# Sector Of A Circle

What is Circle?

A circle is a locus of points equidistant from a given point located at the center of the circle. The common distance from the center of the circle to its point is called radius.

Thus, the circle is defined by its center (o) and radius (r)

What is Sector of a circle?

A circular sector is the portion of a disk enclosed by two radii and an arc.

A sector divides the circle into two regions namely Major and Minor Sector.

The smaller area is known as Minor Sector whereas the region having Greater area is known as Major Sector.

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Area of a sector :

In a circle with radius r and center at O, let ∠POQ = θ (in degrees) be the angle of the sector. Then, the area of a sector of circle formula is calculated using the unitary method.

For the given angle the Area of a sector is represented by:

Angle of the sector is 360°, area of the sector i.e. the Whole circle = $πr^2$

When the Angle is 1°, area of sector = $\frac{πr^2}{360°}$

So, when the angle is θ, area of sector, OPAQ,

$A$ = $\frac{θ}{360°}~×~ πr^2$

Let the angle be 45 therefore the circle will be divided into 8 parts

Now the area of the sector for the above figure can be calculated as ⅛(3.14*r*r)

Thus the Area of a sector is calculated as  $\frac{\theta}{360} \times \frac{22}{7} \times r^{2}$.

Similarly, length of the arc (PQ) of the sector with angle θ,

$l$ = $\frac{θ}{360°}~×~ 2πr$

If the length of the arc of the sector is given instead of the angle of the sector, there is a different way to calculate the area of the sector. Let the length of the arc be l. For the radius of a circle equal to r units, an arc of length r units will subtend 1 radian at the center. It can be hence concluded that an arc of length l will subtend $\frac{l}{r}$ angle at the center. So, if l is the length of the arc, r is the radius of circle and θ is the angle subtended at center,

$θ$ = $\frac{l}{r}$, where θ is in radians

When angle of the sector is 2π, area of the sector i.e. the whole circle = $πr^2$

When the angle is 1, area of the sector = $\frac{πr^2}{2π}$ = $\frac{r^2}{2}$

So, when the angle is θ, area of the sector = $θ~×~\frac{r^2}{2}$

= $\frac{l}{r}~×~\frac{r^2}{2}$

= $\frac{lr}{2}$

Some examples for better understanding are discussed from here on.

Example 1: If the angle of the sector with radius 4 units is 45°, area = $\frac{θ}{360°}~×~ πr^2$

= $\frac{45°}{360°}~×~\frac{22}{7}~×~4~×~4$

= $\frac{44}{7}$ square units

The length of the same sector = $\frac{θ}{360°}~×~ 2πr$

= $\frac{45°}{360°}~×~2~×~\frac{22}{7}~×~4$

= $\frac{22}{7}$ units

Example 2:Find the area of the sector when radius of the circle is 16 units and length of the arc is 5 units.

Solution: If the length of the arc of a circle with radius 16 units is 5 units, the area of the sector corresponding to that arc = $\frac{lr}{2}$ = $\frac{5~×~16}{2}$ = $40$ square units.

#### Practise This Question

Find the area of the given figure.