Do Now: 1) Find the area of the shaded region in terms of Π :

9

7.5

2) The radius of the following circle is 36 cm, and the m≮ABC = 60˚. What is the arc length of AC?

______ < arc length < ______

The possible range of lengths of any arc length is:

Aim #14: What is a radian and how do we measure arc length using radians? CC Geometry H

An arc length must be a fraction of the circle's circumference.

An angle can be measured in either degrees or radians.

1 degree = 1/360 of a complete rotation.

In general, the radian measure of the central angle θ = ____________________

θ = or S = r

length of intercepted arcradius

S

What is 180o in radians? A straight angle, with vertex at the center, has a semi-circle for its intercepted arc. The length of a semi-circle is half the circle, or S = ½(2πr) = πr.

= rTherefore, the radian measure of a straight angle (180o) is π.π radians and 180o are measures of the same angle.

180o = π radians

πr

r

πr

π

1o = _____ radians1 radian = _____o

360o =

O

To convert degrees to radians:

degrees π 180

To convert radians to degrees:

radians 180 π

ex: Convert 90° to radians. ex: Convert π to degrees. 3

1) What is the radian measure of central angle AOC?

2) Express an angle of 135° in radians. 3) Express an angle of in degrees. π6

4) Express an angle of 300° in radians. 5) Express an angle of in degrees. 2π 9

6) Circle O has a diameter of 12. Central angle θ intercepts an arc with a length of 4π. What is the measure of angle θ: a) In degrees? b) In radians?

7) The area of sector AOB in the following image is 28π. a) Find the measurement of the central angle labeled x˚.

b) Express the central angle in radians.

72

8

8) a) Find the length of the radius. b) Find the central angle in radians and degrees.

9) In a circle whose radius measures 5 feet, a central angle intercepts an arc of length 12 feet. Find the radian measure of the central angle.

10) The central angle of circle O has a measure of 4.2 radians and it intercepts an arc whose length is 6.3 meters. What is the length, in meters, of the radius of the circle?

11) Find the radius of circle A, as well as x, y and z. (leave angle measures in radians and arc length in terms of π). Note that C and D do not lie on a diameter.

HW#14Name: _____________________

1. Express an angle of 80° in radians. 2. Express an angle of in degrees. 2π 3

3. Circle O has a diameter of 20. Central angle θ intercepts an arc with a length of 2π. What is the measure of angle θ: a) In radians? b) In degrees?

4. The radius of a circle O is 8 inches. What is the length of the arc intercepted by a central angle of the circle if the measure of the angle is 2.5 radians?

5. a. Given the concentric circles with center A and with m A = 60˚, calculate the arc length intercepted by A on each circle. The inner circle has a radius of 10, and each circle has a radius 10 units greater than the previous circle.

b. An arc, again of degree measure 60˚, has an arc length of 5π cm. What is the radius of the circle on which the arc sits?

CC Geometry HDate: ____________

6. The concentric circles all have center A. The measure of the central angle is 45°. The arc lengths are given. Find the radius of the inner, middle, and outer circle. Then determine the ratio of the arc length to the radius of each circle, and interpret its meaning.

D

F

EC

Review:1) In the figure, the radii of two concentric circles are 24 cm and 12 cm, mDFE = 120o. If chord DE of the larger circle intersects the smaller circle at C, find the area of the shaded region in terms of π.

132o

132o

2) Find the area of the shaded region, to the nearest hundredth, if the diameter is 32 inches.

3) Given circle A, find the following (round to the nearest hundredth, if necessary).

a. m≮CAD in degrees.

b. Area of sector CAD