Shape and Space 9
4 marks · HardA square-based pyramid has a base of side 10 cm. The vertex V is vertically above the centre of the base and is 14 cm above the centre. Explain, step by step, how trigonometry can be used to calculate the angle between a slant edge (V to one corner of the base) and the base. You may use a diagram to support your answer.
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
(See marking scheme and solution.)
Must show: Identifying right triangle with height and half-diagonal, Using trigonometric ratios to find angle
Also accept: Correct structured step-by-step explanation with/without diagram drawn, correct reasoning
Common mistakes: Omitting how to find base diagonal; Missing use of trigonometric ratios; Stating only Pythagoras, not angle
Show mark scheme and worked solution
Marking points
M1 · 1 mark · Identifies and draws the correct vertical right-angled triangle (from vertex, centre of base, to corner of base).
M2 · 1 mark · Finds the distance from the centre to a corner of the base using Pythagoras’ theorem.
M3 · 1 mark · States the trigonometric ratio needed (e.g., ) to find the angle between the slant edge and the base.
A1 · 1 mark · Shows how to use the inverse tangent (or equivalent) to obtain the angle.
Worked solution
Step 1: Draw a right-angled triangle formed by the pyramid’s height (14 cm), the slant edge from the vertex to a base corner (unknown), and the line from the centre of the base to the corner.
Step 2: The base is 10 cm square. The distance from the centre to any corner is cm.
Step 3: Recognise that the slant edge, the height, and the centre-to-corner distance form a right-angled triangle, with the slant edge as the hypotenuse.
Step 4: The angle between the slant edge and the base (θ) is found using:
Step 5: Use the inverse tangent:
This gives the required angle.