3D Pythagoras and Trigonometry - Worksheets, Questions and Revision

18 original exam-style questions with a full mark scheme - free printable PDF.

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GCSE · Geometry - 3D Pythagoras and Trigonometry

G6 3D Pythagoras and Trigonometry

AQA 8365 · Calculator allowed · about 100 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A cuboid has length a, width b and height c. Which of the following gives the length of the space diagonal connecting two opposite vertices of the cuboid?
  • A) √a2 + b2
  • B) √a2 + b2 + c2
  • C) a + b + c
  • D) √a x b x c
(Total for Question 1 is 1 mark)
2
A cone has base radius r and slant height l. Which of the following gives the vertical height h of the cone?
  • A) √l2 + r2
  • B) √l2 - r2
  • C) √r2 - l2
  • D) l - r
(Total for Question 2 is 1 mark)
3
A cuboid has length 3 cm, width 4 cm and height 12 cm. Calculate the length of the space diagonal connecting two opposite vertices of the cuboid.
(Total for Question 3 is 3 marks)
4
A cuboid ABCDEFGH has a rectangular base ABCD with AB = 5 cm and BC = 12 cm. The height of the cuboid, AE, is 9 cm.
(a)Find the length of the diagonal AC of the base.(2)
(b)Hence find the length of the space diagonal AG, giving your answer correct to 3 significant figures.(2)
(Total for Question 4 is 4 marks)
5
A cuboid has a rectangular base with length 8 cm and width 6 cm, and height 5 cm. AC is a diagonal of the base and AG is the space diagonal from the same base vertex A to the opposite top vertex G.
(a)Find the length of AC.(2)
(b)Hence find the angle between AG and the base plane, giving your answer correct to 1 decimal place.(2)
(Total for Question 5 is 4 marks)
6
A vertical flagpole DF stands at the corner D of a horizontal rectangular courtyard ABCD, where AB = 18 m and AD = 24 m. The angle of elevation of the top of the flagpole, F, from corner B, is 32 degrees.
(a)Find the length BD (the diagonal of the courtyard).(2)
(b)Hence calculate the height of the flagpole, DF, giving your answer correct to 3 significant figures.(3)
(Total for Question 6 is 5 marks)
7
A square-based pyramid has a square base ABCD with side length 10 cm. The apex, V, is directly above the centre of the base, M. Each sloping edge VA, VB, VC and VD has length 13 cm.
(a)Find the length of the diagonal AC of the base, giving your answer in exact surd form.(2)
(b)Hence find the vertical height VM of the pyramid, giving your answer correct to 3 significant figures.(3)
(Total for Question 7 is 5 marks)
8
D is a point vertically above A, with DA = 15 cm perpendicular to the horizontal plane containing triangle ABC. In triangle ABC, AB = 7 cm, AC = 9 cm and angle BAC = 65 degrees.
(a)Use the cosine rule to find the length BC, giving your answer correct to 3 significant figures.(3)
(b)Calculate the length DC, giving your answer correct to 3 significant figures.(3)
(Total for Question 8 is 6 marks)
9
A tent has the shape of a triangular prism. The cross-section is an isosceles triangle with base BC = 6 m and each sloping edge AB = AC = 5 m, where A is the ridge point directly above the midpoint of BC.
(a)Find the perpendicular height of the ridge A above the base BC.(3)
(b)Calculate the angle each roof panel makes with the horizontal ground, giving your answer correct to 1 decimal place.(3)
(Total for Question 9 is 6 marks)
10
A cone has base radius 6 cm and vertical height 8 cm.
(a)Find the slant height of the cone.(2)
(b)Find the semi-vertical angle of the cone (the angle between the slant height and the vertical axis), giving your answer correct to 1 decimal place.(2)
(c)The cone is cut by a plane parallel to the base, at a height of 4 cm from the base, creating a smaller cone on top. Find the radius of the circular cross-section where the cut is made.(2)
(Total for Question 10 is 6 marks)
11
A cuboid has length 9 cm, width 12 cm and height 8 cm. ABCD is the base, with AB = 9 cm and BC = 12 cm, and EFGH is the top face directly above, with E above A, F above B, G above C and H above D. AG is the space diagonal from base vertex A to the opposite top vertex G.
(a)Find the length of the diagonal AC of the base.(2)
(b)Find the length of the space diagonal AG.(2)
(c)Calculate the angle between AG and the base plane ABCD, giving your answer correct to 1 decimal place.(2)
(d)Calculate the angle between AG and the vertical edge AE, giving your answer correct to 1 decimal place.(2)
(Total for Question 11 is 8 marks)
12
A cuboid has length a, width b and height c. Show that the length, d, of the space diagonal connecting two opposite vertices is given by d = √a2 + b2 + c2.
(Total for Question 12 is 3 marks)
13
A pyramid has a triangular base PQR, with PQ = 11 cm, QR = 14 cm, and angle PQR = 78 degrees. The apex, V, is positioned such that VQ is perpendicular to the base plane PQR, with VQ = 9 cm.
(a)Use the cosine rule to find the length PR, giving your answer correct to 3 significant figures.(3)
(b)Calculate the length VP, giving your answer correct to 3 significant figures.(2)
(Total for Question 13 is 5 marks)
14
A cuboid has a square base ABCD with side length 6 cm and height 8 cm. Points E, F, G, H lie directly above A, B, C, D respectively. M is the centre of the top face EFGH (the point where the diagonals EG and FH cross). AM is the line from base vertex A to M.
(a)Find the horizontal distance from A to the point directly below M (the centre of the base ABCD), giving your answer in exact surd form.(2)
(b)Hence calculate the length AM, giving your answer correct to 3 significant figures.(2)
(c)Calculate the angle that AM makes with the base, giving your answer correct to 1 decimal place.(2)
(Total for Question 14 is 6 marks)
15
A cube has side length a. The angle between a vertical edge of the cube and the space diagonal from the same vertex is θ. Show that tan(θ) = √2.
(Total for Question 15 is 3 marks)
16
A square-based pyramid has base ABCD with side length 8 cm, and apex V vertically above the centre M of the base, with height VM = 6 cm. N is the midpoint of BC.
(a)Find the length MN.(1)
(b)Hence calculate the angle between the sloping face VBC and the base, giving your answer correct to 1 decimal place.(3)
(Total for Question 16 is 4 marks)
17
A cuboid box has length 30 cm, width 20 cm and height 12 cm. A spider is at one bottom corner, A, and wants to crawl over the surface of the box to reach the opposite top corner, G. Calculate the shortest distance the spider must crawl, giving your answer correct to 3 significant figures.
(Total for Question 17 is 5 marks)
18
O, A, B and C are points such that OA, OB and OC are mutually perpendicular edges meeting at O, with OA = 6 cm, OB = 8 cm and OC = 9 cm.
(a)Find the length AB.(2)
(b)Find the length AC, giving your answer correct to 3 significant figures.(2)
(c)Find the length BC, giving your answer correct to 3 significant figures.(2)
(d)Using exact (unrounded) values, calculate the size of angle BAC, giving your answer correct to 1 decimal place.(3)
(Total for Question 18 is 9 marks)
Mark scheme · G6 3D Pythagoras and Trigonometry

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18