Perimeter and Area of Sector

A part of a circular region enclosed between two radii of the corresponding circle is called a sector of the circle.

The shaded region OAPB is a sector of the circle with centre O. ∠AOB is called the central angle or simply the angle of the sector.

APB is the corresponding arc of this sector. The part OAQB (unshaded region) is also a sector of this circle. OAPB is called the minor sector and OAQB is called the major sector of the circle (with major arc AQB).

Perimeter of Sector

The perimeter of the sector OAPB is equal to OA + OB + length of arc APB.

Let radius OA (or OB) be r, length of the arc APB be l and ∠AOB be θ.

You can find the length l of the arc APB as follows:

Circumference of the circle = 2πr

Now, for total angle 360° at the centre, length = 2πr

So, for angle θ,

$$l = \frac{2 \pi r}{360} \times \theta$$

Area of Sector

Area of the circle = πr2

Now, for total angle 360°, area = πr2

So, for angle θ,

$$\text{area} = \frac{\pi r^2}{360} \times \theta$$

Example 1: Find the perimeter and area of the sector of a circle of radius 6 cm and length of the arc of the sector as 22 cm.

Perimeter of the sector = 2r + length of the arc

= (2 × 6 + 22) cm = 34 cm

For area, first find the central angle θ.

πrθ/180° = 22

θ = 210°

So, area of the sector = πr2θ/360°

= 66 cm2