Find, in radians, the value of the angle $\theta$, as indicated on the diagram.

Find the total length of the perimeter of the logo.

METHOD 1

${\text{area}} = \pi {2^2} - \frac{1}{2}{2^2}\theta \;\;\;( = 3\pi )$     M1A1

Note:     Award M1 for using area formula.

$\Rightarrow 2\theta  = \pi  \Rightarrow \theta  = \frac{\pi }{2}$     A1

Note:     Degrees loses final A1

METHOD 2

let $x = 2\pi  - \theta$

${\text{area}} = \frac{1}{2}{2^2}x\;\;\;( = 3\pi )$     M1

$\Rightarrow x = \frac{3}{2}\pi$     A1

$\Rightarrow \theta  = \frac{\pi }{2}$     A1

METHOD 3

Area of circle is $4\pi$     A1

Shaded area is $\frac{3}{4}$ of the circle     (R1)

$\Rightarrow \theta  = \frac{\pi }{2}$     A1

[3 marks]

${\text{arc length}} = 2\frac{{3\pi }}{2}$     A1

${\text{perimeter}} = 2\frac{{3\pi }}{2} + 2 \times 2$

$= 3\pi  + 4$     A1

[2 marks]

Total [5 marks]

Good methods. Some candidates found the larger angle.

Generally good, some forgot the radii.