Area of a Sector
•
Section 4.3
95
4.3 Area of a Sector
r
θ
Figure 4.3.1
In geometry you learned that the area of a circle of radius
r
is
π
r
2
. We
will now learn how to find the area of a
sector
of a circle.
A sector is
the region bounded by a central angle and its intercepted arc, such as the
shaded region in Figure 4.3.1.
Let
θ
be a central angle in a circle of radius
r
and let
A
be the area of its
sector. Similar to arc length, the ratio of
A
to the area of the entire circle
is the same as the ratio of
θ
to one revolution. In other words, again using
radian measure,
area of sector
area of entire circle
=
sector angle
one revolution
⇒
A
π
r
2
=
θ
2
π
.
Solving for
A
in the above equation, we get the following formula:
In a circle of radius
r
, the area
A
of the sector inside a central angle
θ
is
A
=
1
2
r
2
θ
,
(4.5)
where
θ
is measured in radians.
Example4.8
Find the area of a sector whose angle is
π
5
rad in a circle of radius 4 cm.
Solution:
Using
θ
=
π
5
and
r
=
4 in formula (4.5), the area
A
of the sector is
A
=
1
2
r
2
θ
=
1
2
(4)
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 Fall '11
 Dr.Cheun
 Calculus, Geometry, Arc, Radian, Grad

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