Sector of a Circle (original) (raw)
Last Updated : 23 Jul, 2025
**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. The sector of a circle is a section of a circle formed by the arc and its two radii and it is produced when a section of the circle's circumference and two radii meet at both extremities of the arc. From a slice of pizza to a region between two fan blades, we can see sectors of the circle in our daily lives everywhere.
In this article, we will explore the **geometric shape of the sector which is derived from the circle in detail including its areas, perimeter, and all the formulas related to the sector of a circle.
Table of Content
- What is Sector of a Circle?
- Sector of a Circle Examples
- Sector of a Circle Area
- Formula for Area of a Sector
- Derivation of Formula for Area of a Sector
- Area of Minor Sector
- Area of Major Sector
- Arc Length of Sector of a Circle
- Formula for Arc Length of a Sector
- Derivation of Formula for Arc Length of a Sector
- Sector of a Circle Perimeter
- Perimeter of a Sector Formula
- Sample Problems Sector of a Circle
- Summarizing Important Formulas of Sector of a Circle
What is Sector of a Circle?
A sector is a segment of a circle that includes an arc and the two radii that connect the arc's endpoints to the circle's centre. It represents a fraction of the circle, defined by the arc—part of the circle's perimeter—and the radii at the arc's ends. Visually, a sector resembles a piece of pizza or pie, highlighting its nature as a portion of the whole 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.
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.
Sector of a Circle Examples
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:
Sector of a Circle Area
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.
Semi-Circle As a Sector
The semi-circle is likewise a sector of a circle; in this case, a circle has two equal-sized sectors.
Illustration of Semi-Circle As a Sector
Formula for Area of a Sector
Formula for the area of a sector is given as follows:
**A = (θ/360°) × πr 2
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 × r 2 **× θ
**Read More,
**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°) × πr 2
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
\bold{\text{Area of the Minor Sector} = \frac{\theta}{360}\times πr^2}
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
\bold{\text{Area of the Major Sector } = \frac{360-\theta}{360} \times πr^2}
Arc Length of Sector of a Circle
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.
**Read More,
Sector of a Circle Perimeter
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.
Perimeter of a Sector Formula
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 - 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.
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**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.
Practice Problems
**1. Find the area of the sector for a circle with a radius of 10 cm and a central angle of 45°.
**2. Calculate the arc length of a sector in a circle with a radius of 8 cm and a central angle of 60°.
**3. Determine the perimeter of a sector with a radius of 12 cm and a central angle of 90°.
**4. Find the area of the minor sector of a circle with a radius of 15 cm and a central angle of 30°.
**5. Calculate the area of the major sector of a circle with a radius of 18 cm and a central angle of 120°.
**6. If the area of a sector is 100 square cm and the central angle is 90°, find the radius of the circle.
**Important Maths Related Links
Summarizing Important Formulas of Sector of a Circle
- **Formula for Area of a Sector: A = (θ/360°) × πr2
- **Formula for Arc Length of a Sector: Arc Length = θ°/360° × 2πr
- **Formula for Perimeter of Sector of a Circle: P = (θ/360) × 2πr + 2 × r