Formula to find perimeter of the sector is

**= l + 2r**

where 'l' is the length of the minor arc AB.

Formula to find length of the arc is

l = θ/360° ⋅ 2∏r

Formula to find area of sector is

A = θ/360° ⋅ ∏r^{2 }square units

or

A = rl / 2 square units

**Example 1 :**

Find the perimeter of the sector PQR shown below.

**Solution : **

Length of the arc is

l = θ/360° ⋅ 2∏r

Substitute θ = 60°, r = 42 and ∏ = 22/7.

l = 60°/360° ⋅ 2 ⋅ 22/7 ⋅ 42

l = 1/6 ⋅ 2 ⋅ 22 ⋅ 6

l = 1 ⋅ 2 ⋅ 22

l = 44 cm

Perimeter of sector is

= l + 2r

Substitute l = 44 and r = 42.

= 44 + 2(42)

= 44 + 84

= 128 cm

So, length of the arc is 128 cm.

**Example 2 :**

Find the perimeter of the sector AOB shown below.

**Solution : **

Length of the arc is

l = θ/360° ⋅ 2∏r

Substitute θ = 120°, r = 21 and ∏ = 22/7.

l = 120°/360° ⋅ 2 ⋅ 22/7 ⋅ 21

l = 1/3 ⋅ 2 ⋅ 22 ⋅ 3

l = 1 ⋅ 2 ⋅ 22

l = 44 cm

Perimeter of sector is

= l + 2r

Substitute l = 44 and r = 21.

= 44 + 2(21)

= 44 + 42

= 86 cm

So, length of the arc is 86 cm.

**Example 3 :**

Find the perimeter of sector whose area is 324 square cm and the radius is 27 cm.

**Solution :**

Area of sector = 324 cm^{2}

rl / 2 = 324

Multiply each side by 2.

rl = 648

Substitute 27 for r.

27l = 648

Divide each side by 27.

l = 24 cm

Perimeter of sector is

= l + 2r

= 24 + 2(27)

= 24 + 54

= 78 cm

**Example 4 :**

Find the length of arc, if the perimeter of sector is 45 cm and radius is 10 cm.

**Solution :**

Perimeter of sector = 45 cm

l + 2r = 45

Substitute 10 for r.

l + 2 (10) = 45

l + 20 = 45

l = 45 - 10

l = 35 cm

**Example 5 :**

Find the radius of sector whose perimeter of the sector is 30 cm and length of the arc is 16 cm.

**Solution :**

Perimeter of sector = 30 cm

l + 2r = 30

Substitute 16 for l.

16 + 2r = 30

Subtract 16 from each side.

2r = 14

Divide each side by 2.

r = 7 cm

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