Area of Sector

What is a Sector?

A sector is much the same like 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.

  • Crack the Area of a Sector in Simple Few Steps

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


  • What are the Parts of a Circle?

A Midpoint - It is the most crucial one as it is the connecting dot positioned right in the circle's center.

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.

  • Types of Sectors                                              

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 area. 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 being computed. Hence, for the area of sector of a circle, it's the median angle of 360°, times the area of the circle, 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, area of a sector of a circle formula will be: \[A = (\frac{\text{sector angle}}{360})\times (\pi\times r^{2})\]

Solved Example

Ques. – Given is a circle with radius 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 = 30/360 fraction of circle.

= \[\frac{1}{12}\]

 Area of sector = \[\frac{1}{12}\] * Total Area

                           = \[\frac{1}{12}\] * 4π

                           = \[\frac{\pi}{3}\] (pi π – 3.14)

  1. 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 π/2 radians, the portion of the circle to be measured will be π/2 divided by 2π, or 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 = \[\frac{\text{sector angle}}{(2\times \pi)}\times (\pi\times r^{2})\]

  1. 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 = \[{\text{(portion of the circle)}}(\pi\times r^{2})\].

FAQ (Frequently Asked Questions)

1. How to Measure the Area of a Sector of a Circle?

A sector is simply a pie, portion or wedge of a circle. A sector always has its origin at the midpoint of the circle. The area of a sector is thus a fraction of the area of the circle. This area is equivalent to the median angle.

Typically, the formula to calculate area of sector of a circle is A = pi *r2. But, to find a part of the circle’s area—that part can be identified using the sector's angle.