

a. How many degrees are swept through when the rotating side rotates:

• of a revolution? ______

• revolutions? ______

• n revolutions? ______

b. How many revolutions are made when the angle rotates:

• 60° ______

• 480° ______

• n° ______

c. Complete the following:

• 1 revolution = ______degrees

• 1 degree = ______revolution

II. Radius, Circumference, and the Real Number Line

Working in groups of no more than four, use the circular object and paper strip provided to make a tape for measuring. The unit used to measure will be 1 unit, and this measure is equivalent to the radius of your circular figure.

1. Place your circular object in the center of a piece of paper, and trace its circumference on the paper.

2. Find the center of the circle.

· Describe the method you used.

· Explain why your method works.

3. With a straightedge, draw a radius of the circle. Place one end of your tape on the center of the circle, and put a mark where the tape reaches the end of the radius. Place a 1 at that mark, since it measures one radius.

4. Use your radius again and mark a 2 on your tape. Continue marking unit lengths along the tape until no room is left for another unit.

5. Does your tape look like a number line? Since you will use it as a measuring tape, mark more points and label them.

· Divide the units on your tape into halves and label.

· Divide the units on your tape into fourths and label.

6. Describe how you marked and labeled these points accurately without using a ruler?

7. Use your tape to locate the point that represents the distance around your circular object. When measuring circular distance, what are we really measuring?

· Mark the point on your tape that corresponds to the distance around your circular object. What number or value would you use to label this point on your tape?

· Compare the length you found to that of other groups. Describe any patterns observed and explain why they occur.

8. Locate the point on your tape that represents half the circumference of your circular object.

a. How did you find this value?

b. What is the unit measure of this point? ______

c. How would this value be represented another way?______

9. If the value from 8c represents half the circumference of the circle, how would you represent the circumference in terms of ? Explain your reasoning.

10. Is the value of the point on your number line that represents the circumference of your circle equivalent to the actual circumference for your circle? Explain your reasoning.

III. Radian Measure and the Unit Circle

Revolutions and degrees are two units by which you can measure angles. In scientific work and in advanced work in mathematics, it is usually more convenient to use radian measure. In this next activity you will investigate the radian measure of angles and how they relate to the unit circle.

1. Using your circle where the radius is 1 unit in measure (the unit circle):

· Draw a pair of perpendicular lines, one horizontal and one vertical that intersect at the center of the circle.

· Label the lines x and y to indicate and x-y coordinate system.

· Mark the intersection of the circle with the positive x-axis as the point A, this will represent the initial side of the angle of rotation.

A directed angle of a unit circle lies on the coordinate plane in standard position with its vertex at

O (0, 0) and its initial side on the positive x-axis where A (1, 0). The terminal side rotates about the origin, B (x, y). The distance the terminal side travels around the circle creates an arc, whose measure has a special relationship with the measure of the radius and the angle formed when the terminal side is rotated.

Radius = r units

Length of arc AB = r units = radius

Measure of angle AOB = 1 radian

A radian is the measure of an angle determined by joining the center of a circle to the endpoints of an arc equal in length to the radius of the circle.

2. Suppose a circle with center O has radius r. What will the radian measure of the angle of rotation if the arc is r unit long?

3. If an arc measures 5 times that of the radius, what would be the radian measure of the angle of rotation?

4. What is the radian measure of the angle of rotation whose arc measure is 32 cm in a circle with a radius that measures 4 cm?

5. Describe the relationship that exists in every circle between the measure of the angle of rotation, the length of the corresponding arc, and the length of the radius.

6. Find is the radian measure of an angle that intercepts an arc of length 22 inches, with a radius of 4 inches?

7. Using your unit circle, find and label, in terms of , the radian measure of the directed angle whose arc measure corresponds to:

• of a revolution ______

• 1 revolution ______

8. Consider dividing radians into smaller angle measures, while also considering the fact that a circle is made up of two angles measuring radians, which is equivalent to 2 radians.

Without using a ruler, identify and label the radian measures of each angle, in terms of , when each radian is divided as indicated below. Continue around the circle also labeling angles whose measures are greater than radians.

· Halves

· Fourths

· Thirds

· Sixths

9. Complete this table of degree/radian measure equivalents. After completing the table, complete the diagram of the unit circle with the angle measures in both degrees and radians.

Degree/Radian Equivalents
Degrees / 0° / 30° / 45° / 90° / 135° / 150°
Radians / / /
Degrees / 210° / 240° / 270° / 300° / 315° / 360°
Radians / /

10. Write a paragraph describing the similarities and differences of degree measure and radian measure.