## What Is the Area of a Sector?

**The area of a sector is the area of the region enclosed by an arc and two radii of a ****circle****. It represents a part of the area of a circle. Area of a sector is measured in square units, depending on the unit of the radius.**

**What Is the Sector of a Circle?**

A sector is a part of a circle made of the arc of the circle along with its two radii. In the given diagram, the yellow region represents the sector of the circle. It is formed by two radii and an arc. The angle is the angle made by the sector at the center.

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## Area of a Sector: Formulas

1) The formula to calculate the **area of a sector** **of a circle** when θ is in degrees is given by:

**Area of a sector **$= \frac{θ}{360^{\circ}} \times \pi r^{2}$

where:

- θ is the angle of the sector in degrees (angle subtended by the arc at the center)
- r is the radius of the circle

*(Note: In the above diagram, the term ‘degrees’ simply signifies that this formula for the area of a sector is to be used when the central angle is given in degrees.)*

2) The formula to calculate the **area of a sector** **of a circle** when θ is in radians is given by:

**Area of a sector **$=(\frac{θ}{2}) \times r^{2}$

where:

- θ is the angle of the sector in radians
- r is the radius of the circle

*(Note: In the above diagram, the term ‘radians’ simply signifies that this formula for the area of a sector is to be used when the central angle is in radians.)*

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## How to Calculate the Area of a Sector

Consider these steps to determine a sector’s area:

**Step 1:** Note down the radius and the central angle () of the sector.

**Step 2: **If the angle is in degrees, then substitute the values in the formula

Area $= (\frac{θ}{360^{\circ}}) \times \pi r^{2}$

**Step 3: **If the angle is in radians, then substitute the values in the formula

Area $= (\frac{θ}{2}) \times r^{2}$

**Step 4:** The area is measured in square units. Assign the appropriate unit.

## Area of a Sector Formula: Derivation

**When angle is in degrees:**

The angle made by a complete circle around the center is $θ = 360^{\circ}$**.**

Area of circle with radius $r = \pi r^{2}$

There’s a direct relationship between the central angle and area.

Central angle () | Area |
---|---|

$360^{\circ}$ | $\pi r^{2}$ |

θ | $A = ?$ |

$\frac{Area\; of \;a\; sector}{Area \;of\; a \;circle} = \frac{Central \;Angle}{360^{\circ}}$

$\frac{A}{ \pi r^{2}} = 360^{\circ}$

$A = \frac{θ}{360^{\circ}} \times \pi r^{2}$

**When angle is in radians:**

The angle made by a complete circle around the center is $θ = 2\pi$**.**

Area of circle with radius $r = \pi r^{2}$

Central angle () | Area |
---|---|

$2\pi$ | $\pi r^{2}$ |

θ | A = ? |

$\frac{Area \;of \;a \;sector}{Area\; of \;a \;circle} = \frac{Central\; Angle}{2\pi}$

$\frac{A}{\pi r^{2}} = \frac{θ}{2\pi}$

$A = \frac{θ}{2\pi} \times \pi r^{2}$

$A = \frac{θ}{2} \times r^{2}$

## Area of a Sector in Degrees

The area of a sector when the central angle (θ) is expressed in degrees, is given by

$Area = (\frac{θ}{360^{\circ}}) \times \pi r^{2}$

**Example: Find the area of the given sector.**

$r = 5$ units

$= 45^{\circ}$

Area $= (\frac{θ}{360^{\circ}}) \times \pi r^{2}$

Area $= (\frac{45^{\circ}}{360^{\circ}}) \times \pi (5)^{2}$

Area $= \frac{1}{8} \times 3.14 \times 25$

Area $= 9.8125$** **square units

## Area of a Sector in Radians

The area of a sector when the central angle (θ) is expressed in degrees, is given by

Area** **$= (\frac{θ}{2}) \times r^{2}$

**Example: Find the area of the sector of the circle. (Use **$\pi = 3.14$**).**

Area** **$= (\frac{\pi}{4 \times 2}) \times (7)^{2}$

Area** **$= (\frac{3.14}{4 \times 2}) \times (7)^{2}$

Area** **= 19.2325 square units

## Facts about Area of a Sector

- The area of a sector is directly proportional to the central angle. The area of a sector increases as the central angle increases.
- The area of a sector is always proportional to the square of the radius.
- The maximum area a sector can have is when the central angle is 360 degrees, which corresponds to the entire circle. In this case, the area of the sector is equal to the area of the whole circle. Thus, a circle is a sector with a central angle of 360 degrees and area of
- $\pi r^{2}$
**.** - A semicircle is a sector with a central angle of 180 degrees. Its area if half the area of a circle, given by $\frac{\pi r^{2}}{2}$.

## Conclusion

In this article, we learned about the area of a sector, its formulas, and also the derivation of the formulas. Let’s solve a few examples and MCQs for practice.

## Solved Examples on Area of a Sector

**Example 1: Find the area of the sector in terms of **π** if the circle has a radius of 8 units and a central angle of 45 degrees.**

**Solution: **

Using the formula for sector area, we write

Area** **$= \frac{θ}{360^{\circ}} \times \pi r^{2}$

Here, $r = 8$ units, $θ =45^{\circ}$

Area $= \frac{45^{\circ}}{360^{\circ}} \times π \times 8^{2}$

Area $= (\frac{1}{8}) \times π \times 64$

Area $= 8π$ square units

**Example 2: A sector has a radius of 12 units and a central angle of 90 degrees. Calculate the area of the sector.**

**Solution: **

Area** **$= \frac{θ}{360^{\circ}} \times \pi r^{2}$

Here, $r = 12$ units, $θ = 90^{\circ}$

Area $= \frac{90^{\circ}}{360^{\circ}} \times π \times 12^{2}$

Area $= \frac{1}{4} \times π \times 144$

Area $= 36π$

Area = 113.04 square units

**Example 3: A sector has an area of 16π square units and a radius of 4 units. Find the central angle of the sector in radians.**

**Solution: **

$r = 4$ units

Area of sector $= 16π$ square units

Area of sector $= (\frac{θ}{2}) \times r^{\circ}$

$16π = (\frac{θ}{2}) \times (4)^{2}$

$16π = (\frac{θ}{2}) \times 16$

\frac{θ}{2} = π$

$θ = 2π$

Central angle is 2π. It means that the given area is the area of the whole circle with radius 4 units.

**Example 4: Find the area of a sector with a central angle of 60 degrees and a radius of 10 units.**

**Solution: **

Area $= (\frac{θ}{360^{\circ}}) \times \pi r^{2}$

Area $= (\frac{60^{\circ}}{360^{\circ}}) \times π \times 10^{2}$

Area $= (\frac{1}{6}) \times π \times 100$

Area $= 52.33$ square units

## Practice Problems on Area of a Sector

## Area of Sector of a Circle - Definition, Formula, Examples, FAQs

### A sector has a radius of 12 units and a central angle of 60 degrees. What is the area of the sector?

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

Area $= \frac{60^{\circ}}{360^{\circ}} \times \pi \times 144$

Area $= 24π$ square units

### The area of a sector when the central angle is given in degrees is given by

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

### A sector has an area of 8π square units and a radius of 4 units. What is the central angle of the sector?

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

$8π = \frac{\theta}{360^{\circ}} \times \pi \times 16$

$\frac{1}{2} = \frac{\theta}{360^{\circ}}$

$\theta = 180^{\circ}$

### The area of a quarter circle is ___ times the area of a circle.

The area of a quarter circle is $\frac{1}{4}$ times the area of a circle.

## Frequently Asked Questions on Area of a Sector

**What distinguishes the areas of a sector and a circle, respectively?**

A sector’s area is a fraction of the circle’s overall area. It is the area that the circle’s arc and two radii enclose. On the other hand, the term “**area of a sector of a circle**” describes the entire region that the circle’s circumference encloses.

**Is it possible for a sector’s size to exceed that of the complete circle?**

No, a sector cannot have a larger area than the complete circle. Always a piece or proportion of the entire circle, a sector. As a result, its surface area is never greater than or equal to the circles.

**How is the formula for a sector’s area derived?**

The **area of sector formula** is produced by considering the percentage of the circle’s angle that it encloses. It is calculated by dividing the angle’s fraction by the circle’s whole area. This relationship is denoted by the formula $A_{sec} = (\frac{\theta}{360})\times \pi \times r^{2}$, where the sector’s central angle and r is the circle’s diameter.

**Can a sector’s area be negative?**

A sector’s area cannot be negative. A region’s area defines its size, which is always a positive number. Area is a non-negative quantity.