PreCalc H ­ Aim #37 ­ arc length area of sector.notebook
Unit 6
Introduction to Trigonometry
Aim #37: How do we determine the arc length and the area
of a sector of a circle?
Do Now:
Draw a central angle of 225
o
o
Convert 225 to radians.
Give one positive and one negative
coterminal angle.
Buzz words:
angle - rotation of a ray from its initial side to its terminal side
central angle - an angle whose vertex is at the center of a circle
positive angle - counterclockwise rotation
negative angle - clockwise rotation
coterminal - having the same terminal side
1
PreCalc H ­ Aim #37 ­ arc length area of sector.notebook
What is a Radian?
An angle in standard position has a measure of 1radian if the
length of the intercepted arc is equal to the radius of the
circle.
How many radians in θ?
θ
2
6
3
2
θ
θ
3
3
More Buzz Words:
radian - the measure of an angle that intercepts an arc
on the unit circle whose length is r
θ - theta - a Greek letter commonly used to represent
an angle
2
PreCalc H ­ Aim #37 ­ arc length area of sector.notebook
The Unit Circle in Degrees & Radians
Arc Length
If s = arc length, then
therefore
s = rθ
3
PreCalc H ­ Aim #37 ­ arc length area of sector.notebook
eg. Through how many degrees does the minute hand of a clock turn
in 45 minutes?
Through how many radians?
If the minutes hand is 5 cm long, how far does its end travel in 45
minutes?
eg. Glasgow, Montana, is due north of Albuquerque, New Mexico.
o
Find the distance between Glasgow (48 9' north latitude) and
o
Albuquerque (35 5' north latitude), assuming that the radius of
the Earth is 3960 miles.
903 miles
4
PreCalc H ­ Aim #37 ­ arc length area of sector.notebook
Area of a Sector
If A = area of a sector, then
therefore
eg. Given circle O with radius OA = 8 cm and the measure of
angle AOB =
, find:
a) The exact length of arc AB
b) The exact area of the shaded region
5
PreCalc H ­ Aim #37 ­ arc length area of sector.notebook
eg. Given circle O with radius OB = 20 cm and the
o
measure of angle COB = 38 , find:
a) The exact length of arc BC
b) The exact area of sector COB
Wrap it Up!
Find the area of the dark blue sector shown below. The radius
of the circle is 4 units and the length of the arc measures 7.85
units. Express your answer to the nearest tenth of a square
unit.
6