# Area of Sector

The area of a sector of a circle is the amount of space enclosed within the boundary of a sector. A sector always originates from the center of the circle. A sector of a circle is defined as the portion of a circle that is enclosed between its two radii and the arc adjoining them. The semi-circle is the most common sector of a circle, which represents half of a circle. Let's learn about the area of a sector how to calculate it.

1. | What is Area of a Sector? |

2. | Area of Sector of a Circle Formula |

3. | Real-Life Example of Area of Sector |

4. | FAQs on Area of Sector |

## What is Area of a Sector of Circle?

The area enclosed by the sector of a circle is called the area of a sector of a circle. For example, a pizza slice is an example of a sector that represents a fraction of a pizza. There are two types of sectors: minor and major sectors. A minor sector is a sector that is less than a semi-circle, whereas a major sector is a sector greater than a semi–circle.

The figure shown below represents a sector in a circle. The shaded region shows the area of the sector OAPB. Here, ∠AOB is the angle of the sector. In fact, the unshaded region is also a sector of the circle. So, the shaded region is the area of the minor sector and the unshaded region is the area of the major sector.

### Real-Life Example of Area of a Sector of Circle

One of the most common real-life examples of the area of a sector is a slice of a pizza. The shape of slices of a circular pizza is like a sector. A pizza of 7 inches radius is sectioned into 6 equal slices as shown in the below figure. Each slice is a sector.

The area of the sector formed by each slice can be calculated by using the area of the sector formula. Now, calculate the area of the sector shape pizza slice by using the area of a sector formula. Area of Pizza slice = (θ/360º) × πr^{2} = (60º/360º) × (22/7) × 7^{2} = 1/6 × 22 × 7 = 77/3 = 25.67 square units.

Let's check out the area of a sector formula and its derivation.

## Area of a Sector of Circle Formula

The area of a sector of circle formula can be calculated to find the total space enclosed by the said part. The area of the sector can be calculated using the following formulas,

- Area of a Sector of Circle = (θ/360º) × πr
^{2}, where, θ is the angle subtended at the center, in degrees, and r is the radius of the circle. - Area of a Sector of Circle = 1/2 × r
^{2}θ, where, θ is the angle subtended at the center, in radians, and r is the radius of the circle.

### Formula Derivation

Let's apply the unitary method to derive the formula of the area of a sector of circle. We know, the degree measure of the complete circle is 360º. The area of a circle with an angle measuring 360º at the center is given by πr^{2}, where r is the radius of the circle.

If the angle at the center of the circle is 1º, the area of the sector is πr^{2}/360º. So, if the angle at the center is θ, the area of the sector is, Area of a Sector of Circle = (θ/360º) × πr^{2}, where,

- θ is the angle subtended at the center, given in degrees.
- r is the radius of the circle.

If the angle at the center is θ in radians, the area of the sector is, area of a sector of circle = (1/2) × r^{2}θ, where,

- θ is the angle subtended at the center, given in radians.
- r is the radius of the circle.

**Semi-circle** and **quadrant **form sectors of a circle and are special types of the sector of a circle with angles 180º and 90º respectively.

**Example: **The central angles made by the red, green, and blue color in a circle of radius 6 units are 100º, 160º, and 100º respectively. Find the area of sector of each color.

**Solution:**

The angle made by red color is θ = 160º. Therefore, the area of the sector having red color is, area(red) = (θ/360º) × πr^{2} = (160º/360º) × (22/7) × 6^{2} = 4/9 × 22/7 × 36 = 352/7 = 50.28 square units.

The angle made by blue color is θ = 100º. Therefore, the area of the sector having blue color is, area(blue) = (θ/360º) × πr^{2} = (100º/360º) × (22/7) × 6^{2} = 5/18 × 22/7 × 36 = 220/7 = 31.43 square units.

The angle made by green color is the same as that of blue color. Thus, the area of the sector having green color is equal to the area of the sector having a blue color. Therefore, the area covered by green color = 50.28 square units.

### Important Notes

Given below are a few important points that would help you to solve the area of sector problems.

- The area of a sector of circle is the fractional area of the circle.
- The area of a sector of circle with radius r is given by Area = (θ/360º) × π r
^{2} - The arc length of the sector of radius r is given by Arc Length of a Sector = r × θ

## Solved Examples on Area of a Sector

**Example 1: AB is a chord of a circle that subtends an angle of 60° at the center of a circle. The radius of the circle is 7 inches. Can you find the area of the minor sector of this circle?****Solution:**The radius of the circle is 7 inches. We will use the formula of the area of a sector of circle. The area of minor sector = (θ/360°) × π r

^{2 }= (60°/360°) × (22/7) × 7^{2}= 77/3 = 25.67 square units. Therefore, the area of the minor sector is 25.67 square units-
**Example 2: An umbrella has equally spaced 8 ribs. If viewed as a flat circle of radius 7 units, then what would be the area between two consecutive ribs of the umbrella.****Solution:**The radius of the flat umbrella would be 7units. There are 8 ribs in the umbrella. The angle of each sector of the umbrella is 45° because the complete angle is divided into 8 parts. Thus, the area of sector = 360°× π r

^{2}= (45°/360°) × 22/7 × 7^{2}= 77/4 = 19.25 square units. Therefore, the area between the two consecutive ribs of her umbrella is19.25 square units. -
**Example 3: A circle-shaped brooch is divided into 10 sectors by 5 wires. The length of the diameter is 2 units. Can you determine the area of each sector of the brooch?****Solution:**The diameter of the brooch is 2 units. Since the radius is half of the diameter, the radius of the brooch is 1 unit. The angle of each sector of the brooch is 36° because the complete angle is divided into 10 parts. Area of Sector = (θ/360°) × πr

^{2}= 36°/360° × 22/7 × 1 = 11/35. Therefore, the area of each sector of the brooch is 11/35 square units.

## FAQs on Area of Sector of Circle

### What is Meant by the Area of a Sector?

The area enclosed by the sector of a circle is called the area of a sector of a circle. The portion or part of the circumference region enclosed by two radii and the corresponding arc is called a sector of the circle.

### What is the Formula for Area of a Sector?

The area of a sector of the circle formula can be calculated to find the total space enclosed by the said part. Thus,

- Area of a Sector of Circle = (θ/360º) × πr
^{2}, where, θ is the angle subtended at the center, given in degrees. r is the radius of the circle. - Area of a Sector of Circle = 1/2 × r
^{2}θ, where, θ is the angle subtended at the center, given in radians. r is the radius of the circle.

### What is Arc Length of a Sector?

The** **arc length of a sector is defined as the interspace between the two points along a section of a curve. The arc length of a circle can be calculated using the formula,

- Arc Length = θ × r, when θ is in radian.
- Arc Length = θ × (π/180) × r, when θ is in degree.

### How to Calculate Area of a Sector using Degrees?

When the angle subtended at the center is given in degrees, The area of a sector can be calculated using the following formula, area of a sector of circle = (θ/360º) × πr^{2}, where, θ is the angle subtended at the center, given in degrees, r is the radius of the circle.

### What do you Mean by Sector of a Circle?

A sector is defined as the portion of a circle that is enclosed between its two radii and the arc adjoining them. The semi-circle is the most common sector of a circle, which represents half of a circle.

### What do you Mean by the Arc of a Circle?

A part of a curve or a part of a circumference of a circle is called Arc. Many objects have a curve in their shape. The curved portion of these objects is mathematically called an arc.

### How the Area of a Sector of Circle Formula is Derived?

The derivation for area of sector of circle formula can be derived in the following way,

- Apply the unitary method to derive the formula of the area of a sector of circle.
- We know, the degree measure of the complete circle is 360º. The area of a circle with an angle measuring 360º at the center is given by πr
^{2}, where r is the radius of the circle. - If the angle at the center of the circle is 1º, the area of the sector is πr
^{2}/360º. So, if the angle at the center is θ, the area of the sector is, Area of a Sector of a Circle = (θ/360º) × πr^{2}, where, θ is the angle subtended at the center, given in degrees, r is the radius of the circle.