definition
Sector of a circle: 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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Circle
areacircle1
(full circle)
areac1
central ∠ = 360º
fractional part:
360/360 = 1

Semi-circle
areacircle2
(half circle = half of area) areac3
central ∠ = 180º
fractional part:
180/360 = 1/2
Quarter-circle
areacircle3
(¼ of circle = ¼ of area) areac2
central ∠ = 90º
fractional part:
90/360 = 1/4
Any Sector
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.

When finding the area of a sector, you are actually finding a fractional part of the area of the entire circle. The fraction is determined by the ratio of the central angle of the sector to the "entire central angle" of 360 degrees.   [frational part = formula; where n = central angle]

It is also possible to find the area of a sector by expressing the fraction as the
ratio of the
arc length (s) to the entire circumference.   sector1 

Exs

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
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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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

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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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