Question

A spiral is made up of successive semicircles, with centers alternately at $A$ and $B$, starting with center at $A$, of radii $0.5cm,1.0cm,1.5cm,2.0cm$, . . . as shown in Fig. What is the total length of such a spiral made up of thirteen consecutive semicircles

Answer 1

Circumference of first semicircle $=$ $\pi r=0.5\pi$

Circumference of second semicircle $=$ $\pi r=\pi$

Circumference of third semicircle $=$ $\pi r=1.5\pi$

It is clear that $a=0.5\pi$, $d=0.5\pi$ and $n=13$

Hence, length of spiral can be calculated as follows:

$S=\frac{n}{2}[2a+(n-1\left)d\right]$

$=$$\frac{13}{2}(2\times 0.5\pi +12\times 0.5\pi )$

$=$ $\frac{13}{2}\times 7\pi$

$=$ $\frac{13}{2}\times 7\times \frac{22}{7}$

$=143$ cm