A circle has always been an important shape among all geometrical figures. There are various concepts and formulas related to a circle. The sectors and segments are perhaps most useful of them. In this article, we shall focus on the concept of a sector of a circle, especially on calculating the perimeter of a sector.

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Let us first understand the terminologies related to the sector of a circle.

A sector is said to be a part of a circle made of the arc of the circle along with its two radii. It is a portion of the circle formed by a portion of the circumference (arc) and radii of the circle at both endpoints of the arc as shown in the following diagram.

The shape of a sector of a circle can be compared with a slice of pizza or a pie.

The perimeter of something is said to be the length of the boundary. In the same way, the perimeter of sector the represents the measurement of its boundary. The perimeter of the sector of a circle is the length of two radii along with the arc that makes the sector. In the following diagram, a sector is shown by white color.

The perimeter should be calculated by doubling the radius and then adding it to the length of the arc.

The formula for the perimeter of the sector of a circle is given below :

Perimeter of sector = radius + radius + arc length

Perimeter of sector = 2 radius + arc length

Arc length is calculated using the relation :

Arc length = radius $\times$ central angle

Therefore

**Perimeter of sector = 2 Radius + (radius $\times$ central angle)**In the form of symbols, this formula is represented more conveniently as under :

**P = 2 r + r $\theta$**
Let us find the Perimeter of Sector of a Circle With Central Angle:

In order to find the perimeter of a sector, the central angle must be known. This is an angle inclined on the center by a given arc of a circle. Central angle in surrounded by two radii and an arc. It is an angle at the center of the sector as demonstrated in the picture below.

The calculation of finding the perimeter of a sector of a circle using central angle can be done in following steps.

(i) Determine radius of a circle, denotes it as r.

(ii) Determine central angle $\theta$.

(ii) Calculate value of the length of the arc by finding the product of central angle and radius. Remember that central angle must be in radians. If it is given in degrees, convert it to radians by multiplying the degree value with $\frac{\pi}{180}$.

(iii) Double the radius and add it in arc length. This gives required perimeter.
The examples based on the perimeter of a sector of a circle are given below.

**Example 1:** A circular arc whose radius is 12 cm, makes an angle of 30$^{\circ}$ at center. Find the perimeter of sector formed. Use $\pi$ = 3.14

Solution : Given that r = 12 cm,

$\theta$ = $30^{\circ}$ = $30^{\circ}\ \times$ $\frac{\pi}{180^{\circ}}$ = $\frac{\pi}{6}$

P = 2 r + r $\theta$

P = 2 . 12 + 12 .$\frac{\pi}{6}$

P = 24 + 2 $\pi$

P = 24 + 6.28 = 30.28

Perimeter of sector is 30.28 cm**Example 2 :** Find the perimeter of a sector of a quarter circle of radius 7 cm. Use $\pi$ = $\frac{22}{7}$.

Solution : A quarter circle subtends a right angle at the center of a circle.

Thus, r = 7 cm, $\theta$ = $90^{\circ}$ = $90^{\circ} \times$ $\frac{\pi}{180^{\circ}}$ = $\frac{\pi}{2}$

P = 2 r + r $\theta$

P = 2 . 7 + 7 .$\frac{\pi}{2}$

P = 14 + 7 .$\frac{22}{7.2}$

P = 14 + 11 = 25

The Perimeter of the sector is 25 cm.

A sector is said to be a part of a circle made of the arc of the circle along with its two radii. It is a portion of the circle formed by a portion of the circumference (arc) and radii of the circle at both endpoints of the arc as shown in the following diagram.

The shape of a sector of a circle can be compared with a slice of pizza or a pie.

The perimeter of something is said to be the length of the boundary. In the same way, the perimeter of sector the represents the measurement of its boundary. The perimeter of the sector of a circle is the length of two radii along with the arc that makes the sector. In the following diagram, a sector is shown by white color.

The perimeter should be calculated by doubling the radius and then adding it to the length of the arc.

The formula for the perimeter of the sector of a circle is given below :

Perimeter of sector = radius + radius + arc length

Perimeter of sector = 2 radius + arc length

Arc length is calculated using the relation :

Arc length = radius $\times$ central angle

Therefore

In order to find the perimeter of a sector, the central angle must be known. This is an angle inclined on the center by a given arc of a circle. Central angle in surrounded by two radii and an arc. It is an angle at the center of the sector as demonstrated in the picture below.

(i)

(ii)

(ii)

(iii)

Solution :

$\theta$ = $30^{\circ}$ = $30^{\circ}\ \times$ $\frac{\pi}{180^{\circ}}$ = $\frac{\pi}{6}$

Perimeter of sector is given by the formula

P = 2 r + r $\theta$

P = 2 . 12 + 12 .$\frac{\pi}{6}$

P = 24 + 2 $\pi$

P = 24 + 6.28 = 30.28

Perimeter of sector is 30.28 cm

Solution :

Thus, r = 7 cm, $\theta$ = $90^{\circ}$ = $90^{\circ} \times$ $\frac{\pi}{180^{\circ}}$ = $\frac{\pi}{2}$

P = 2 r + r $\theta$

P = 2 . 7 + 7 .$\frac{\pi}{2}$

P = 14 + 7 .$\frac{22}{7.2}$

P = 14 + 11 = 25

The Perimeter of the sector is 25 cm.

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