Before knowing about a sector of a circle, let’s know how the area of a circle is calculated. When it comes to the area, it is always related to two-dimensions. Anything which is two dimensional can form a plane. So, any two-dimensional figure will have area. What about a circle?

*Definition 1*: A circle is the collection of all the points in a plane which are at a fixed distance from a fixed point. The fixed point is known as the center of the circle and the fixed distance is known as the radius of the circle.

*Definition 2*: If all the points which lie inside and on the circle are taken together, the plane constructed is known as a disk.

A disk is basically the region bounded by a circle. So, the area of a circle will always be that of the disk. The area, A of the circle with radius r is given by

\(A\)

*Definition 3*: The portion of the circle enclosed by two radii and the corresponding arc is known as the sector of a circle.

Basically, a sector is the portion of a circle. It would hence be right to say that a semi-circle or a quarter-circle is a sector of the given circle. In fig.1, *OPAQ* is called the minor sector and *OPBQ* is called the major sector because of lesser and greater areas. The angles subtended by the arcs *PAQ* and *PBQ* are equal to the angle of the sectors *OPAQ* and *OPBQ* respectively. When the angle of the sector is equal to 180°, there is no minor or major sector.

Area of 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.

When 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,*

\(A\)

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 \(\frac{l}{r}\)

\(θ\)

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 = \(\frac{πr^2}{2π}\)

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 = \(\frac{θ}{360°}~×~ πr^2\)

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

= \(\frac{44}{7}\)

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

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

= \(\frac{22}{7}\)

*Example 2:* 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 = \(\frac{lr}{2}\)

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