A cuboid has a rectangular base measuring 6 cm by 8 cm and a height of 10 cm. Calculate the angle between a space diagonal of the cuboid and the base.

The angle between a space diagonal and the base is measured between the diagonal itself and its projection onto the base, which is the diagonal of the rectangular base. Finding it takes two stages.

1. Base diagonal \(=\sqrt{6^{2}+8^{2}}\) [M1] \(=\sqrt{36+64}=\sqrt{100}=10\) cm [A1], using the 6, 8, 10 right-angled triangle.
2. The space diagonal, the base diagonal and the vertical edge form a right-angled triangle, with the right angle where the vertical edge meets the base. The vertical edge of 10 cm is opposite the required angle \(\theta\), and the base diagonal of 10 cm is adjacent to it, so \(\tan\theta=\dfrac{10}{10}\) [M1].
3. \(\tan\theta=1\), so \(\theta=45^{\circ}\) [A1] cao.

The result is exact and needs no calculator, because a tangent of 1 corresponds to an isosceles right-angled triangle in which the opposite and adjacent sides are equal. That is exactly the situation here, since the height happens to equal the base diagonal.

The step most often mishandled is choosing which length is adjacent. It is not one of the base edges, 6 cm or 8 cm, but the base diagonal, because that is the line directly beneath the space diagonal.