O LevelMathsCambridgeExam QuestionsPythagoras & Trigonometry3D Pythagoras & Trigonometry

3D Pythagoras & Trigonometry (Cambridge O Level Maths)

Exam Questions

2 hours33 questions
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1
Sme Calculator3 marks

cie-igcse-2020-oct-nov-p2-tz1-q21

The diagram shows a cuboid P Q R S T U V W.
P V space equals space 17.2 space cm
The angle between the line P V and the base T U V W of the cuboid is 43 degree.

Calculate P T.

P T space equals ........................................... cm

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2
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The diagram shows a cuboid A B C D E F G H.

q19-4ma1-1h-qp-jan20-paper1-igcse-maths

E H space equals space 9 space cm comma space space space space H G space equals space 5 space cm  and     G B space equals space 6 space cm.

Work out the size of the angle between space A H space and the plane E F G H.
Give your answer correct to 3 significant figures.

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3
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The diagram shows a triangular prism.

q19-paper2h-june2018-edexcel-igcse-maths

A F space equals space 10 space cm comma space space A B space equals space 24 space cm space andspace B C space equals space 8 space cm.
Anglespace F A B space = angle A D C = angle B C D equals 90 degree

Work out the size of the angle between the line B E and the plane A B C D.
Give your answer correct to 1 decimal place.

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4
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The diagram shows cuboid A B C D E F G H.

q21-paper2h-june2019-edexcel-igcse-maths

For this cuboid

the length of A B : the length of B C : the length of C F = 4 : 2 : 3

Calculate the size of the angle between A F and the plane A B C D.
Give your answer correct to one decimal place.

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5
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The length of a diagonal of a cuboid is 20 cm

The diagonal makes an angle of 24° with the base.

The area of the base is 150 cm2

q24-paper-3h-june-2019-aqa-gcse-maths

Work out the volume of the cuboid.

...................................... cm3

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6
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cie-igcse-2020-oct-nov-p4-tz3-q6a

ABCDEFGH is a cuboid.
AB = 8 cm, BC = 5 cm and CG = 11 cm.

i)
Calculate angle CAB.
 
Angle CAB = ................................................ [2]
ii)
Calculate angle GAC.
 
Angle GAC = ................................................ [2]

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7
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7nbfuCQj_cie-igcse-2018-may-jun-3-p4-q6b


The diagram shows a cuboid.
A B = 8 cm, B C = 4 cm and C R = 5 cm.

Calculate the angle between the diagonal A R and the plane B C R Q.

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8
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A B C D is a horizontal rectangular field.

q19-4ma1-2hr-qp-jan22-paper2-igcse-maths

A vertical pole, A E, is placed at the corner A of the field.

A E equals 12 space straight m      A B equals 18 space straight m A D equals 8 space straight m

Calculate the size of the angle between E C and the plane A B C D
Give your answer correct to one decimal place.

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9
Sme Calculator4 marks

The diagram shows a right-angled triangular prism ABCDEF.

q18-paper4-nov2019-ocr-gcse-maths

Length AD = 11 cm, length CD = 10 cm and length CF = 6 cm.

Calculate the exact length of AF.

..................................................... cm

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10
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A B C D E F G H space spaceis a cuboid.

q1-hard-4-12-3d-pythogoras-and-trigonometry-edexcel-gcse-maths

A B space equals space 7.3 space cm C H space equals space 8.1 space cm Angle space B C A space equals space 48 degree

Find the size of the angle between A H and the plane A B C D.
Give your answer correct to 1 decimal place.

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11
Sme Calculator8 marks

q4b-june-2021-4024-21-cambridge-o-level-maths-d

The diagram shows a cuboid.
 T W space equals space 15 cm, W V space equals space 10 cm and R V space equals space 8 cm. 

i)
Show that T R space equals space 19.7 cm, correct to 1 decimal place.

[3]

ii)
X is the midpoint of P Q.
Calculate T R with hat on top X .
T R with hat on top X = ................................................ [5]

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1
Sme Calculator5 marks

cie-igcse-2020-may-june-p2-tz1-q18

The diagram shows cuboid A B C D E F G H of length 20 cm and width 5.5 cm.
The volume of the cuboid is 495 cm3 .

Find the angle between the line A G and the base of the cuboid A B C D.

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2
Sme Calculator4 marks

cie-igcse-2019-may-jun-p2-tz2-q24

A B C D E F G H is a cuboid.
A B equals 18 space text cm end text, B C equals 7 space text cm end text and C G equals 12 space text cm end text.

Calculate the angle that the diagonal A G makes with the base A B C D. 

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3
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cie-igcse-2020-mary-jun-p2-tz3-q20

The diagram shows a pyramid with a square base ABCD.
The diagonals AC and BD intersect at M.
The vertex V is vertically above M.
AB = 11 cm and AV = 18.6 cm.
 
Calculate the angle that AV makes with the base.

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4
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cie-igcse-2018-oct-nov-p2-tz2-q22

The diagram shows a cuboid with dimensions 5.5 cm, 8 cm and 16.2 cm.

Calculate the angle between the line AB and the horizontal base of the cuboid.

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5
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cie-igcse-2018-oct-nov-p2-tz3-q23

The diagram shows a triangular prism.
AB = 12 cm, BC = 6 cm, PC = 4 cm, angle BCP = 90° and angle QDC = 90°.
 
Calculate the angle between AP and the rectangular base ABCD.

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6a
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cie-igcse-2019-oct-nov-p4-tz2-q4b

The diagram shows a prism ABCDEF.
The cross-section is a right-angled triangle BCD.
BD = 10 cm, BC = 5.2 cm and ED = 18 cm.

Calculate angle BEC.

 

Angle BEC = ...................................................

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6b
Sme Calculator3 marks

The point G lies on the line ED and GD = 7 cm.

Work out angle BGE.

 

Angle BGE = ................................................

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7
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cie-igcse-2019-oct-nov-p4-tz3-q4

The diagram shows a field, ABCD, on horizontal ground.

There is a vertical post at C.
From B, the angle of elevation of the top of the post is 19°.

Find the height of the post.


................................................ m

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8a
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cie-igcse-2020-march-p4-tz2-q8b

The diagram shows an open box A B C D E F G H in the shape of a cuboid.
A B equals 20 cm, B C equals 18 cm and A E equals 16 cm.
A thin rod A G X rests partly in the box as shown.
The rod is 40 cm long.

Calculate G X, the length of the rod which is outside the box.


G X equals ............................................ cm

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8b
Sme Calculator3 marks

Calculate the angle the rod makes with the base of the box.

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9
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The diagram shows a cuboid A B C D E F G H.

q2-hard-4-12-3d-pythogoras-and-trigonometry-edexcel-gcse-maths

A B space equals space 7 space cm comma space A F equals space 5 space cm space and space F C equals space 15 space cm.

Calculate the volume of the cuboid.
Give your answer correct to 3 significant figures.

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10a
Sme Calculator4 marks

The diagram shows a pyramid.

q3a-hard-4-12-3d-pythogoras-and-trigonometry-edexcel-gcse-maths

B C D E spaceis a square with sides of length 10 cm.
The other faces of the pyramid are equilateral triangles with sides of length 10 cm.

Calculate the volume of the pyramid.
Give your answer in the form a over b square root of 2 space cm cubed where a and b are integers to be found.

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10b
Sme Calculator2 marks

Find the size of angle D A B.

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11a
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q24-specimen-paper2-2025-4024-02-cambridge-o-level-maths-d

A B C D E spaceis a rectangular-based pyramid.
space A C spaceand space B D spaceintersect at F.
space E F spaceis perpendicular to F C.

A D space equals space 12 cm comma space D C space equals space 8 cm space and space E C space equals space 16 cm.

Show that E F = 14.3cm correct to 1 decimal place.

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11b
Sme Calculator2 marks

Calculate the angle between the line space E C spaceand the base A B C D.

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1
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cie-igcse-2019-may-june-1-q21

The diagram shows a pyramid with a square base ABCD of side length 8cm.
The diagonals of the square, AC and BD, intersect at M.
V is vertically above M and VM = 10 cm.

Calculate the angle between VA and the base.

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