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O LevelMathsCambridgeExam QuestionsPythagoras & TrigonometrySine, Cosine Rule & Area of Triangles

Sine, Cosine Rule & Area of Triangles (Cambridge O Level Maths)

Exam Questions

5 hours46 questions
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1a
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q13-paper4-spec-2025-cie-igcse-further-maths

The diagram shows a quadrilateral, A B C D, formed from two triangles, A B C and A C D.
A B C is a right-angled triangle.

Calculate angle B A C.

Angle B A C = ..............................................

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1b
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Calculate B D.

B D = ......................................... cm

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2
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q17-paper4-spec-2025-cie-igcse-further-maths

The area of the triangle is 50cm2.
Calculate the value of sin x.

sin space x space= ..............................................

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3a
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cie-igcse-2020-oct-nov-p4-tz1-q6

The diagram shows a field, ABCD, on horizontal ground.
BC = 192 m, CD = 287.9 m and BD = 168 m. 

Angle CBD  rounds to 106.0°, correct to 1 decimal place.

Calculate the area of triangle BCD.


 ............................................ m2

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3b
Sme Calculator2 marks

Tomas buys the triangular part of the field, BCD.
The cost is $35 750 per hectare.

Calculate the amount he pays.
Give your answer correct to the nearest $100.
[1 hectare = 10000m2]

   
$ ................................................

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4a
Sme Calculator3 marks

cie-igcse-2020-may-june-p4-tz1-q7

The diagram shows the positions of three points A, B and C in a field.

Show that B C is 118.1 m, correct to 1 decimal place.

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4b
Sme Calculator3 marks

Calculate angle A B C.

Angle A B C equals ................................................

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5a
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cie-igcse-2020-may-jun-p4-tz2-q4

The diagram shows two triangles.

Calculate QR.
 

QR = ............................................ m

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5b
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Calculate RS.
 

RS = ............................................ m

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5c
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Calculate the total area of the two triangles.
 

 ............................................ m2

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6
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cie-igcse-2019-oct-nov-p2-tz1-q13

Calculate the area of the triangle.

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

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7
Sme Calculator4 marks

cie-igcse-2019-oct-nov-p4-tz1-q5a

Use the cosine rule to find angle BAC.

Angle BAC = ...................................................

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8
Sme Calculator7 marks

cie-igcse-2019-oct-nov-p4-tz2-q6a


The diagram shows triangle ABC.

i)
Use the cosine rule to calculate BC.

 

BC = ............................................... m [4]

ii)
Use the sine rule to calculate angle ACB.

 

Angle ACB = ................................................... [3]

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9a
Sme Calculator4 marks

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

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

Use the cosine rule to find angle BAC.


Angle BAC = ...................................................

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9b
Sme Calculator3 marks

Use the sine rule to find angle CAD.

Angle CAD = ...................................................

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9c
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Calculate the area of the field.


............................................... m2

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10
Sme Calculator2 marks

cie-igcse-2018-oct-nov-p4-tz1-q7a
 Calculate the area of triangle PQR.
 

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

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11
Sme Calculator3 marks

cie-igcse-2018-may-jun-1-13
Use the sine rule to find angle ABC.

Angle ABC = ..............................................

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12
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cie-igcse-2018-may-jun-1-p2-q20
Calculate angle LMN.

Angle LMN = ..............................................

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13
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cie-igcse-2018-may-jun-3-p2-q23

Calculate the value of x.

x equals................................................

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14
Sme Calculator3 marks

q2-medium-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

Calculate the length of P R.
Give your answer correct to 3 significant figures.

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1
Sme Calculator4 marks

cie-igcse-2020-oct-nov-p4-tz1-q6

The diagram shows a field, ABCD, on horizontal ground.
BC = 192 m, CD = 287.9 m and BD = 168 m.

Calculate angle CBD and show that it rounds to 106.0°, correct to 1 decimal place.

  

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2a
Sme Calculator3 marks

cie-igcse-2020-oct-nov-p2-tz2-q19a

Calculate the value of x.

space x space equals space................................................

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2b
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Calculate the area of the triangle.

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

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3
Sme Calculator4 marks

cie-igcse-2020-oct-nov-p4-tz2-q8c

AD  = 10 cm, BC  = 8 cm, BX  = 5 cm, CX  = 7 cm.   

The diagonals of the cyclic quadrilateral ABCD intersect at X.  

Calculate angle BXC.
 

 Angle BXC = ................................................

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4
Sme Calculator10 marks

cie-igcse-2020-march-p4-tz2-q8a

The diagram shows a quadrilateral P Q R S formed from two triangles, P Q S and Q R S.

Calculate

i)
Q R,


Q R equals ............................................ cm [3]

ii)
P S,


P S equals ............................................ cm [3]

iii)
the area of quadrilateral P Q R S.


........................................... cm2 [4]

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5a
Sme Calculator4 marks

cie-igcse-2020-specimen-p4-q9a

The diagram shows a trapezium A B C D.
A B is parallel to D C.
A B space equals space 55 space straight m, B D space equals space 70 space straight m, angle A B D space equals space 40 degree and angle B C D space equals space 32 degree.

Calculate A D.
 

A D space equals ........................................... m

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5b
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Calculate B C.


B C space equals ........................................... m

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5c
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Calculate the area of A B C D.


 .......................................... m2

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6
Sme Calculator3 marks

cie-igcse-2019-oct-nov-p2-tz1-q16

Calculate the obtuse angle x in this triangle.

x = ...................................................

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7
Sme Calculator8 marks

cie-igcse-2019-may-jun-p4-tz2-q8a

The diagram shows two triangles A B D and B C D.
A D equals 16.5 space text cm end text and B D equals 12.4 space text cm end text.
Angle A D B equals 64 degree, angle B D C equals 53 degree and angle D B C equals 95 degree.

i)
Find A B. 
 
A B = ........................................ cm [4]
ii)
Find B C. 
 
B C = ........................................ cm [4]

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8
Sme Calculator8 marks

cie-igcse-2019-may-jun-p4-tz3-q9a

In the diagram, B C is a vertical wall standing on horizontal ground A B.
D is the point on A B where A D = 58 m.
The angle of elevation of C from A is 26°.
The angle of elevation of C from D is 72°.

i)
Show that A C = 76.7 m, correct to 1 decimal place.
 
[5]
ii)
Calculate B D.
 
B D = .......................................... m [3]

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9a
Sme Calculator3 marks

cie-igcse-2019-march-p2-q24

Calculate SR.
 

SR = ............................................... cm

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9b
Sme Calculator4 marks

Calculate RQ.
 

RQ = ............................................... cm

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10
Sme Calculator8 marks

cie-igcse-2018-oct-nov-p4-tz1-q7b
 

In the diagram, AB = 18 cm, BC = 21.3 cm and BD = 11.6 cm.
Angle BDC = 123.5° and angle ABC is a right angle.

i)
Calculate angle BCD.
 
 

Angle BCD = ............................................... [3]

ii)
Calculate AD.
 
 

AD = ......................................... cm [5]

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11a
Sme Calculator4 marks

The diagram shows the positions of three cities, Geneva (G), Budapest (B) and Hamburg (H).

cie-igcse-2018-oct-nov-p4-tz2-q8

Use the cosine rule to calculate the distance from Geneva to Budapest.
 

.......................................... km

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11b
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The bearing of Budapest from Hamburg is 133°.

Calculate the bearing of Budapest from Geneva.

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12a
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cie-igcse-2018-oct-nov-p4-tz3-q6
 

Calculate angle ACB.
 

Angle ACB = .................................................

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12b
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Calculate angle ACD.
 

Angle ACD = ..................................................

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12c
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Calculate the area of the quadrilateral ABCD.
 

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

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13a
Sme Calculator4 marks

cie-igcse-2018-may-jun-3-p4-q6a
In the pentagon A B C D E, angle A C B = angle A E D = 90°.
Triangle A C D is equilateral with side length 12 cm.
D E = B C = 6 cm.

Calculate angle B A E.

Angle B A E = .............................................

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13b
Sme Calculator5 marks

Calculate.

i)
A B.

A B = ......................................... cm [2]
ii)
A E.

A E = ......................................... cm [3]

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13c
Sme Calculator4 marks

Calculate the area of the pentagon.

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

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14
Sme Calculator5 marks

q2-hard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

A B C spaceis a triangle.

A B space equals space 8.7 space cm. space Angle space A B C space equals space 49 degree. space Angle space A C B space equals space 64 degree.

Calculate the area of triangle A B C.
Give your answer correct to 3 significant figures.

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15
Sme Calculator5 marks

The diagram shows triangle L M N.

q3-hard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

Calculate the length of L N.
Give your answer correct to 3 significant figures.

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16
Sme Calculator5 marks

q4-hard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

A B C spaceis a triangle.
D is a point on A C.
Angle space B A D space= 45°
Angle space A D B space= 80°
A B = 7.4cm
D C = 5.8 cm

Work out the length of B C.
Give your answer correct to 3 significant figures.

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17
Sme Calculator5 marks

A B C spaceis a triangle.

q5-hard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

A C space= 8.4m
Angle space A C B space= 40°

The area of the triangle = 100 m2.

Work out the length of A B.
Give your answer correct to 3 significant figures.
You must show all your working.

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18
Sme Calculator5 marks

q1-hard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

Work out the length of  A D.
Give your answer correct to 3 significant figures.

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19a
Sme Calculator3 marks

q19-specimen-paper2-2025-4024-02-cambridge-o-level-maths-d

A B C D spaceis a quadrilateral with A B space equals space 500 straight m, B C space equals space 650 straight m and D A space equals space 350 straight m.
Angle D A B space equals space 135 degree and angle B D C space equals space 52 degree.

Calculate D B.

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19b
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Calculate the obtuse angle B C D.

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1
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Find the area of a regular hexagon with side length 7.4 cm.


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

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2
Sme Calculator6 marks

cie-igcse-2020-mary-jun-p2-tz2-q27

The diagram shows a cuboid.
AB = 8 cm, AD = 6 cm and DH = 6 cm.
 
Calculate angle HAF.
 

Angle HAF = .................................................

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3a
Sme Calculator4 marks

cie-igcse-2020-may-jun-p4-tz3-q6a

The diagram shows triangle ABC with point G inside.
CB = 11 cm, CG = 5.3 cm and BG = 6.9 cm. 
Angle CAB = 42° and angle ACG = 54°.

Calculate the value of x.
 

x equals................................................ 

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3b
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Calculate AC.
 

AC = ........................................... cm

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4
Sme Calculator9 marks

cie-igcse-2019-oct-nov-p4-tz2-q6b

The area of triangle DEF is 70 m2.

i)
Show that  4 x squared plus 11 x minus 300 equals 0.

 

[4]

ii)
Use the quadratic formula to solve 4 x squared plus 11 x minus 300 equals 0.
Show all your working and give your answers correct to 2 decimal places.

 

x = .......................... or x = .......................... [4]

iii)
Find the length of DE.

 

DE = ............................................... m [1]

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5a
Sme Calculator4 marks

cie-igcse-2020-may-jun-p4-tz1-q3

The diagram shows a field ABCDE.

Calculate the perimeter of the field ABCDE.
 

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

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5b
Sme Calculator4 marks

Calculate angle ABD.
 

Angle ABD =  ..........................................................

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5c
Sme Calculator4 marks
i)
Calculate angle CBD.
 
Angle CBD = .................................................... [2]
ii)
The point C is due north of the point B.
 
Find the bearing of D from B.
 
[2]

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5d
Sme Calculator4 marks

Calculate the area of the field ABCDE.
Give your answer in hectares.
[1 hectare = 10 000m2]
 

...................................... hectares

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6a
Sme Calculator4 marks

Triangle E F G has an area of 70 m2.
E F colon F G equals 1 colon 2 and angle E F G=40°.

Calculate E F.
 

E F = .......................................... m

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6b
Sme Calculator1 mark

A different triangle P Q R also has an area of 70 m2.
P Q colon Q R equals 1 colon 2 and P Q equals E F
 
Find angle P Q R.
 

Angle P Q R = ..............................................

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7a
Sme Calculator4 marks

cie-igcse-2018-may-jun-2-p4-q5

The diagram shows a design made from a triangle AOC joined to a sector OCB.
AC = 8cm, OB = OC = 7 cm and angle ACO = 78°.

Use the cosine rule to show that OA = 9.47 cm, correct to 2 decimal places.

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7b
Sme Calculator3 marks

Calculate angle OAC.

Angle OAC = ................................................

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7c
Sme Calculator5 marks

The perimeter of the design is 29.5 cm.

Show that angle COB = 41.2°, correct to 1 decimal place.

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7d
Sme Calculator4 marks

Calculate the total area of the design.

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

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8a
Sme Calculator4 marks

cie-igcse-2018-march-p4-q8

The diagram shows two ports, L and P, and a buoy, M.
The bearing of L from P is 201° and LP = 248 km.
The bearing of M from P is 127°.
Angle PML = 42°.

Use the sine rule to calculate LM.
 

LM = ......................................... km

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8b
Sme Calculator3 marks

A ship sails directly from L to P.

Calculate the shortest distance from M to LP.
 

 .......................................... km

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9
Sme Calculator5 marks

q1-vhard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

The area of triangle space A B C space is 6 square root of 2 spacem2.

Calculate the value of x.
Give your answer correct to 3 spacesignificant figures.

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10
Sme Calculator5 marks

A B C spaceand space A D C spaceare triangles.

q2-vhard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

The area of triangle space A D C spaceis 56 m2

Work out the length of A B.
Give your answer correct to 1 decimal place.

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11
Sme Calculator5 marks

A B C D spaceis a parallelogram.

q3-vhard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

A C = 9 cm
D C space = 11 cm
Angle D A C = 100°

Calculate the area of the parallelogram.
Give your answer correct to 3 significant figures.

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12
Sme Calculator6 marks

q4-vhard-4-11-sine-cosine-rule-and-area-of-triangge-edexcel-gcse-maths

A B C spaceis an acute-angled triangle.
space B A space= 7cm
B C  = 8cm

The area of triangle space A B C space is 18 cm2.

Work out the size of angle B A C.
Give your answer correct to 3 significant figures.
You must show all your working.

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13a
Sme Calculator4 marks

q10a-june-2018-4024-2-cambridge-o-level-maths-d

In triangle A B C, space A B with hat on top C= 45° and space B A with hat on top C space= 65°.
A C spaceis 5 cm shorter than B C.

i)
Show that B C space equals fraction numerator 5 space sin space 65 over denominator sin space 65 space minus space sin space 45 end fraction

[3]

ii)
Find the length of B C.

B C= ..................cm[1]

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13b
Sme Calculator4 marks

q10b-june-2018-4024-2-cambridge-o-level-maths-d

In triangle P Q R, space P Q space= 13 cm, space Q R space= 6 cm and space R P space= 10 cm.
 Q R spaceis produced to S.
 

i)
Find the value of cos space P R with hat on top Q, giving your answer as a fraction in its simplest form.

[3]

ii)
Hence write down the value of cos P R with hat on top S

[1]

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