Area of a Sector

The area of a sector is a portion of the area of a circle, defined by a central angle. Imagine a slice of pizza; that slice represents a sector of the whole pizza (circle). The size of the sector depends on the angle it subtends at the center of the circle. The larger the angle, the larger the area of the sector. This concept is fundamental in geometry and is crucial for understanding more advanced topics like trigonometry and calculus.

Formula

The formula to calculate the area of a sector is:

Area of Sector = $\frac{\theta}{360^\circ} \times \pi r^2$

Where:

• $\theta$ (theta) is the central angle of the sector, measured in degrees.
• $r$ is the radius of the circle.
• $\pi$ (pi) is a mathematical constant, approximately equal to 3.14159.

Examples

Example-1: Calculate the area of a sector with a central angle of 60 degrees and a radius of 5 cm.

Solution:

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

Area = $\frac{1}{6} \times \pi \times 25 \text{ cm}^2$

Area ≈ 13.09 $\text{cm}^2$

Example-2: A circle has a radius of 8 inches. Find the area of a sector with a central angle of 120 degrees.

Solution:

Area = $\frac{120^\circ}{360^\circ} \times \pi (8 \text{ in})^2$

Area = $\frac{1}{3} \times \pi \times 64 \text{ in}^2$

Area ≈ 67.02 $\text{in}^2$

Common mistakes by students

Students often make the following mistakes:

• Using radians instead of degrees: The formula requires the central angle to be in degrees. Make sure to convert radians to degrees (using the conversion factor $\frac{180}{\pi}$) if given.
• Incorrectly calculating the radius: Confusing the radius with the diameter or a chord. Remember the radius is the distance from the center of the circle to any point on the circumference.
• Forgetting the units: Always include the correct units (e.g., cm², in²) in your answer.
• Incorrect arithmetic: Errors in calculation, especially when using a calculator or dealing with $\pi$.

Real Life Application

The concept of the area of a sector has many real-world applications, including:

• Designing and manufacturing: Used in designing circular objects, such as pizzas, cakes, and watch faces, determining the area of segments or slices.
• Architecture: In the design of curved structures like archways and curved windows, requiring the calculation of sector areas.
• Irrigation: Sprinkler systems in agriculture often cover sectors of circular areas, necessitating sector area calculations.
• Navigation: Calculating the area covered by a radar scan which scans a sector.

Fun Fact

The concept of sectors extends beyond circles. It’s also used in other geometric shapes, and similar area calculations arise in areas of mathematics and computer science related to optimization problems or for the areas scanned by rotating sensors (e.g., radar).

Recommended YouTube Videos for Deeper Understanding

Q.1 A sector of a circle has a radius of 7 cm and a central angle of 60 degrees. What is the area of the sector?

Check Solution

Ans: C

Area of sector = $(\frac{\theta}{360^\circ})\times\pi r^2 = (\frac{60^\circ}{360^\circ})\times\pi (7)^2 = \frac{1}{6}\times49\pi = \frac{49\pi}{6}$

Q.2 If the area of a sector is $12\pi$ cm$^2$ and the radius of the circle is 6 cm, what is the central angle of the sector?

Check Solution

Ans: D

$12\pi = (\frac{\theta}{360^\circ})\times\pi (6)^2$. Solving for $\theta$, we get $\theta = 120^\circ$

Q.3 A sector has a central angle of 45 degrees and an area of $9\pi$ square inches. What is the radius of the circle?

Check Solution

Ans: B

$9\pi = (\frac{45^\circ}{360^\circ})\times\pi r^2$. Solving for $r$, we get $r^2 = 72$, therefore $r = \sqrt{72}= 6\sqrt{2}$.

Q.4 A circular pizza is cut into 8 equal slices. If the diameter of the pizza is 16 inches, what is the area of one slice?

Check Solution

Ans: B

The radius is 8 inches. Each slice is a sector with an angle of $360^\circ / 8 = 45^\circ$. Area = $(\frac{45}{360})\times\pi(8^2) = 8\pi$

Q.5 A sector has a radius of 10 cm. If the area of the sector is one-fourth of the area of the entire circle, what is the central angle?

Check Solution

Ans: B

If the sector’s area is 1/4 of the circle, then the central angle is 1/4 of 360 degrees, thus $\theta = 90^\circ$.

Next Topic: Area of a Segment