Lesson Area of a sector

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Area of a sector


Definition

A  sector  of the circle is the plane figure restricted by the arc of the circle and by two radii connecting the center of the circle with endpoints of the arc  (Figure 1).

A sector is characterized by two parameters:  the radius of the circle and the central angle which leans on this arc.

Theorem 1

The area of a sector of the circle is half the product of the radius of the circle  r               
and the radian measure of its central angle  alpha

S = 1%2F2alphar%5E2                (1)

Proof

Let us consider the ratio of the area of the sector  S  to the area  S%5B0%5D  of
the circle of the same radius r,  which is  S%5B0%5D = pir%5E2.
Based on the general properties of the area this ratio should be equal to  alpha%2F%282pi%29:

S%2FS%5B0%5D = alpha%2F%282pi%29.


      Figure 1. A sector

From this proportion S = alpha%2F%282pi%29.S%5B0%5D = alpha%2F%282pi%29.pir%5E2 = 1%2F2alphar%5E2.

It is exactly what the formula  (1)  states.


Example 1

Find the area of a  72°-sector of the circle which has the radius of  10 cm.

Solution

Use the formula  (1)  above for the area of a sector.  You have                                               

S = 1%2F2.alpha.r%5E2 = 1%2F2%2872%2A2%2Api%29%2F360.10%5E2 =

                       = 1%2F2*1.2566*100 = 62.832 cm%5E2 (approximately).


Figure 2. To the Example 1

Answer.  The area of the segment of the circle is  62.832 cm%5E2 (approximately).


Problem 1

Find the area of the figure which is restricted by the contour of a square                      
with the side of  10 cm  from the bottom and from the lateral sides and by
a semicircle from the top  (Figure 3a).

Solution

Let us draw the horizontal line at the top of the square  (Figure 3b).
Then our figure is the combination of the square with the side  a = 10 cm
and the semicircle with the radius  r = 5 cm.

The area of this combined figure is the sum of the area of the square
and the semicircle:

S = a%5E2 + 1%2F2.pir%5E2 = 10%5E2 + 1%2F2*3.14159*5%5E2 = 100 + 39.27 =
                                                            = 139.27 cm%5E2 (approximately).

Answer.  The area of the combined figure is  139.27 cm%5E2 (approximately).



Figure 3a. To the Problem 1      




Figure 3b.  To the solution
    of the Problem 1



Problem 2

Find the area of the figure which is obtained from a square with the side                      
of  10 cm  after cutting off a semicircle of the radius of  6 cm  as shown
in the  Figure 4.

Solution

The area of the figure under consideration is the difference of the area
of the square with the side  a = 10 cm  and the semicircle with the
radius  r = 3 cm.  So,

S = a%5E2 - 1%2F2.pir%5E2 = 10%5E2 - 1%2F2*3.14159*3%5E2 = 100 - 14.14 =
                                                            = 85.86 cm%5E2 (approximately).



Figure 4. To the Problem 2      

Answer.  The area of the figure under consideration is  85.86 cm%5E2 (approximately).


My other lessons on the area of a circle,  the area of a sector and the area of a segment of the circle in this site are
    - Area of a circle  and
    - Area of a segment of the circle
under the current topic  Area and surface area  of the section  Geometry,  and
    - Solved problems on area of a circle,
    - Solved problems on area of a sector,
    - Solved problems on area of a segment of the circle  and
    - Solved problems on area of a circle, a sector and a segment of the circle
under the topic  Geometry  of the section  Word problems.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

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