Three-dimensional Pythagoras and trigonometry

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 3 marks
A cuboid measures 3 cm by 4 cm by 12 cm.
(a) Work out the length of the longest straight rod that fits inside the cuboid. (3)
Question 2Calculator · 5 marks
A cuboid has horizontal base 6 cm by 8 cm and vertical height 24 cm.
(a) Find its space diagonal. (3)
(b) Find the angle between the space diagonal and the base, to 1 decimal place. (2)
Question 3Calculator · 4 marks
(a) A right pyramid has a rectangular base 10 cm by 16 cm. Its apex is vertically above the centre of the base. Its volume is 640 cm³. Work out its total surface area, including the base. Give your answer to 3 significant figures. Volume of a pyramid $= \frac{1}{3} \times$ area of base $\times$ perpendicular height You must show your working. (4)

Answers and marks

(a) 13 cm

(a) $26$ cm

(b) $67.4$ °

P1 In the vertical cross-section, opposite is height and adjacent is base diagonal.
A1 Correct answer: $67.4$ °

(a) $512$ cm²

Three-dimensional Pythagoras and trigonometry

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