Find the length of the arc in terms of π that subtends an angle of 30∘ at the centre of a circle of radius 4 cm.

Solution:

Consider the circle of radius ‘r’ below showing a sector containing an angle θ at the center. 

The arc formed by the segments OA and OB is S and the angle it subtends at the centre is θ. The angle  θ is given by the relationship:

Θ  =  S/r

Where r is the radius of the circle and is represented by the segments OA or OB. Hence, arc length

S = rΘ

Where Θ is typically in radians. In the given problem Θ = 30° and r = 4 cm

180° =   π radians

Hence 30°, when expressed in radians, will be equal to :

 π  × 30°/180°  = π × 1/6  = π/6 radians

Therefore the length of the arc S will be :

S = 4 × π/6 = 2π/3 cm-radians

Find the length of the arc in terms of π that subtends an angle of 30∘ at the centre of a circle of radius 4 cm.

Summary:

 The length of the arc in terms of π that subtends an angle of 30∘ at the centre of a circle of radius 4 cm is 2π/3 cm-radians