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
Sme Calculator4 marks

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
Sme Calculator4 marks

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
Sme Calculator4 marks

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
Sme Calculator4 marks

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
Sme Calculator2 marks

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
Sme Calculator4 marks

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
Sme Calculator4 marks

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
Sme Calculator4 marks

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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2
Sme Calculator4 marks

The diagram shows a triangular prism.

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

The base, A B C D, of the prism is a square of side length 15 cm.
Angle space A B E spaceand angle space C B E spaceare right angles.
Angle space E A B space= 35°

M spaceis the point on space D A spacesuch that

D M colon M A space equals 2 colon 3


Calculate the size of the angle between space E M spaceand the base of the prism.
Give your answer correct to 1 decimal place.

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3
Sme Calculator5 marks

V A B C D spaceis a solid pyramid.

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

A B C D spaceis a square of side 20 cm.

The angle between any sloping edge and the plane space A B C D spaceis 55°

Calculate the surface area of the pyramid.
Give your answer correct to 2 significant figures.

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4
Sme Calculator6 marks

V A B C D is a square-based pyramid.

The horizontal base A B C D has side length 10 cm and centre M.

Angle V M A is 90°
Angle V A M spaceis 68°

q27-paper3h-nov2017-aqa-gcse-maths

Volume space of space pyramid space equals space 1 third cross times area space of space base space cross times space perpendicular space height

Work out the volume of the pyramid.

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

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5
Sme Calculator6 marks

The diagram shows the prism A B C D E F with cross section triangle A B C.

q21-4ma1-1hr-qp-jan20-paper1-igcse-maths

Angle space B E C space= 40° and angle space A C B spaceis obtuse.
A C = 6 cm and space C E space= 13cm

The area of triangle space A B C spaceis 22 cm squared

Calculate the length of A B.
Give your answer correct to one decimal place.

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

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6
Sme Calculator5 marks

cie-igcse-2020-oct-nov-p4-tz2-q9a

The diagram shows a prism with a rectangular base, ABFE.
The cross-section, ABCD, is a trapezium with AD = BC.
AB = 8 cm, GH = 5 cm, BF = 12 cm and angle ABC = 70°.

The perpendicular from G onto EF meets EF at X. 

i)
Show that EX = 6.5 cm.
 
[1]
ii)
Calculate AX.
 
AX = ........................................... cm [2]
iii)
Calculate the angle between the diagonal AG and the base ABFE.
 
[2]

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7
Sme Calculator4 marks

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

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

Ivana has a pencil of length 13cm.

Does this pencil fit completely inside the cuboid?
Show how you decide.

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8
Sme Calculator3 marks

cie-igcse-2019-oct-nov-p4-tz1-q4d

The diagram shows a pyramid with a square base ABCD.
DB = 8 cm.
P is vertically above the centre, X, of the base and PX = 5 cm.

Calculate the angle between PB and the base ABCD.

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9
Sme Calculator3 marks

The diagram shows a solid prism A B C D E F G H.

q4-veryhard-3-11-3d-pythogoras-trigonometry-cie-igse-maths-extended

The trapezium A B C D, in which A D is parallel to B C, is a cross section of the prism.
The base A D E H of the prism is a horizontal plane.
A D E H and B C F G are rectangles.
The midpoint of B C is vertically above the midpoint of A D so that B A space equals space C D.


space A D space equals space 37 space cm space   G F space equals space 28 space cm    D E space equals space 24 space cm


The perpendicular distance between edges A D and B C is 20 cm.

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

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10
Sme Calculator6 marks

The diagram shows a solid pyramid A B C D E with a horizontal base.

q23-paper2hr-june2019-edexcel-igcse-maths

The base, B C D E , of the pyramid is a square of side 10 cm.

The vertex A of the pyramid is vertically above the centre O of the base so that  A B space equals space A C space equals space A D space equals space A E

The total surface area of the pyramid is 360 cm squared

Work out the size of the angle between A C and the base B C D E.
Give your answer correct to 3 significant figures.

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11a
Sme Calculator4 marks

Alvin has a crate in the shape of a cuboid.
The crate is open at the top.
The internal dimensions of the crate are 46cm long by 46cm wide by 55cm high.

q18-paper6-june2017-ocr-gcse-maths

Alvin has a stick of length 95cm.
Alvin places the stick in the crate so that the shortest possible length extends out above the top of the crate.

Calculate the length of the stick that extends out of the crate.

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

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11b
Sme Calculator3 marks

Calculate the angle the stick makes with the base of the crate.

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