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


sectorbanner

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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definition
Segment of a Circle: The segment of a circle is the region bounded by a chord and the arc subtended by the chord.           

While a sector looks like a "pie" slice, a segment looks like the "pie" slice with the triangular portion cut off (it's somewhat like the "crust section of the pizza slice"). The segment is the small partially curved figure left when the triangular portion of the sector is removed.
segment

segmentbanner

You can probably "guess" what the formula for finding the area of a segment involves.
A segment is a sector with the triangular portion removed.
Formula: Area of Segment = Area of Sector - Area of Triangle

ex

Find the area of a segment of a circle with a central angle of 120 degrees and a radius of 8 cm. Express answer to the nearest integer.

segmentA

Solution:
Start by finding the area of the sector.
segmentAf

Now, find the area of the triangle. Dropping the altitude from the center forms a 30-60-90 degree triangle. Using the 30-60-90 rules (or trigonometry), find the altitude, which is 4, and the other leg, which is 4rad3 or 6.92820323.

segmentA3

segment8

segmentA2
AAT



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