WAEC 2014 · Paper 2 · Q5

The diagram shows a right pyramid with a rectangular base $WXYZ$ and vertex $O$ . If $|WX| = 8\text{ cm}$ , $|ZW| = 6\text{ cm}$ and $|OX| = 13\text{ cm}$ , calculate the:

Not to scale.

(a)
height of the pyramid;
(b)
value of $\angle OXZ$ , correct to the nearest degree;
(c)
volume of the pyramid.

Worked solution (try it first)

(a)

The vertex $O$ is directly above the centre $M$ of the base, where the diagonals cross.
The diagonal $|ZX| = \sqrt{8^2 + 6^2} = 10$ cm, so $|MX| = 5$ cm.
Triangle $OMX$ is right-angled at $M$ : height $|OM| = \sqrt{13^2 - 5^2}$
$= \sqrt{144}$
$= 12\text{ cm}$ .

(b)

$\angle OXZ$ is the angle at $X$ in triangle $OMX$ .
$MX = 5$ is adjacent and $OX = 13$ is the hypotenuse: $\cos\angle OXZ = \frac{5}{13}$
$\approx 0.3846$ .
So $\angle OXZ \approx 67^\circ$ .

(c)

Volume of a pyramid $= \frac13 \times \text{base area} \times \text{height}$
$= \frac13 \times (8 \times 6) \times 12$
$= 192\text{ cm}^3$ .

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