Theory Of Area Of Sector

The area of a segment of a circle is the size of space that is enclosed within the perimeter that defines the area. The sector always begins from the central point of the circle. The term “circle sector” is defined as the area of a circle divided by its two radiuses as well as the arc which is adjacent to them. 

The semi-circle is the most popular sector of a circle that is a half-circle. Learn more about the size of a sector as well as its formula and how to determine an area for a particular sector by using degrees and radians. Use the area calculator for calculation.

What is the Area of the Sector of a Circle?

The area that is enclosed by the sector of a circle is known as the “sectional area. For instance, pizza slices are an illustration of a sector that represents a portion of the pizza. Minor sectors are the sector that is smaller than a semi-circle. In contrast, major sectors are an industry that is greater than the semi-circle. Using the land area calculator, you can make the calculation.

The illustration below illustrates the circles of sectors. The shaded region represents the size in the segment OAPB. This time, AOB is the angle of the sector. It is worth noting that the region which is not shaded is also a part of the circular. Thus, the shaded region is the portion in the sector of a minor, and the region that is not shaded is the main sector.

A circle represents the size of a sector.

Let’s now learn about the subject of a sector formula as well as its formula for derivation.

Area of Sector Formula

In order to determine the space that is contained in the area of the segment, apply the area of the formula for a sector. The size of a particular sector can be determined using the following formulas:

The area of a Sector of Circle = (th/360o) pr2 where the angle of the sector is that is subtended by the arc in the center in degrees and ‘r’ represents the radius of the circle.

The area of a sector of a Circle is 1/2 x r2th. In this equation, this is the sector angle that is subtended by a central arc in radians, and “r” refers to the circumference of the circle.

Area of Sector Formula Derivation

Let’s use the unitary method in order to calculate an equation for measuring the area of the sector in the circle. We know that a full circle is 360o. The size of a circle with a 360o angle in the center is calculated by pr2 in which ‘r’ is its radius.

If the angle in its center is one degree, the area of the circle is pr2/360o. Also If that angle of the circle is th then it is the surface of the circle is: the area of a sector of a Circle equals (th/360o) + pr2 where pr2 is the value.

Th is the angle that is subtended at the center measured in degrees.

In the sense that pr2 is the total area of a circle. th/360o also tells us how much area is covered in the section.

Area of a sector of an arc

If the angle in the middle is called thin degrees the area of the circle of a circle is (1/2) + r2th where,

The angle that is subtended in the center is expressed in the form of radians.

R is the radius of the circle.

It is important to note that quadrants and semi-circles are distinct types of circles that have angles of 90deg and 180deg respectively.

Area of Sector Using Degrees

Use these formulas to learn how to determine the area of the part of the circle where the subtended angle is calculated in degrees, with the aid of an illustration.

• Example 

A circle is split into three sectors. The central angles formed of the circumference are 160deg, 100deg, and 100deg, respectively. Find the total area of three of the sectors.

Solution:

The angle created in the initial sector 160deg. Thus, the size of the first sector is (th/360deg) + pr2 = (160deg/360deg) (160deg/360deg) (22/7) + 62 = 4/9 x 22/7 36 = 352/7 50.28 Square units.

The angle formed in the 2nd sector 100deg. The size of the second segment is (th/360deg) pr2 = (100deg/360deg) (100deg/360deg) (22/7) + 62 = 5/18 x 21.7 x 36 = 220/7 31.43 sq. units.

The angle created by this sector is exactly identical to that in the first sector (the equals 100 degrees). So, the size that is the area of the second sector equivalent to the size of the third. Thus, the total area in sector 3 is 31.43 cubic units.

Area of Sector in Radians

If we want to determine the area of a specific sector when the angle is stated in radians, we can use the formula Area of the sector = (1/2) 2nd x r2th where the angle is that is subtended to the center which is expressed in radians “r” represents the diameter of the circular area. We must understand how the formula came to be. 

We are aware of this formula: the surface of a segment (in degrees) is (th/360o) pr2 since it is the fraction of the circumference of a circle. The same principle applies in the case of formulas when you wish to translate them into the radians. We just have to substitute 360deg for 2p, since 2p (in radians) equals 360deg. This means that the area for the entire sector measured in the radians is (th/2p) + pr2. In simplifying the formula we can calculate the area of the sector as (th/2) + r2 or (1/2) + 2nd. Let’s look at how to calculate the size of a particular sector in radians by using an example.

• Example: 

Calculate the surface of the segment when its radius is six units and the angle that is subtended in the center is 2p/3.

Solution: 

Given that Radius = 6 units The measurement of angle (th)equals 2p/3

The area of the sector can be determined using the formula Sector’s Area (in Radians) is (th/2) (th/2) x 2. When we substitute the values in the formulaabove, you will get the area of the Sector (in Radians) equals [2p/(3×2)[2p/(3×2)] x (62) = (p/3) 36x = 12p.

Thus, the area of the specified sector in radians is calculated as twelve square units.

Real-Life Example of Area of Sector of Circle

The most well-known real-world example of the size of an area is the slice of pizza. Slices of pizza that are circular are much like a segment. Take a look at the image below, which shows the pizza cut into six equal slices in which each slice is a sector and the pizza’s radius of 7 inches. 

Let us now determine the size of the section that is formed by each slice employing the surface of the sector formula. It is important to note that because the pizza is split into 6 equal slices it is a sector with an angle of 60 degrees. Area of Pizza slice = (th/360deg) x pr2 = (60deg/360deg) x (22/7) x 72 = 1/6 x 22 x 7 = 77/3 = 25.67 square units.

The Area of a Sector of the Circle: A Real Life Example – Pizza using slices

Tips on Area of Sector

Here’s an overview of some crucial points to consider to solve sectoral issues.

An area or segment of an arc can be defined as the fractional size of the circle.

A segment of a circle that has a radius “r” is calculated using the formula the area of a segment = (th/360o) * the p-value of r2.

The length of the arc in the radius of the sector is calculated using the formula Arc Length for the Sector = the radius r times the length of the sector.