Shape and Space 9

5 marks · Hard

The diagram shows a cuboid with length 12 cm, width 7 cm, and height 20 cm.

Cuboid: Length = 12 cm (from A to B), width = 7 cm (from B to C), height = 20 cm (from A to E vertically above A).

(a) Calculate the length of the diagonal from A to the opposite vertex G (the space diagonal).

(b) Calculate the angle between the diagonal AG and the base ABCD. Give your answer to 3 significant figures.

(a)

Calculate the length of the space diagonal AG.

[2 marks]

(b)

Calculate the angle between AG and the base ABCD. Give your answer to 3 significant figures.

[3 marks]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

(a) 23.3 cm

(b) 60.1° degrees

Common mistakes: Using wrong dimensions in diagonal calculation; Incorrect triangle for angle

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Identifies and uses the 3D Pythagoras’ theorem for space diagonal.

A1 (a) · 1 mark · Obtains correct value for diagonal (to 3 s.f.).

M2 (b) · 1 mark · Finds the length of relevant right triangle in the base (e.g., diagonal of the base).

M3 (b) · 1 mark · States or uses correct trigonometric ratio for angle calculation.

A2 (b) · 1 mark · Correct angle (to 3 s.f.).

Worked solution

(a) Use 3D Pythagoras for AG:

(b) Find the length of the base diagonal AC:

Now form a right-angled triangle with AG (hypotenuse), vertical height (20 cm), and AC (side on base):