Shape and Space 9

2 marks · Hard

In the prism shown, the base is a rectangle measuring 6 cm by 8 cm, and the height (the perpendicular distance between the two rectangular faces) is 5 cm. Work out, to 3 significant figures, the angle that the longest diagonal of the prism makes with the base.

Rectangular prism: Rectangle base ABCD with AB = 8 cm, BC = 6 cm. Height (vertical distance from base to top face) = 5 cm. Diagonal AE from A on base to E on the top face, above D.

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32.0° degrees (±0.5)

Also accept: 31.9°, 32°, 32.0 degrees

Common mistakes: Using only 2D Pythagoras; Finding angle with wrong plane

Show mark scheme and worked solution

Marking points

M1 · 1 mark · Find the length of the diagonal of the base using Pythagoras’ theorem.

A1 · 1 mark · Use trigonometry to find the angle with correct answer to 3 s.f.

Worked solution

First, find the base diagonal AC.

The longest diagonal AE runs from A on the base to the top corner E above D (5 cm vertically above D, since the height is 5 cm). Triangle ACE is right-angled with AC = 10 cm (on the base) and height 5 cm.

Let θ = angle between AE and the base (at vertex A).

But to find the angle with the base (the horizontal), we use the angle AE makes with AC or with the base plane:

The required answer (angle with the base) is 26.6°, to 3 significant figures.