[Maths Class Notes] on Area of Sector Pdf for Exam

A sector is much the same as a “pie” or a “pizza slice”. It typically involves a province (sector) encompassed by 2 Radii and 1 Arc lying between the radii. To better understand what a sector of a circle is, here is a specimen from the real-life examples of food. A variety of foods that we eat resembles the sector of a circle, typically because they replicate circles. 

 

Find the Sector in the Circle

A circle consists of an interior, its area. Divide the circle into pieces and you will get a sector. Similarly, cut the cake into Slices and you will get the sector of a circle.

To ace over the calculation of area of sector, understanding the anatomy of the circle is the key. 

A Midpoint – It is the most crucial one as it is the connecting dot positioned right in the circle’s centre.

Radius – A line that runs through the midpoint to the exterior of the circle

Diameter – A line extending across the circle and passes through its midpoint. 

Take Notice of Diameter – it is actually two radii combined. Therefore, a circle’s radius is invariably half the length of its diameter and its diameter is always double the length of its radius.

A semicircle (half circle) and a Quadrant (Quarter of a circle) are two main types of Sector:

Besides, There are primarily two “portions” of a circle: Of which, the “pizza” slice is known as the Sector. Whereas, a cut carried off from between two points on the circle by a “chord” is called a Segment. This means Chunks cut off by joining any two points on the circle are segments. 

Seeing that, both sectors and segments are part of a circle’s interior, both have areas. We can measure their area using formulas.

Easy Formulas to Calculate Area of a Sector of a Circle

1. Calculating the Area of Sector of a Circle Using Degrees

A whole of a circle surrounds 360°, thus the ratio of the sector’s angle calculation to 360° is directly proportional to the fraction of the circle’s area is computed. Hence, for the area of the sector of a circle, it’s the median angle of 360°, times the area of the circle. The sector of a circle that subtends an angle of 180 degrees at the centre is called a semicircle. For instance, if the sector’s angle happens to measure 180°, and the two radiuses creating it are 10 inches, you would divide 180 by 360 (180/360) or 1/2 to obtain the portion of the circle.

In numeric terms, the area of a sector of a circle formula will be = [frac {theta}{360} times pi r^{2}]

Hence, the area of the semicircle will be = [frac {1}{2} times pi r^{2}]

This is because the angle is 180 degrees.

Solved Example

Question: Given is a circle with a radius of 2 cm and ∠ A = 30°. Find the area of the sector of the circle below? 

Solution: Area of circle = πr2 =  π 22 = 4π

 

Total degrees in a circle = 360°

 

Given that the central angle is 30 degrees and the radius is 2cm,  

 

Therefore, 30° slice = [frac {30}{360}] fraction of circle. 

 

= [frac {30}{360} times pi r^{2}]. 

 

 Area of sector = [frac {theta}{360}]* Total Area 

 

                           = [frac {theta}{360} times pi r^{2}]

 

                           =  [frac {1}{12} times frac {22}{7} times 4]

 

                          = 1.047 square cm                                      

2. Calculating the Area of Sector of a Circle Using Radians

If the question is available with radians as a replacement for degrees to calculate the area of sector angle, the usual method of finding the sector’s area remains the same. Radians are units that are utilized for computing angles. A circle on a whole circumscribes 2 π radians, so the portion of 2 π is the fraction of the circle’s area that is being measured.

To put it simply, if the sector’s angle is [frac {pi}{2}] radians, the portion of the circle to be measured will be [frac {pi}{2}] divided by 2π, or [frac {1}{4}] of the circle. 

So, to find the area, multiply the circle’s area by the fraction of the circle that is being dealt with. Just make use of radians instead of degrees. The sector of a circle formula in radians is:  

A =  [frac{text{angle in radian}}{pi} times pi r^{2}]  

For example, if a sector contains an angle of [frac {pi}{3}] at the centre. Then the area of the sector would be  [frac {pi}{3}]

= [frac{frac{pi}{3}}{pi} times pi r^{2}] 

 = [frac {1}{3} times pi r^{2}]

3. Calculating the Area of Sector Using the Known Portions of a Circle

In cases where the portion of a circle is known, don’t divide degrees or radians by any value.

 

For example, if the known sector is 1/4 of a circle, then just multiply the formula for the area of a circle by ¼, and you are good to go to find the area of the sector. 

 

Thankfully, this sector of a circle formula would work with any stated fraction of a circle. Even if the known portion is 1/120 of a circle, simply multiply the formula for the area of a circle by 1/120.

 

So, when given the portion of the circle, the sector of a circle formula is:

 

A = (portion of the circle) (π x r2)

 

For example, if we have to find the area of the quadrant of a circle, then it is already known to us that the angle subtended by the quadrant at the centre of the circle is always 90 degrees and it divides the circle into four parts of equal area.

 

Hence, we can also say that the area of a single quadrant of a circle is ¼ of the area of the circle. 

 

Area of Quadrant =[frac {90}{360} times pi r^{2}] 

 

= [frac {1}{4} times pi r^{2}] 

 

So if the radius of a circle is 10 cm then the area of its one of the quadrant would be :

 

= [frac {1}{4} times pi times 100]

 

= 78.57 square cm.

 

If the angle at the centre that is θ wou
ld be in radians, area of the sector of a circle = [frac {1}{2} times r^{2} theta], where,

Here, θ will be the angle subtended at the centre, given in radians and r is the radius of the circle.

It should be noted that semicircles and quadrants are special types of sectors of a circle. They made an angle of 180° and 90° respectively with the centre.

Area of Sector Using Degrees

Now let us discuss the area of the sector of a circle whose angle is given to you in degrees with the help of the example given here.

Example: Determine the area of a sector if the radius of the circle is 8 units, and the angle subtended at the centre = [frac {pi}{3}]. 

Solution: Given, radius = 8 units; Angle measure (θ)= [frac {pi}{3}]

The area of the given sector can be calculated with the given formula, 

[A = (frac {text{sector angle}}{360}) times (pi times r^{2}) ]

Hence, Area of sector would be  = [(frac {theta}{360}) times pi r^{2}]

When we substitute the given values  we get the Area of the sector  

= [frac{frac{pi}{3}}{360} times frac {22}{7} times 64]

=12π

Hence, the area of the given sector in radians is expressed as 12π square units