Geometry Honors Section 5.3 Circumference and Area of Circles
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Transcript of Geometry Honors Section 5.3 Circumference and Area of Circles
Geometry Honors Section 5.3
Circumference and Area of Circles
While the distance around the outside of a polygon is known as
the ________, the distance around the outside of a circle is called the
____________.
perimeter
circumference
For any circle, the ratio of the circumference to the diameter, , is
the same. This ratio is approximately equal to
___________. We use the Greek letter ____ to represent this
irrational number. A fractional approximation is _____.
Cd
3.141592656
227
Once again, = , so
C = ____
or in terms of the radius
C = ______
Cd
d
2 r
Activity 2 on page 316 explains how the formula for the area of a
circle is derived.
A = ______ 2r
Example: Find the circumference and area of a circle with a diameter of 12. Give an exact answer and an answer rounded to the nearest 1000th.
2 6 12 37.699C 2 26 36 113.097A r
6r
Example: Find the area of a circle with a circumference of .18
rrC
218
2
9r
8192
2
A
rA
Example: Find the area of the shaded region. Give an exact answer and an answer rounded to the nearest 1000th.
circlesquare AA
22 rs 22 510
25100
460.21
A sector of a circle is the region bounded by two radii and the arc
joining there outer endpoints.
Example: Find the area of sector AQB.
21236060
24
As you can see from the previous example, the
area of a sector =
OR
2
360M r
2 360Area of Sector M
r
Example: A circle has a diameter of 30 feet. If the area of a sector in this circle has a measure of ft2, find the measure of arc determining this sector.
2
15r
2 360Area of Sector M
r
360152
2
M
3602252 M
02.3
720225
M
M
A similar formula can be used to find the length of an arc. Length of an arc =
OR
2360M r
2 360length of arc M
r
Example: A circle has a radius of 6 cm. If an arc has a measure of 800, find the length of the arc.
2 360length of arc M
r
36080
12
L
6.2
960360
L
L
Example: An arc has a measure of 300 and a length of inches. What is the radius of the circle in which this arc is found?
18
2 360length of arc M
r
36030
218
r
108648060
rr
Note: The “measure of an arc” and the “length of an arc” are not the same thing. The measure of an arc is given in _______ and refers to ___________________ The length of an arc is given in __________ and refers to _______________________
degreesa fraction of the circle.
in / cm / ftthe distance along the arc.
While degree is certainly the unit of angle measure that you are
most familiar with, another commonly used unit for measuring
angles is the radian.
A radian is a unit of angle measure
radius. thelength toin equal is arc dintercepte
whosecircle a ofcenter at the anglean toequal
radian 1
Since the circumference of any circle is equal to ______, then there must be _____ arcs of length ___ on any circle.
Thus, the radian measure of a full circle is _____. We also know the degree measure
of a full circle is _______. Therefore, _______________________
or
2 r
2
r2
360360 radians 2
180 radians
Example: Convert 72o to radians
72180x
72180 x
18072
x
52
x
Example: Convert radians to degrees
x8
180
x
8
180
x 5.22 x5.22