#### Question

A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take `pi = 22/7` )

#### Solution

Semi-perimeter of circle = πr

_{I1} = π(0.5) = `pi/2 `

I_{2} = π(1) = π cm

I_{3} = π(1.5) = `(3pi)/2 `

Therefore, I_{1}, I_{2}, I_{3} ,i.e. the lengths of the semi-circles are in an A.P.,

`pi/2, pi, (3pi)/2, 2pi ........`

`a = pi/2`

`d = pi - pi/2=pi/2`

*S*_{13} =?

We know that the sum of *n* terms of an a A.P. is given by

`S_n = n/2[2a+(n-1)d]`

`= 13/2[2(pi/2)+(13-1)(pi/2)]`

`= 13/2[pi+(12pi)/2]`

`=(13/2)(7pi)`

`=(91pi)/2`

`= (91xx22)/2xx7 = 13xx11`

= 143

Therefore, the length of such spiral of thirteen consecutive semi-circles will be 143 cm