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# Sector|Definition & Meaning

## Definition

A sector of a circle is a closed region bounded by an arc and any two radii connecting the ends of that arc to the center of the circle. It looks like a slice of cake or pizza. The area of a sector is always less than the circleâ€™s area. The sectorâ€™s arc forms an angle at the center of the circle (between the two bounding radii). The greater the arcâ€™s length, the larger this angle.

A sector of a circle is a section of a circle that has the form of a **pie** and is created by combining the **arc** of a circle with two of its radii. A sector is formed when a segment of the **circumference** of a circle, also referred to as an **arc**, and two of the circle’s **radii** meet at both **endpoints** of the arc.

A sector of a circle seems like a piece of pie or **pizza** when you view it closely. A circle is considered to be one of the most flawless forms in geometry.

The most basic shape in all of **geometry** is that of a sector cut out of the **middle** of a circle. It possesses a variety of distinct **components** of its own. For example, the **diameter**, the **radius**, the **circumference**, the **segment**, and the **sector**.

## Area of a Sector

The total **space** that is contained within the **perimeter** of a **sector** of a circle is referred to as the **area** of that sector of that circle. Every beginning point for a sector is located at the circle’s **epicenter**. The Semi-circle of any circle can also be considered a **sector. **In that scenario, however, a circle will be said to have two sectors where both sectors will possess the same amount of **area**.

Now let us get familiar with the **formula** for determining the total area of a sector. Let us assume that **â€˜râ€™** is the radius of the circle whose sectorâ€™s area we want to determine. Let us also assume that **â€˜Î¸â€™** is the angle of the sector whose are we want to find. Then the **formula** below can be employed to find the area of the sector:

Area of Sector = A = (Î¸/360Â°) Ã— Ï€r^{2}

Here, â€˜Î¸â€™ refers to the angle of the sector in degrees, and â€˜râ€™ refers to the radius of the circle.

## The Area of the Sector in Relation to the Length of the Arc

There is an **additional** method for determining the area of a sector that can be utilized in the event that the length of the **arc** of the sector rather than the angle of the sector is provided. Let’s say that the **length** of the arc is denoted by the letter **‘L.’** If the radius of a circle equals r units, then an arc that is exactly r units in length will form exactly one radian at the circle’s **center**.

Therefore, one can draw the conclusion that an arc with a length ‘L’ will **subtend** an angle with a measure of r/L, which is the angle at the center. Therefore, if ‘L’ represents the length of the arc, r represents the **radius** of the circle, and Î¸ represents the angle that is formed at the center, then:

Î¸ = L/r

**â€˜Î¸â€™** here refers to the angle in radians.

Now let us incorporate the value of **Î¸** in the area of the sector:

Area of Sector = A = (Î¸) Ã— Ï€r^{2}

A = (L/r) Ã— (r^{2}/2)

A = (L Ã— r)/2

## Perimeter Formula for a Sector

As we know that perimeter refers to the summation of the length of boundaries of any geometrical shape. Using the same concept, we can derive the formula for the perimeter of a Sector of any circle. Below we have done the derivation for the formula of the perimeter of a sector.

If we assume that â€˜râ€™ is the radius of the circle, â€˜Î¸â€™ is the angle subtended by the sector on the circle center, and ‘L’ is the length of the arc, then:

The perimeter of a sector = 2r + arc length

Since we know that:

Arc length of a sector = L = (Î¸/360) Ã— 2Ï€r

Thus:

The perimeter of a sector = 2r + ((Î¸/360) Ã— 2Ï€r )

## Sector Proportion

The area of one sector of a circle in relation to the total area of the circle is what is meant by the term “sector proportion.” It is denoted by the symbol pi (from the Greek alphabet), and its value is about 3.14159. The formula for calculating the proportion of a sector is as follows:

(Area of the sector of a circle) / (Area of the complete circle) = (Î¸/360) Ã— Ï€

Here â€˜Î¸â€™ refers to the value of the **central** angle in degrees.

For instance, if a sector has a central angle of 45 degrees, the sector proportion can be calculated as:

Sector proportion = (45/360) Ã— Ï€

Sector proportion = 0.125 Ï€

This indicates that the **area** of the sector constitutes 12.5 percent of the total area of the circle.

The notion of sector proportion is **essential** in maths as well as in other domains since it enables a **contrast** of the area of a sector to the area of the **complete** circle. This analysis can be helpful in real applications such as **computing** the percentages of a pie chart, as well as when dealing with **calculations** of angles and **circular** parameters.

## Applications of Sectors

The use of sectors of circles can be found in a **diverse** array of contexts and fields. The following are some of the more **typical** applications:

### Engineering

Sectors are utilized in the field of **engineering** to describe and analyze a wide variety of data, including **pulley** systems and **gear** teeth, amongst other things. Calculating the strength and **tension** of circular constructions such as **pipes** and **discs** is another application for these equations.

### Physics

In the field of physics, sectors are a useful tool for **representing** and **analyzing** a wide variety of phenomena, including **angular** momentum and **centripetal** force, amongst others. They are also employed to **compute** the area of a sector as well as the **arc length**, which is something that can be helpful when trying to solve problems relating to **physics**.

### Mathematics

Sectors are a useful tool for representing and analyzing a wide variety of **geometric **figures and forms in the field of **mathematics**. Calculating the **size **and **arc length **of a sector, which may be helpful when attempting to solve **geometric **issues, requires the use of these tools.

### Graphing and Data Visualization

Sectors are frequently used to **depict **the **proportion **of various **data sets **when **pie charts, **as well as related types of data **visualization, **are utilized.

### Navigation

In navigation, sectors are used to both depict the various parts of **maps **and to assist **navigators **in **locating **themselves in relation to the map.

### Biology

In the field of biology, separate parts of **organisms**, such as individual sections of **leaves **or the petals of a **flower**, are referred to as “**sectors**,” and they are represented by the term.

## Examples of Sectors

### Example 1

Calculate the arc length of a sector if the angle of the sector is **60 degrees **and the radius of the circle is 6 meters.

### Solution

Since we know that:

Arc length of a sector = **L **= (Î¸/360) Ã— 2Ï€r

According to the given data:

Î¸ = 60^{0}

r = 6 m

Now:

Arc length of a sector = L = (60/360) Ã— 2 Ã— Ï€ Ã— 6

L = 2 Ã— Ï€

**L = 44/7 m**

### Example 2

Calculate the **area **of a sector of a circle whose radius is **3 m** and the length of the sector’s arc is **10 m**.

### Solution

Since we know that:

A = (L Ã— r)/2

According to the given data:

L = 10 m

r = 3 m

Now:

A = (10 Ã— 3)/2

**A = 15 m ^{2}**

*All images/mathematical drawings were created with GeoGebra.*