IGCSEMaths: ExtendedCIEExam Questions3. Geometry3.11 3D Pythagoras & Trigonometry

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First teaching 2021

Last exams 2024

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3D Pythagoras & Trigonometry (CIE IGCSE Maths: Extended)

Exam Questions

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3.11 3D Pythagoras & Trigonometry

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3.11 3D Pythagoras & Trigonometry

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1
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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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2
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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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3
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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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4
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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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5
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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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1a
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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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1b
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The point G lies on the line ED and GD = 7 cm.

Work out angle BGE.

 

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

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2
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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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3a
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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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3b
Sme Calculator3 marks

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

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4
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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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5a
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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 correct to 3 significant figures.

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5b
Sme Calculator2 marks

Find the size of angle D A B.

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1
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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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2
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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.

Ivana has a pencil of length 13cm.

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

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3
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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4
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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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5
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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6a
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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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6b
Sme Calculator3 marks

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

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