WAEC 2023 · Paper 2 · Q11

(a)
Two regular polygons $P$ and $Q$ are such that the number of sides of $P$ is twice the number of sides of $Q$ . The difference between the exterior angles of $Q$ and $P$ is $45^\circ$ . Find the number of sides of $P$ .
(b)
The area of a semi-circle is $32\pi\text{ cm}^2$ . Find, in terms of $\pi$ , the circumference of the semi-circle.

Worked solution (try it first)

(a)

Let $Q$ have $n$ sides, so $P$ has $2n$ .
The exterior angle of a regular polygon is $\frac{360^\circ}{\text{number of sides}}$ : $Q$ 's is $\frac{360}{n}$ and $P$ 's is $\frac{360}{2n} = \frac{180}{n}$ .
Their difference is $45^\circ$ : $\frac{360}{n} - \frac{180}{n} = \frac{180}{n}$
$= 45$ , so $n = 4$ .
$P$ has $2 \times 4 = 8$ sides.

(b)

$\frac12\pi r^2 = 32\pi$ , so $r^2 = 64$ and $r = 8$ cm.
The boundary of the semicircle is the curved half, $\pi r = 8\pi$ , plus the diameter, $2r = 16$ : $(8\pi + 16)$ cm.

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