# Sector of a Circle: Definition, Formula, Area, and Perimeter (2023)

Sector of a Circle is one of the components of a circle like a segment which students learn in their academic years as it is one of the important geometric shapes. From a slice of pizza to a region between two fan blades, we can see sectors of the circle in our daily life everywhere. In this article, we will explore this geometric shape of the sector which is derived from the circle in detail including its areas, perimeter, and all the formulas related to it. Other than that, we will also learn how to solve problems related to it using all the formulas we will learn in this same article. So, let’s fill our brains with the knowledge of the concept with the name “Sector of a Circle”.

## Sector of a Circle Definition

A sector of a circle is a portion of a circle that is enclosed by two radii and the arc that they form.

In other words, a sector of a circle is a pie-shaped section of a circle formed by the arc and its two radii and it is produced when a section of the circle’s circumference (also known as an arc) and two radii meet at both extremities of the arc. A semi-circle, which represents half of a circle, is the most frequent sector of a circle. We can see in the above illustrated diagram, that there are two sectors formed in the circle always.

• Major Sector: The sector with a larger arc length is called the major sector.
• Minor Sector: The sector with a smaller arc length is called the minor sector.

### Sector Angle

The angle subtended by the arc at the centre of the circle is known as the sector angle or central angle of the sector. In the above diagram, we can see that, the angle subtended by the minor sector is θ, thus θ is the sector angle for the minor sector. As we know total angle subtended at any point is 360°, thus the angle subtended by the major sector is 360° – θ.

### Examples of Sector of a Circle

Some examples of sectors of circles are slices of pizza or pie, a clock face, a fan blade etc. Some examples of sectors of the circle are shown in the following illustration: ## Area of a Sector

The area of a sector of a circle is the amount of space occupied inside a sector of a circle’s border. A sector always begins at the circle’s centre. The semi-circle is likewise a sector of a circle; in this case, a circle has two equal-sized sectors.

### Formula for Area of a Sector

Formula for the area of a sector is given as follows:

A = (θ/360°) × πr2

Where,

• θ is the sector angle subtended by the arcs at the center (in degrees),
• r is the radius of the circle.

Another Formula

If the subtended angle θ is in radians, the area is given by,

A = 1/2 × r2 × θ

### Derivation of Formula for Area of a Sector

Consider a circle with centre O and radius r, suppose OAPB is its sector and θ (in degrees) is the angle subtended by the arcs at the centre. We know, the area of the whole circular region is given by, πr2.

If the subtended angle is 360°, the area of the sector is equal to that of the whole circle, that is, πr2.

Apply the unitary method to find the area of the sector for any angle θ.

If the subtended angle is 1°, the area of the sector is given by, πr2/360°.

Hence, when the angle is θ, the area of the sector, OAPB = (θ/360°) × πr2

This derives the formula for the area of a sector of a circle.

### Area of Minor Sector

The formula derived in the above section is generally used as the area of the minor sector. As θ is mostly the general representation of the angle of the minor sector. Thus ### Area of Major Sector

As sector angle for the major sector is generally represented by 360° – θ. Thus, the area of the major sector is given by ## Arc Length of a Sector

The arc length of a sector is the length of the arc that is enclosed by the sector. In other words, an arc is the sub-length of the circumference of the circle. It is a general belief that arc length is the perimeter of the sector but it is only the circular part of the sector not the complete perimeter. We will discuss the perimeter in the article ahead.

### Formula for Arc Length of a Sector

The formula for the arc length of a sector with θ sector angle is given as follows:

Arc Length of a Sector = θ°/360° × 2πr

Where,

• θ is the sector angle subtended by the arcs at the centre (in degrees),
• r is the radius of the circle.

### Derivation of Formula for Arc Length of a Sector

Consider a circle with centre O and radius r. Let OAPB be a sector of the circle, and θ° be the angle subtended by the arc at the centre O. We know that the circumference of the whole circle is given by 2πr. If the subtended angle is 360°, the arc length of the sector is equal to the circumference of the whole circle, which is 2πr.

To find the arc length for any angle θ, we can set up a proportion using the unitary method:

If the subtended angle is 360°, the arc length of the sector is 2πr.

If the subtended angle is θ°, the arc length of the sector is x.

Using proportions we get

θ°/360° = x/2πr

⇒ x = θ°/360° × 2πr

x = θ°/360° × πd

Where d = 2r is the diameter of the circle.

This derives the formula for the arc length of a sector of a circle.

## Perimeter of a Sector

The perimeter of any geometric shape is its boundary. Thus, for the sector of a circle perimeter is also the boundary of the circle which include the arc length as well as the radius of the circle which encloses the sector.

### Formula for Perimeter of a Sector

The formula for the perimeter of a circle is given by:

Perimeter of Sector = Arc Length + 2 × r

Perimeter of Sector = (θ/360) × 2πr + 2 × r

Where,

• θ is the measure of the central angle in degrees,
• π is a mathematical constant (π≈3.14), and
• r is the radius of the circle.

## Summary of Sector of a Circle

• Sector is the region enclosed by two radii and arc length in the circle.
• Angle subtended by the arc on the centre is known as the central angle.
• Area of a sector of the circle is
• Arc length of the sector of the circle is
• Perimeter of the sector of the circle is

Some Key points about Sector of a Circle are:

• Sum of the angles of any sector of a circle is always 360 degrees.
• Area of a sector is always less than the area of the entire circle.
• Arc length of the sector is also always less than the circumference of the circle.
• Perimeter of a sector can be more than the circumference of the entire circle.

• Equation of a Circle
• Area of a Circle
• Circumference of Circle

## Sample Problems Sector of a Circle

Problem 1: Find the area of the sector for a given circle of radius 5 cm if the angle of its sector is 30°.

Solution:

We have, r = 5 and θ = 30°.

Use the formula A = (θ/360°) × πr2 to find the area.

A = (30/360) × (22/7) × 52

⇒ A = 550/840

⇒ A = 0.65 sq. cm

Problem 2: Find the area of the sector for a given circle of radius 9 cm if the angle of its sector is 45°.

Solution:

We have, r = 9 and θ = 45°.

Use the formula A = (θ/360°) × πr2 to find the area.

A = (45/360) × (22/7) × 92

⇒ A = 1782/56

⇒ A = 31.82 sq. cm

Problem 3: Find the area of the sector for a given circle of radius 15 cm if the angle of its sector is π/2 radians.

Solution:

We have, r = 15 and θ = π/2.

Use the formula A = 1/2 × r2 × θ to find the area.

A = 1/2 × 152 × π/2

⇒ A = 1/2 × 225 × 11/7

⇒ A = 2475/14

⇒ A = 176.78 sq. cm

Problem 4: Find the angle subtended at the centre of the circle if the area of its sector is 770 sq. cm and its radius is 7 cm.

Solution:

We have, r = 7 and A = 770.

Use the formula A = (θ/360°) × πr2 to find the value of θ.

=> 770 = (θ/360) × (22/7) × 72

=> 770 = (θ/360) × 154

=> θ/360 = 5

=> θ = 1800°

Problem 5: Find the area of a circle if the area of its sector is 132 sq. cm and the angle subtended at the centre of the circle is 60°.

Solution:

We have, θ = 60° and A = 132.

Use the formula A = (θ/360°) × πr2 to find the value of θ.

=> 132 = (60/360) × (22/7) × r2

=> 132 = (1/6) × (22/7) × r2

=> r2 = 252

=> r = 15.87 cm

Now, Area of circle = πr2

= (22/7) × 15.87 ×15.87

= 5540.85/7

= 791.55 sq. cm

Problem 6: Calculate the arc length when r = 9 cm and θ = 45°.

Solution:

Given,

• r = 9 cm
• θ = 45°

L = (45/360) × 2π × 9

L = (1/8) × (2 × 22/7) × 9

L = (1/8) × (44/7) × 9

L = (1/8) × 44 × 9

L = 44/8 × 9

L = 99/2 cm (rounded to two decimal places)

Therefore, the arc length of the sector is 49.5 cm.

## FAQs on Sectors of a Circle

### Q1: What are Sectors of a Circle?

The sectors of a circle are parts or portions of the circle that are bounded by two radii and the corresponding arc between them.

### Q2: What is a Central Angle in a Circle Sector?

A central angle is an angle with its vertex at the centre of a circle and its sides extending to the endpoints of an arc. It determines the size of the sector and is measured in degrees or radians.

### Q3: How is Area of a Sector of a Circle Calculated?

The area of a sector can be calculated using the formula as follows:

Area of Sector = (θ/360) × πr2

Where,

• θ is the measure of the central angle in degrees,
• π is a mathematical constant (π≈3.14), and
• r is the radius of the circle.

### Q4: What is Arc Length of a Sector?

The arc length of a sector is the distance along the circumference of the circle that forms the arc.

### Q5: What is the formula for Arc length of a Sector?

Arc length of a sector is given by the following formula:

Arc Length of Sector = (θ/360) × 2πr

Where,

• θ is the measure of the central angle in degrees,
• π is a mathematical constant (π≈3.14), and
• r is the radius of the circle.

### Q6: How is Perimeter of a Circle’s Sector Calculated?

The perimeter of a circle sector is the sum of the length of the arc and the lengths of the two radii that form the sector. The formula for the perimeter of a circle is given by:

• Perimeter of Sector = Arc Length + 2 × r
• Perimeter of Sector = (θ/360) × 2πr + 2 × r

Where,

• θ is the measure of the central angle in degrees,
• π is a mathematical constant (π≈3.14), and
• r is the radius of the circle.

### Q7: Can Area of Sector be Larger than Area of Whole Circle?

No, the area of any sector can’t be larger than the area of the whole circle as it is the part of the circle and it can maximum be equal to the area of a circle as the largest possible sector is a full circle.

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