GCSEMathsOCRExam Questions10. Mensuration3D Pythagoras & Trigonometry

3D Pythagoras & Trigonometry (OCR GCSE Maths)

Exam Questions

3 hours36 questions
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1
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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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2
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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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3
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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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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 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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6
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Here is a cube with edge length space x spacecm

One diagonal is shown.

q16-paper-2h-nov-2021-aqa-gcse-maths

Circle the length, in centimetres, of the diagonal.

square root of 3 space x cube root of 3 x squared end root square root of x cubed end root cube root of 3 x

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7
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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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8
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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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9
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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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10
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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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1
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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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2
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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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3a
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The diagram shows a pyramid.

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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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3b
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Find the size of angle D A B.

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4
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The length of the longest diagonal of a cube is 25cm.

Calculate the total surface area of the cube.

................................................... cm2

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5
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The diagram shows a right-angled triangular prism ABCDEF.

q20-paper4-nov2018-ocr-gcse-maths

Calculate angle AFB.

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6
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The diagram shows a cuboid ABCDEFGH.

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Calculate angle BHF.

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7
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A B C is an isosceles triangle in a horizontal plane.
The point T is vertically above B.

q22-4ma1-2hr-qp-jan21-paper2-igcse-maths

Angle A B C space equals space 140 degree
A B space equals space B C space equals space 8 c m
T B space equals space 10 space c m
M is the midpoint of A C.


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

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8
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The diagram shows cuboid 

q18-paper2h-nov2020-edexcel-igcse-maths-1

table row cell A B space end cell equals cell space 5 space cm space end cell row cell A H space end cell equals cell space 4 space cm end cell end table
The size of the angle between C H and the plane A B C D is 35 degree

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

..............................................cm cubed

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9
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Here is a cube A B C D E F G H.

q20-paper1hr-jan2019-edexcel-igcse-maths

M is the midpoint of the edge G H.

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

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10
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The diagram shows a triangular prism A B C D E F with a horizontal base A B E F.

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A C space equals space B C space equals space F D space equals space E D space equals space 12 space cm       A B space equals space 10 space cm           B E space equals space 15 space cm

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

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11a
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The diagram shows a prism A B C D E F G H with a horizontal base.

q17-paper1hr-nov2020-edexcel-igcse-maths

The base of the prism, E F G H, is a square of side 12 cm.
Trapezium A D E F is a cross section of the prism where A F and D E are vertical edges.

table row cell D E space end cell equals cell space C H space equals space 10 space cm space end cell row cell A D space end cell equals cell space B C space equals space 15 cm end cell end table

Work out the size of the angle between C F and the base E F G H.
Give your answer correct to one decimal place.

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11b
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Work out the length of B E.
Give your answer correct to one decimal place.

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

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12a
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A B C D E F is a triangular prism which represents part of a hill.

      A B C F is the horizontal rectangular base.
      D is vertically above C.

q25-paper-3h-june-2018-aqa-gcse-maths

Work out the height C D.

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

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12b
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Jamil walks in a straight line from A to D.

q25b-paper-3h-june-2018-aqa-gcse-maths

Work out the size of angle D A C.
You must show your working.

........................degrees

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13a
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Rectangle A B C D is the horizontal base of a triangular prism A B C D E F.

A E space equals space B E

E is vertically above M, the midpoint of A B.

A B space equals space 16 cm    A E space equals space 17 spacecm    B C space equals space 30 cm

q25-paper3h-june2017-aqa-gcse-maths

Show that E M space equals space 15 cm

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13b
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Work out the size of angle E C M.

...........................degrees

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14
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Right-angled trianglespace A B C spaceis the cross section of a prism.

A B space equals space 5 space cm space space space space space space B C space equals space 12 space cm
M is the midpoint of C F.
A M space equals space 16 space cm

q23-paper-2h-nov-2020-aqa-gcse-maths

Work out the volume of the prism.

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

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1
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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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2
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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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3a
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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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3b
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Calculate the angle the stick makes with the base of the crate.

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4
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The diagram shows the prism A B C D E F G H J K with horizontal base A E F G

q21-january-2022-1h-edexcel-igcse-maths

A B C D E is a cross section of the prism where

A B D E is a square
B C D is an equilateral triangle

E F equals 2 cross times A E 
M spaceis the midpoint of G F so that J M is vertical.

Angle M A J equals y degree
Given that            tan space y degree space equals space T
find the value ofspace T, giving your answer in the form fraction numerator square root of p plus square root of q over denominator 17 end fraction where p and q are integers.

T = ...........................

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5
Sme Calculator3 marks

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

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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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6
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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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7
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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8
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°

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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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9
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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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10
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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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11
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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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12
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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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