definition
A sector of a circle is the portion of a circle enclosed by two radii and an arc. It resembles a "pizza" slice.


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Common sectors in circles:    ("A" represents AREA of sector):

Full
Circle
areacircle1
(full circle)
areac1
central ∠ = 360º
fractional part:
360/360 = 1

Semi-circle Sector
areacircle2
(half circle = half of area) areac3
central ∠ = 180º
fractional part:
180/360 = 1/2
Quarter-circle Sector
areacircle3
(¼ of circle = ¼ of area) areac2
central ∠ = 90º
fractional part:
90/360 = 1/4
Any Sector
(nº central angle)
areacircle4
(fractional part of circle)
areac4
n = number of degrees in central angle of sector.
areasectorarclength
where s is the arc length of the sector.

The area of a sector is found using proportionality (establishing a "fractional part").

Method 1: Calculating the Area of a Sector when the Central Angle is known:

When finding the area of a sector (using the central angle and radius), you are actually finding
a
"fractional part" of the area of the entire circle.

The "fractional part" is the ratio of the central angle of the sector
to the
"entire circle of 360 degrees".  

[fractional part = formula; where n = central angle]




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Method 2: Calculating the Area of a Sector when the Arc Length and Radius are known:

When finding the area of a sector (using the arc length and radius),
the
"fractional part" becomes the ratio of the arc length (s) to the entire circumference.   

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Exs

Method 1:

1.
Find the area of a sector with a central angle of 40º and a radius of 12 cm. Express answer to the nearest tenth.
Solution:
area8
Method 2:

2.
Find the area of a sector with an arc length of 30 cm and a radius of 10 cm.

Solution:
area7


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derivebanner

We have seen that the area of a sector is a fractional part of the area of the entire circle. The area of a sector can be expressed using its central angle or its arc length.

The following proportions are true regarding the sector:
sectorcircle2

sectorcircle

We are going to derive the formula for the area of a sector using the sector's arc length.

Using the first and last ratios in the proportion shown above, the following is true.
segmentcircle3

If A = area of the sector, and s = arc length, then segmentcircle4.

Solving the proportion for A gives:
segmentcircle7


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