**Arc measure and arc length :**

An arc measure is the measure of an angle that the arc creates in the center of a circle. This measure can be given in degrees.

Arc length of a sector POQ is the length of the portion on the circumference of the circle intercepted between the bounding radii (OP and OQ) and is denoted by l.

** Length of arc = (θ/360) x 2Πr**

**Example 1 :**

The radius of a sector is 42 cm and its sector angle is 60°. Find its arc length

**Solution :**

**Radius of the sector = 42 cm**

**Sector angle **θ = 60°

To find the arc length, we need to use the formula

l = (θ/360) x 2Πr

= (60/360) x 2 x (22/7) x 42

= (1/6) x 2 x 22 x 6

= 44 cm

**Example 2 :**

The arc length of a sector is 66 cm and the central angle is 30°. Find its radius.

**Solution :**

**Arc length of sector = 66 cm**

**Sector angle **θ = 30°

l = (θ/360) x 2Πr

Applying the values of arc length and central angle in the formula, we get

66 = (30/360) x 2 x (22/7) x r

66 = (1/12) x 2 x 22 x r

r = (66 x 12)/(2 x 22)

= 126 cm

**Example 3 :**

The length of arc of a sector is 22 cm and its radius is 10.5 cm. Find its central angle.

**Solution :**

**Arc length of sector = 22 cm**

**radius = 10.5 cm **

l = (θ/360) x 2Πr

Applying the values of arc length and central angle in the formula, we get

22 = (θ/360) x 2 x (22/7) x 10.5

22 = (θ/360) x 2 x 22 x 1.5

θ = (22 x 360)/(2 x 22 x 1.5)

= 180/1.5

θ = 120°

Hence the required central angle is 120°.

**Example 4 :**

A pendulum swings through an angle of 30° and describes an arc length of 11 cm. Find the length of the pendulum.

**Solution :**

**Arc length of sector = 11 cm**

**sector angle = 30**°

If the pendulum swings once, then it forms a sector and the radius of the sector is the length of the pendulum. So,

l = (θ/360) x 2Πr

Applying the values of arc length and central angle in the formula, we get

11 = (30/360) x 2 x (22/7) x r

11 = (1/12) x 2 x (22/7) x r

r = (11 x 7 x 12)/(2 x 22)

r = 7 x 3

r = 21 cm

Hence the length of pendulum is 21 cm

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