Area of Sector

In geometry, the circle is one of the most fundamental shapes, representing perfect symmetry and balance. It is a common figure in various fields, from mathematics to engineering, and even in everyday life. Among the different parts of a circle, the sector stands out as a unique and significant portion, often compared to a 'slice' of a pie. Understanding the area of a sector is crucial for many practical applications, including calculating the portion of a circular garden, determining the coverage of a radar, or even in pie charts used in statistics.

The Area of a Sector of a Circle is the measure of the region enclosed by the two radii and the corresponding arc. It depends on the angle subtended by the sector at the center of the circle and the circle's radius.

1.0Sector of Circle

A Sector of a Circle is a region enclosed by two radii and the arc connecting them. It is essentially a 'wedge' of the circle, where the central angle (the angle formed at the circle's center by the two radii) defines the size of the sector. Depending on the size of the central angle, a sector can either be a minor sector or a major sector:

• Minor Sector: This is the smaller portion of the circle, where the central angle θ is less than 180° (or π radians). It represents less than half of the circle.
• Major Sector: This is the larger portion of the circle, where the central angle θ is more than 180° (or π radians). It covers more than half of the circle.

The area of a sector is a measure of the space occupied by this 'slice' of the circle. It is directly proportional to the central angle—the larger the angle, the bigger the sector, and consequently, the greater the area. The relationship between the angle and the area is captured in the formulas used to calculate the area of a sector, making it possible to determine how much of the circle's total area is occupied by the sector.

2.0Area of a Sector Formula

The area of a sector formula is based on the proportion of the central angle to the total angle of the circle. For a circle with radius r and a central angle θ:

1. Area of Sector in Degree

When θ is in degrees:

Area of sector = $\frac{\theta }{360^o}\times \pi r^2$

2. Area of Sector in Radian

When θ is in radians:

Area of sector = $\frac{\theta }{2\pi }\times \pi r^2$

Area of sector = $\frac{1}{2}\times r^2\times \theta$

3.0Area of Major Sector vs. Area of minor Sector 

In a circle, a major sector is the larger portion, and a minor sector is the smaller portion. The formula for both remains the same, but the central angle differs:

• Area of Minor Sector: When $\theta$  < 180^o (or $\theta <\pi radians$).
• Area of Major Sector: When $\theta$  > 180^o (or $\theta$ > $\pi$ radians).

To find the area of a major sector, you can subtract the area of the minor sector from the total area of the circle.

4.0Area of Sector Formula with Arc Length 

The arc length is another critical element in sector calculations. If the length of the arc l is known, the area of the sector can be calculated using:

Area of Sector = $\frac{1}{2}\times r\times l$

This formula is useful when direct information about the arc length is provided instead of the angle.

5.0Area of Sector Solved Examples

Example 1: Find the area of a sector with a 5 cm radius and a central angle of 60°. 

Solution: 

r = 5 cm, θ = 60°

So, by formula of area of sector = $\frac{\theta }{360^o}\times \pi r^2$

Area = $\frac{60^o}{360^o}\times \pi (5{)}^2$

_{Area =} $\frac{1}{6}\times 25\pi$

Area = 13.09 cm²

So, the area of a sector is 13.09 cm². 

Example 2: Determine the area of a sector with a 7 cm radius and an angle of $\frac{\pi }{3}$ radians.

Solution:  

r = 7 cm, $\theta$ = $\frac{\pi }{3}$

So, by formula of area of sector = $\frac{1}{2}\times r^2\times \theta$

Area = $\frac{1}{2}\times 7^2\times \frac{\pi }{3}$

Area = $\frac{49\pi }{6}$

Area = 25.61 cm²

So, the area of a sector is 25.61 cm². 

Example 3: Find the area of a sector of a circle with a radius of 6 cm and a central angle of 90°.

Solution:

r = 6 cm, θ = 90°

So, by formula of area of sector = $\frac{\theta }{360^o}\times \pi r^2$

Area = $\frac{90^o}{360^o}\times \pi (6{)}^2$

Area = $\frac{1}{4}\times 36\pi$

Area = 9π cm²

So, the area of a sector is 9π cm². 

Example 4: Calculate the area of a sector with a radius of 4 cm and a central angle of $\frac{\pi }{2}$ radians.

Solution:

r = 4 cm,  θ = 90°

Area of Sector = $\frac{1}{2}\times (4^2)\times \frac{\pi }{2}$

Area = $\frac{1}{2}\times (4^2)\times \frac{\pi }{2}$

_{Area =} $\frac{8\pi }{2}$

Area = 4π cm²

So, the area of a sector is 4π cm². 

Example 5: A circular pizza is divided into 8 equal slices. If one slice is removed, what is the area of the remaining pizza? The radius of the pizza is 12 cm.

Solution:

First, calculate the area of the entire pizza:

Area of pizza = Area of Circle = πr²

= π × (12)²

= 144 cm² 

Since the pizza is divided into 8 equal slices, each slice represents:

Central angle of one slice =$\frac{360^o}{8}$ = $45^o$

The area of one slice (sector) is  = $\frac{\theta }{360^o}\times \pi r^2$

Area = $\frac{45^o}{360^o}\times 144\pi$

Area = $\frac{1}{8}\times 144\pi$

Area = 18π cm²

So, the area of the remaining 7 slices is:

Remaining area = Total Area of Circle – Area of Sector

= 144π − 18π = 126π cm²

Thus, the area of the remaining pizza is 126π cm².

Example 6: A circular garden has a path along its perimeter, and a fountain is installed in a sector that subtends a 120° angle at the center. If the radius of the garden is 15 meters, what is the area of the sector occupied by the fountain?

Solution:

To find the area of the sector:

Area of sector = $\frac{\theta }{360^o}\times \pi r^2$

Substitute the values r = 15 m, θ = 120°

Area = $\frac{120^o}{360^o}\times \pi (15{)}^2$

_{Area =} $\frac{1}{3}\times 225\pi$

Area of sector = 75π m²

So, the area occupied by the fountain is 75π m²,  which is approximately 235.62 m². 

Example 7: If a sector of a circle has an area of 50 cm² and the radius of the circle is 10 cm, what is the central angle of the sector in radians?

Solution:

We use the formula for the area of a sector when the angle is in radians:

Area of sector = $\frac{1}{2}\times r^2\times \theta$

Given Area = 50 cm² and r = 10 cm, substitute the values:

50 = $\frac{1}{2}\times (10{)}^2\times \theta$

50 = $\frac{1}{2}\times 100\times \theta$

_or $\theta$ _{= 1 radian}

So, the central angle of the sector is 1 radian. 

Example 8: A sector of a circle has a perimeter of 40 cm, and the radius of the circle is 12 cm. What is the area of this sector?

Solution:

The perimeter of a sector consists of two radii and the arc length. Let l be the arc length. Then:

Perimeter = 2r + l

Given r = 12 cm and perimeter is 40 cm:

40 = 2 × 12 + l

l = 40 – 24 = 16 cm

Now, the area of the sector can be calculated using the arc length:

Area of sector = $\frac{1}{2}\times r\times l$

_{Area =} $\frac{1}{2}\times 12\times 16$

Area of sector = 96 cm²

So, the area of the sector is 96 cm². 

Example 9: A sector has an area of 150 cm², and the angle of the sector is $\frac{\pi }{3}$ radians. What is the radius of the circle?

Solution:

Using the formula for the area of a sector:

Area = $\frac{1}{2}\times r^2\times \theta$

Substitute the values Area = 150 cm², $\theta =\frac{\pi }{3}$

150 = $\frac{1}{2}\times r^2\times \frac{\pi }{3}$

Multiply both sides by 2:

300 = $r^2\times \frac{\pi }{3}$

Multiply by $\frac{3}{\pi }$

$r^2=\frac{900}{\pi }$

Take the square root of both sides:

$r=\sqrt{\frac{900}{\pi }}\approx \sqrt{286.48}\approx 16.92cm$

So, the radius of the circle is approximately 16.92 cm.

6.0Area of Sector Practice Questions

• Find the area of a sector with 10 cm radius and an angle of 45°.
• Calculate the area of a sector with a 6 cm radius and a central angle of $\frac{pi}{4}$ radians.
• Determine the area of a major sector of a circle with a radius of 8 cm and a central angle of 240°.
• Given an arc length of 12 cm and a radius of 4 cm, find the area of the corresponding sector.
• A circular track with a radius of 50 meters is designed such that a sector of the track is used for running practice. The sector subtends an angle of 120° at the center. The rest of the track is used for walking. Calculate the area dedicated to running and the remaining area used for walking.
• A sector of a circle has an arc length of 14 cm and a central angle of 45°. Find the radius of the circle. Then, calculate the area of the entire circle and the area of the sector.
• A circular farm is divided into two sectors by a straight fence along the diameter. One sector is reserved for growing crops and subtends an angle of 150° at the center, while the remaining sector is left for grazing. If the radius of the farm is 30 meters, find the area of each sector. How much more area is available for grazing than for growing crops?
• A clock face is divided into 12 equal sectors by its hour marks. If the radius of the clock is 20 cm, find the area of the sector between the 12 o'clock and 2 o'clock positions. Then, determine the total area covered by the sectors between the 12 o'clock and 6 o'clock positions.
• A sector of a circle is part of a decorative fan blade. If the central angle of the sector is $\frac{pi}{4}$ radians and the radius of the blade is 40 cm, find the area of the sector. If the fan has 5 blades, what is the total area covered by all the blades?
• In a science project, a student designs a model of the solar system where each planet's orbit is represented by a sector of a circle. The sector representing Earth's orbit subtends a central angle of 1 radian, and the radius of Earth's orbit is modeled as 100 cm. Calculate the area of the sector representing Earth's orbit.