The diagram shows four identical blades of a fan, whose centre is O. Arc AB on the fan blade forms an angle of $\pi/6$ at the centre O. OC is 5 cm and BC is 25 cm.

1. Find the outer radius and apply the arc length formula

Since $O$, $C$ and $B$ are collinear,
$OB=OC+BC=5+25=30\text{ cm}.$
Arc length $AB=r\theta$:
$AB=30\times \frac{\pi}{6}=5\pi\text{ cm}.$

1. Identify the radii and the straight sides of the blade

Outer radius: $OB=OC+BC=5+25=30$ cm.

Inner radius: $OC=5$ cm.

The two straight sides of the blade each have length $25$ cm.

2. Find the inner arc length

Using arc length $=r\theta$,

inner arc length $=5\times \dfrac{\pi}{6}=\dfrac{5\pi}{6}$ cm.

3. Add all boundary lengths to find the perimeter

From part (i), outer arc $AB=5\pi$ cm.

$\text{Perimeter}=5\pi+\frac{5\pi}{6}+25+25$

$=\frac{35\pi}{6}+50\text{ cm}$