Area of a Sector
- Introduction
- Area of Sector of a Circle
- Formula for the Area of Sector
- Derivation of Area of Sector Formula
- Real-Life Examples of Area of Sector of Circle
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
A circle is a flat shape that's like a pie, with all its points equally far from the middle. This middle point is called the center of the circle. The sector of a circle is a piece of it, like a slice of pie. It's the part between two lines from the middle to the edge and the curve of the circle. To find how much space is inside a sector, you use a special math formula. This formula works whether you measure the angle of the sector in degrees or in radians.
Area of Sector of a Circle
The area enclosed by a circle's sector is known as its sector area. Think of a pizza slice as a sector representing a portion of the whole pizza. Sectors come in two types: minor and major. A minor sector is smaller than a semicircle, while a major sector is larger than a semi circle. In the provided figure, `OAPB` represents the minor sector, where `∠AOB` is the sector angle. It should be noted that `AQBO` also forms a sector of the circle, called the major sector.
Formula for the Area of Sector
`1`. The formula for finding the area of a sector of a circle when the angle `θ` is in degrees can be expressed as follows:
`A = \frac{\theta}{360^\circ} \times \pi r^2`
where,
- \( A \) is the area of the sector
- \( \theta \) is the angle of the sector in degrees
- \( r \) is the radius of the circle
`2`. The formula for finding the area of a sector of a circle when the angle `θ` is in radians can be expressed as follows:
`A = \frac{\theta}{2} \times r^2`
where,
- \( A \) is the area of the sector
- \( \theta \) is the angle of the sector in radians
- \( r \) is the radius of the circle
Derivation of Area of Sector Formula
`1`. When angle is in degrees:
The angle formed by a full rotation around the center of a circle is \( \theta = 360^\circ\).
Formula for circular area `=` \(\pi r^2 \)
There’s a direct relationship between the central angle and the area.
`frac{text{Area of a sector}}{text{Area of a circle}} = frac{text{Central Angle}}{360^{circ}}`
`{A}/(\pi r^{2}) = {\theta}/{360^\circ}`
`A = \{\theta}/{360^\circ} \times \pi r^{2}`
Example: Find the area of the given sector.
(Use \(\pi = 3.14\)).
Solution:
`r = 5` units
`theta = 60^circ`
`A = {\theta}/{360^\circ} \times \pi r^{2}`
`A = \frac{60^\circ}{360^\circ} \times \pi (5)^{2}`
`A = \frac{1}{6} \times 3.14 \times 25`
`A = 13.083` square units
`2`. When angle is in radians:
The angle formed by a full rotation around the centre of a circle is \( \theta = 2\pi\).
Formula for circular area `=` \(\pi r^2 \)
There’s a direct relationship between the central angle and area.
`frac{text{Area of a sector}}{text{Area of a circle}} = frac{text{Central Angle}}{2pi}`
`frac{A}{pi r^{2}} = frac{theta}{2pi}`
`A = frac{theta}{2pi} times \pi r^{2}`
`A = \frac{\theta}{2} times r^{2}`
Example: Find the area of the sector of the circle.
(Use \(\pi = 3.14\)).
Solution:
`r = 4` units
`\theta = \frac{\pi}{3}`
`A = \frac{\theta}{2} \times r^{2}`
`A = \frac{\pi}{3 \times 2} \times (4)^{2}`
`A = \frac{3.14}{6} \times 16`
`A = 1.046 \times 8`
`A = 8.37` square units
Real-Life Examples of Area of Sector of Circle
Pizza: A common illustration of a sector's area in daily life is evident in pizza slices, which often resemble sectors of a circle.
Clock: Within a clock's circular shape, the minute and hour hands occupy specific sectors of the circle.
Solved Examples
Example `1`. A circle with a diameter of `6` units is divided into `8` equal sectors. Determine the area of each sector of the circle? (Use \(\pi = 3.14\)).
Solution:
Step `1`: First, we need to find the radius of the circle.
The diameter of the circle is `6` units, therefore, the radius of the circle is:
`r = \frac{d}{2}`
`r = \frac{6}{2}`
`r = 3`
Step `2`: A complete angle of a circle `= 360°`, to find the angle of each sector of the circle we divide `360` by `8`.
`\theta = \frac{360^\circ}{8}`
`\theta = 45^\circ`
Step 3: Now let’s calculate the area of the sector.
`r = 3` units
`theta = 45^circ`
`A = frac{theta}{360^circ} times pi r^{2}`
`A = frac{45^circ}{360^circ} times pi (3)^{2}`
`A = frac{1}{8} times 3.14 times 9`
`A = 3.5325` square units
Therefore, the area of each sector of the circle is `3.5325` square units.
Example `2`. The angle of a sector of a circle is `90°`, and the radius of the circle is `10` cm. What is the area of the sector of this circle? (Use \(\pi = 3.14\)).
Solution:
`r = 10` cm
`theta = 90^circ`
`A = frac{theta}{360^circ} times pi r^{2}`
`A = frac{90^circ}{360^circ} times pi (10)^{2}`
`A = frac{1}{4} times 3.14 times 100`
`A = 78.5` square cm
Therefore, the area of each sector of the circle is `78.5` square cm.
Example `3`. A sector has an area of `36π` square units and a radius of `6` units. Find the central angle of the sector in radians.
Solution:
`r = 6` units
`A = 12pi` square units
`A = frac{theta}{2} times r^{2}`
`12pi = frac{theta}{2} times (6)^{2}`
`12pi = frac{theta}{2} times 36`
`frac{pi}{3} = frac{theta}{2}`
`theta = frac{2pi}{3}`
Central angle is `\frac{2\pi}{3}`.
Practice Problems
Q`1`. A sector has an area of `50π` square units and a radius of `5` units. Find the central angle of the sector in radians.
- `5π`
- `2π`
- `4π`
- `π`
Answer: c
Q`2`. Find the area of the sector in terms of `π` if the circle has a radius of `12` units and a central angle of `90` degrees.
- `12π` square units
- `24π` square units
- `36π` square units
- `16π` square units
Answer: c
Q`3`. If the angle of a sector of a circle is `45°`, and the radius of the circle is `8` inches, what is the area of the sector of this circle?
- `20.36` square inches
- `43.21` square inches
- `36.34` square inches
- `25.12` square inches
Answer: d
Q`4`. Find the area of the sector in terms of `π` if the circle has a radius of `6` units and a central angle of `120` degrees.
- `12π` square units
- `6π` square units
- `4π` square units
- `10π` square units
Answer: a
Q`5`. A sector has an area of `48π` square units and a radius of `12` units. Find the central angle of the sector in radians.
- `(2π)/3`
- `π/3`
- `(3π)/4`
- `(5π)/6`
Answer: a
Frequently Asked Questions
Q`1`. How to find the area of a sector?
Answer: We can find the area of the sector using the following formula:
`A = \frac{\theta}{360^\circ} \times \pi r^{2}`
Q`2`. How do you find the central angle of a sector?
Answer: We can find the central angle using the following formula:
`\theta = {A \times 360^\circ}/ {\pi r^{2}}`
Q`3`. How can we find the radius of a sector if its area is known?
Answer: We can find the radius using the following formula:
`r = \sqrt{{A \times 360^\circ}/ {\pi \times \theta}}`
Q`4`. How is the area of a sector related to its central angle?
Answer: The area increases with a larger central angle.
Q`5`. Can the area of a sector be larger than the area of the whole circle?
Answer: No, the area of a sector is always less than or equal to the area of the whole circle.