iGCSE Mathematics (0580) :C5.2 Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium and compound shapes derived from these. iGCSE Style Questions Paper 3

Question



The rectangle ABCD shows Mr Liu’s garden.
(a) Mr Liu puts a fence around three sides of his garden, AB, BC and CD.
The fence costs $3.28 per metre.
Calculate the cost of the fence.
(b) (i) Calculate the area of Mr Liu’s garden.

(ii) Mr Liu uses an area of 408\(m^2\) in his garden for a lawn, flowers and vegetables.

He divides this area into three parts, in the ratio
lawn:fl owers:vegetables = 5:3:4.
Calculate the area used for each part.
(c) Mr Liu walks in a straight line across his garden from A to C.
Calculate the distance Mr Liu walks.
(ii) The pond is fi lled with water to a depth of 2 metres.
Calculate the volume of water in the pond.

Answer/Explanation

Answer:

(a) 252.56

(b) (i) 510
(ii) 170
102
136

(c) 34.5

(d) (i) 63.6 or 63.61 – 63.63
(ii) 127 or 127.2…

Question



The diagram shows the positions of three towns A, B and C.
B is 27km north of A and the distance between A and C is 82km.
(a) Calculate BC.
(b) Write down the three fi gure bearing of C from A.
(c) (i) Use trigonometry to calculate angle ABC.
(ii) Work out the bearing of C from B.
(d) (i) Calculate the area of triangle ABC.
(ii) The land forming the triangle ABC is valued at $8400 for each square kilometre.
Calculate the value of this land.

Answer/Explanation

Answer:

(a) 86.3 or 86.33075…..

(b) 090 cao

(c) (i) 71.8 or 71.77492…..
(ii) 108.2 or 108

(d) (i) 1107
(ii) 9 298 800

Question

The diagram shows the cross section of a medal presentation platform.
(a) Show that x = 150.

Answer/Explanation

Ans: x + x + 180 = 480
2x = 300 

(b) Work out the perimeter of the cross section.

Answer/Explanation

Ans: 1060 [cm]

(c) (i) Calculate the area of the cross section.

Answer/Explanation

Ans: 16 500 

(ii) The platform is a prism, 170 cm deep.
Find the volume of the platform.

Answer/Explanation

Ans: 2 805000

(iii) The prism is completely fi lled with a light material.
1 cubic metre of this material has mass 16kg.
Calculate the mass of the material used.

Answer/Explanation

Ans: 44.9 or 44-88

Question

Two congruent quadrilaterals, G and H, and a point P are shown on this 1cm2 grid.
(a) (i) Write down the mathematical name of the shaded quadrilateral.

Answer/Explanation

Ans: Trapezium

(ii) Calculate the area of the shaded quadrilateral.
Give the units of your answer.

Answer/Explanation

Ans: 16 cm2

(b) Describe fully the single transformation that maps quadrilateral G onto quadrilateral H.

Answer/Explanation

Ans: Rotation
90°[anti-clockwise] oe
[centre] (–2, –8) 

(c) On the grid, draw the images of quadrilateral G after the following transformations.
(i) Refl ection in the line y = 0.

Answer/Explanation

Ans: Correct reflection in y = 0 

(ii) Translation by the vector \(\binom{-5}{7}.\)

Answer/Explanation

Ans: Translation 5 left and 7 up 

(iii) Enlargement by scale factor 0.5 with centre P.

Answer/Explanation

Ans: Correct Enlargement 

(d) On quadrilateral H mark, with an arc, an obtuse angle.

Answer/Explanation

Ans: Obtuse angle marked

Question

(a) The angles in a triangle are in the ratio 3:4:8 .
(i) Show that the smallest angle of the triangle is 36°.

Answer/Explanation

Ans: \(\frac{3}{3+4+8}or\frac{180}{3+4+8}\)

\(3\div (15)\times 180 or\frac{180\times 3}{15} (=36)\)

(ii) Work out the other two angles of the triangle.

Answer/Explanation

Ans: 48 [and] 96 

(b) Another triangle ABC has angle BAC = 35° and angle ABC = 65°.
(i) Using a protractor and straight edge complete an accurate drawing of the triangle ABC.
The side AB has been drawn for you.

Answer/Explanation

Ans: Angle BAC = 35 (±2º)

Angle ABC = 65 (±2º) and triangle completed

(ii) Measure the length, in centimetres, of the shortest side of your triangle.

Answer/Explanation

Ans: 4.45cm to 4.85cm

(c) A different triangle has base 7.0cm and height 5.6cm.
Calculate the area of this triangle, giving the units of your answer.

Answer/Explanation

Ans: 19.6 cao 

cm2 oe

Question

(a) 

A, B and C lie on a circle with diameter AC.
AC is extended to D and angle BAC = 63°.
Work out angle BCD.
Give reasons to explain your answer.

Answer/Explanation

Ans: 153

two correct geometrical reasons.

(b) 

The diagram shows a circle with radius 3cm inside a square of side 6cm.
Calculate the shaded area.

Answer/Explanation

Ans: 14.8
or 14.79 to 14.80 

(c) 

FGH is a right-angled triangle.
Calculate
(i) GH,

Answer/Explanation

Ans:  36 cm

(ii) the perimeter of the triangle,

Answer/Explanation

Ans: 108  cm

(iii) the area of the triangle.

Answer/Explanation

Ans: 486  cm2

Question

 (a) A cuboid measures 6cm by 3cm by 2cm.
(i) On this 1 \(cm^2\) grid, complete the net of the cuboid.

(ii) Calculate the volume of the cuboid.
……………………………. \(cm^3\)
(b)

Write down the mathematical name of this shape.
(c)

Mark an obtuse angle on this trapezium.
(d) A regular polygon has an exterior angle of 22.5°.
Work out how many sides this polygon has.
(e)

The diagram shows a shape made from a semi-circle, radius 6cm, and a right-angled triangle.
(i) Show that x = 16.
(ii) Calculate the area of the shape.
…………………………… \(cm^2\)

Answer/Explanation

Answer:

(a) (i) Correct net
(ii) 36

(b) Hexagon

(c) Obtuse angel indicated

(d) 16

(e) (i) \(\sqrt{20^2 – 12^2}\)
(ii) 153 or 152.5 to 152.6

Question

(a) Here are the first four terms of a sequence.
18      25      32      39
(i) Write down the next term.
(ii) Explain how you worked out your answer.
(b) The nth term of another sequence is \(n^2 + 3\).
Write down the first three terms of this sequence.
(c) Simplify.
(i) 6a + 5h − 4a − 8h
(ii) 5(x + 3) + 4(2x − 6)
(d) Factorise.
6g + 15

(e) A rectangle has length (x + 6) cm and width 5cm.
The area of this rectangle is 85cm2.
Find the value of x.
x = ………………………………………..

Answer/Explanation

Answer:

(a) (i) 46
(ii) Add 7 oe

(b) 4,  7,  12

(c) (i) 2a – 3h final answer
(ii) 13x – 9 final answer

(d) 3( 2g + 5) final answer

(e) 11 nfww

Question

(a) Here are the first four terms of a sequence.
18      25      32      39
(i) Write down the next term.
(ii) Explain how you worked out your answer.
(b) The nth term of another sequence is \(n^2 + 3\).
Write down the first three terms of this sequence.
(c) Simplify.
(i) 6a + 5h − 4a − 8h
(ii) 5(x + 3) + 4(2x − 6)
(d) Factorise.
6g + 15

(e) A rectangle has length (x + 6) cm and width 5cm.
The area of this rectangle is 85cm2.
Find the value of x.
x = ………………………………………..

Answer/Explanation

Answer:

(a) (i) 46
(ii) Add 7 oe

(b) 4,  7,  12

(c) (i) 2a – 3h final answer
(ii) 13x – 9 final answer

(d) 3( 2g + 5) final answer

(e) 11 nfww

Question

Jared is building a house.
(a)

The diagram shows the plan of the floor of the house.
(i) Find the area of the floor.
……………………………………… \(m^2\)

(ii) For every square metre of floor area, it costs $2175 to build the house.
Calculate the cost of building the house.
Give your answer correct to 3 significant figures.
$…………………………………………..
(b)

The diagram shows a section of the roof.
Using trigonometry, calculate the value of x.
x = ………………………………………….
(c) Jared invests $50000 for three years at a rate of 2% per year compound interest.
Calculate the total amount Jared receives at the end of the three years.
$ ……………………………..
(d) Jared also built an apartment for $180000.
He sells it for $198 000.
Calculate the percentage profit that he makes.
………………………. %

Answer/Explanation

Answer:

(a) (i) 73.38
(ii) 160 000
(b) 45.8 or 45.80 to 45.81
(c) 53 060.4[0]
(d) 10

Question

 (a) The diagram shows a triangle, A, on a 1 \(cm^2\) grid.

(i) Find the area of triangle A.
…………………………. \(cm^2\)
(ii) On the grid, draw an enlargement of triangle A with scale factor 2.

(i) Describe fully the single transformation that maps triangle B onto triangle C.
(ii) Reflect triangle B in the line y = –1.
(iii) Translate triangle B by the vector \(\begin{pmatrix}
5\\1

\end{pmatrix}\)

Answer/Explanation

Ans:

(a)(i) 7.5
(ii) Correct enlargement

(b) (i) Rotation
[centre] (0,0) oe
180°
(ii) Correct reflection with points
(–3,–3), (–1,–5) and (–6,–6)
(iii) Correct translation with points
(4,4), (2,2) and (–1,5)

Question

Ten students estimate the length and width of their rectangular school hall.
The results are shown in the table.

The first 8 results have been plotted on the scatter diagram.

(a) On the scatter diagram, plot the results for students I and J.
(b) What type of correlation is shown by this scatter diagram?
(c) (i) On the scatter diagram, draw a line of best fit.
(ii) Another student, Pedro, estimates the length of the hall as 31m.
His result for the width is missing.
Use your line of best fit to estimate his result for the width.
……………………….. m
(d) The actual measurements of the hall are length 44m and width 34m.

(i) The teacher says a ‘good estimator’ has both estimates no more than 5m from the actual
measurements.
Write down the letters of the students who are ‘good estimators’.
(ii) Work out the perimeter of the hall.
…………………… m
(iii) Calculate the length of a diagonal of the hall.
…………………………… m
(e) The hall is divided into two areas.

Find the shaded area.
………………………. \(m^2\)

Answer/Explanation

Answer:

(a) I, J correctly plotted
(b) positive
(c) (i) ruled line of best fit
(ii) 16 to 19

(d) (i) D,H, I
(ii) 156
(iii) 55.6 or 55.60 to 55.61

(e) 1020

Question

The diagram shows a field in the shape of a trapezium.
AB = 150m, BC = 90m and CD = 120m.
Angle ABC = angle BCD = 90°.
(a) Calculate the area of the field.
\(……………………………………..m^{2}\)
(b) (i) Show that AD = 95m, correct to the nearest metre .
(ii) A fence is built around the perimeter of the field.
It costs \($48\) to build each 5-metre section of the fence.
Calculate the cost of building this fence.
\($ …………………………………………\)

Answer/Explanation

(a) 12150
(b)\((i)AD=\sqrt{90^{2}+(150-120)^{2}}\)
=94.9 OR 94.8
(ii) 4368

Question

A, B and C are points on the circumference of a circle, centre O.
(a) Write down the mathematical name for
(i) the straight line AC,
…………………………………………
(ii) the straight line AB.
…………………………………………
(b) Give a geometrical reason why angle ABC = 90°.
………………………………………………………………………………………………………………………………………….
(c) AB = 20 cm and AC = 52cm.
(i) Use trigonometry to calculate angle BAC.
Angle BAC = ………………………………………..
(ii) Show that BC = 48cm.
(iii) Work out the area of triangle ABC.
…………………………………..\( cm^{2}\)
(iv) Work out the total shaded area.
………………………………….. \(cm^{2}\)

Answer/Explanation

(a)(i) Diameter
(ii) Chord
(b) Angle [in] semi-circle [is 90]
(c)(i) 67.4 or 67.38…..
(ii)\( (BC)^{2}=\sqrt{52^{2}-20^{2}}\)
(iii) 480
(iv) 582 or 581.8 to 582.0

Question 

 

 
 

Question

(a) The diagram shows a rectangle with length 7a and width 2a.

Write an expression, in its simplest form, for

(i) the perimeter,[2]

(ii) the area.[2]

(b) The nth term of a sequence is n2 + 5.

Find the first three terms of this sequence.

___ , ___ , ___ [2]

(c) (i) Complete the table of values for \(y=\frac{12}{x},x\neq 0.\)[3]

     (ii) On the grid, draw the graph of \(y=\frac{12}{x}\) for \(-6\leqslant x\leqslant -1\) and \(1\leqslant x\leqslant 6\).[4]

     (iii) On the grid, draw the line y = 8. [1]

     (iv) Use your graph to solve \(\frac{12}{x}=8\).

x =  [1]

Answer/Explanation

Ans:

5(a)(i) 18a final answer

5(a)(ii) 14a2 final answer

5(b) 6 9 14

5(c)(i) −4 −6 −12 6 4 3

5(c)(ii) Correct curve

5(c)(iii) Correct ruled line drawn 

5(c)(iv) 1.3 to 1.7

Question

(a) Using a straight edge and compasses only, construct the equilateral triangle ABC. The base AB has been drawn for you.[2]

(b) 

Calculate the area of this trapezium.

m2 [2]

(c) Each interior angle of a regular polygon is 162°.

Calculate the number of sides of the polygon. [3]

(d) 

The area of this triangle is 363cm2.

Calculate the value of h.

h =  [3]

(e)

This shape is drawn using two semicircles that have the same center.

The large semicircle has radius 7cm.

The small semicircle has radius 3cm.

Calculate the area of the shape.

cm2 [3]

Answer/Explanation

Ans:

10(a) correct triangle drawn with arcs

10(b) 280

10(c) 20

10(d) 11

10(e) 62.8 or 62.83 to 62.84

Question

 The diagram shows the net of a triangular prism on a \(1cm^{2}\) grid.


(a) Write down the mathematical name for the type of triangle shown on the grid.
………………………………………….
(b) (i) Measure the perpendicular height of the triangle.
…………………………………….. cm
(ii) Calculate the area of the triangle.
…………………………………… \(cm^{2}\)
(iii) Calculate the volume of the triangular prism.
……………………………………\( cm^{3}\)

Answer/Explanation

(a) Equilateral
(b)(i) 4.1 to 4.5
(ii) 10.25 to 11.25
(iii) 61.5 to 67.5

Question

Tarak has two fields.
He grows wheat, barley and corn in his fields.
(a) 

The diagram shows Tarak’s two triangular fields, PQR and PRS.
Angle RPS = 90° and angle PRS = 53°.
PQ = 174m, QR = 120m and PR = 126m.
(i) Show that angle PRQ = 90°.

Answer/Explanation

Ans: Complete method shown and evaluated 

(ii) Calculate the area of the quadrilateral PQRS.
Give your answer correct to 4 significant figures.

Answer/Explanation

Ans: 18090 cao

(b) (i) The mass, m tonnes, of wheat grown in 2021 is 4.3 tonnes, correct to 1 decimal place.
Complete this statement about the value of m.

Answer/Explanation

Ans: 4.25       4.35 

(ii) In 2020, 2.6 tonnes of barley is grown.
In 2021, 3.25 tonnes of barley is grown.
Show that the percentage increase in barley grown from 2020 to 2021 is 25%.

Answer/Explanation

Ans: Complete method seen 

(iii) In 2019, 2.4 tonnes of corn is grown.
In 2020, 20% more corn is grown than in 2019.
In 2021, 20% less corn is grown than in 2020.
Calculate the amount of corn grown in 2021.

Answer/Explanation

Ans: 2.304 

Question

(a) The diagram shows three quadrilaterals, A, B and C, on a 1cm2 grid.

(i) (a) Write down the mathematical name for quadrilateral B.

Answer/Explanation

Ans: Trapezium 

(b) Work out the area of quadrilateral B.
Give the units of your answer.

Answer/Explanation

Ans: 28
        cm2

(ii) Measure angle w.

Answer/Explanation

Ans: 117

(iii) Describe fully the single transformation that maps
(a) quadrilateral A onto quadrilateral B,

Answer/Explanation

Ans: Enlargement
         [centre] ( 2, -2)
         [scale factor] 2 

(b) quadrilateral A onto quadrilateral C.

Answer/Explanation

Ans: Rotation
         [centre] ( − 2, 4)
        90° clockwise oe 

(b) The diagram shows a parallelogram and a line AB on a 1cm2 grid.

On the grid, complete a triangle, ABC, that has the same area as the parallelogram.

Answer/Explanation

Ans: A correct triangle drawn 

Question

In this question all the measurements are in centimetres.

The diagram shows a triangle with sides of length 2x + 3, 11 – x and 3x.
(a) Explain why x must be less than 11.

Answer/Explanation

Ans: If x is more than 11 then 11 – x
would be negative oe 

(b) Write down an expression, in terms of x, for the perimeter of the triangle.
Give your answer in its simplest possible form.

Answer/Explanation

Ans: 14 + 4x cao
accept 2(2x + 7)

(c) The perimeter of the triangle is 32 cm.
(i) Write down an equation in terms of x and solve it.

Answer/Explanation

Ans: 4.5 cao

(ii) Work out the length of the shortest side of the triangle.

Answer/Explanation

Ans: 6.5

Question

Triangle ABC is drawn on a 1cm2 grid.
E is the point (0, 0).
(a) Write down the gradient of the line AB.

Answer/Explanation

Ans: 2 cao 

(b) The gradient of BC is – 0.5 .
Write down the equation of the line BC in the form y = mx + c.

Answer/Explanation

Ans: –0.5x + 6 

(c) Write down the ratio AE: EC.
Give your answer in its simplest form.

Answer/Explanation

Ans: 1:4

(d) Measure angle ABE.

Answer/Explanation

Ans: 25°–29°

(e) Triangle ABE is similar to triangle BCE.
Explain what the word similar tells you about the triangles ABE and BCE.

Answer/Explanation

Ans: (Corresponding) angles equal oe
          (Corresponding) lengths in same ratio oe

(f) Calculate the area of triangle ABC.

Answer/Explanation

Ans: 45

(g) ABCD is a rectangle.
(i) Mark point D on the grid.

Answer/Explanation

Ans: D correctly marked on grid 

(ii) Write down the co-ordinates of D.

Answer/Explanation

Ans: (9, –6) 

Question

The diagram shows a rectangular field, PQRS.
QR = 120m, PQ = 50m and P is due North of Q.
Bill and Said run from P to R.
Bill runs along the sides PQ and QR.
Said runs directly from P to R.
(a) Calculate how far
(i) Bill runs,

Answer/Explanation

Ans: 170

(ii) Said runs.

Answer/Explanation

Ans: 130

(b) Bill takes 34 seconds to reach R.
Calculate Bill’s average speed.

Answer/Explanation

Ans: 5

(c) Said runs at 4m/s.
Who arrives at R first and by how many seconds?

Answer/Explanation

Ans: Said by 1.5 secs 

(d) (i) Use trigonometry to calculate the size of the angle marked y.

Answer/Explanation

Ans: 67.4° 

(ii) Find the bearing of R from P.

Answer/Explanation

Ans: 113° or 112.6° 

(e) Calculate the area of the field in square kilometres.
Give your answer in standard form.

Answer/Explanation

Ans: 6 × 10–3

Question

The diagram shows a plot of land, ABCD, in the shape of a trapezium.
(a) Show that CD = 19.2m, correct to 1 decimal place.

Answer/Explanation

Ans: (CD2=) (32 – 20)2+ 152  oe
          (CD =) √369 = 19.20 to 19.21 

(b) A fence is built around the perimeter of the plot of land.
The cost of the fence is $35 for each metre.
Calculate the total cost of the fence.

Answer/Explanation

Ans: 3017

(c) Calculate the area of the plot of land.
Give your answer in square metres.

Answer/Explanation

Ans: 390

(d) A house is built on the plot of land.
The area of the plot is divided in the ratio house : grounds = 3 : 7.
Calculate the area of the grounds.

Answer/Explanation

Ans: 273

(e) (i) In the space below, make a scale drawing of the plot of land.
Use a scale of 1 centimetre to represent 4 metres.
The side AB has been drawn for you.

Answer/Explanation

Ans: trapezium constructed
BC = 5 cm, AD = 8 cm
Both 90o to AB

(ii) Measure angle ADC.

Answer/Explanation

Ans: 49 – 53° 

(iii) Use your diagram to find the actual length BD in metres.

Answer/Explanation

Ans: 34.4 – 36.4 m

Question

The shape above is the net of a solid drawn on a 1 cm square grid.

(a) Write down the geometrical name of the solid

(b) Find the perimeter of the net.

(c) Work out

(i) the area of one of the triangles,

(ii) the volume of the solid.

(d) A cuboid of length 4 cm and width 3 cm has the same volume as the solid.
Calculate the height of the cuboid.

Answer/Explanation

(a) Triangular prism

(b) 50.4

(c)(i) Area =\(\frac{1}{2}\times 4 \times 3\)

=6

(ii) Volume= \(Area\times 7\)

= 6.7

=42  \({cm}^3\)

(d) Given:

Volume of solid= 42\({cm}^3\)

Volume of Cuboid= l.b.h

where, l is length

b is breadth

 and h is height

A/Q

Volume of solid= Volume of cuboid

42= l.b.h

42=4.3.h

\(h=\frac{42}{12}\)

h= 3.5 cm

Question

(a) Solve the equation 2(x + 4) = 3(x + 2) + 8 

(b) Make z the subject of za + b = 3 .

(c) Find x when \(2 x^3=54\)

(d) A rectangular field has a length of x metres. The width of the field is (2x – 5) metres.
(i) Show that the perimeter of the field is (6x – 10) metres.

(ii) The perimeter of the field is 50 metres.
Find the length of the field.

Answer/Explanation

(a) 2(x + 4) = 3(x + 2) + 8

2x + 8 = 3x + 6 + 8

2x + 8 = 3x + 14

8 = x + 14

 x=-6

Therefore, the solution to the equation is x = -6

(b)  za + b = 3

za = 3 – b

\(z = \frac{(3 – b)}{ a}\)

(c)  \(2x^3\) = 54

\(x^3\) = 27

x = 3

(d) (i) Perimeter = length + width + length + width

= x + (2x – 5) + x + (2x – 5)

= 6x – 10

(ii) As solved above ,

Perimeter = 6x – 10

Given that the perimeter of the field is 50 meters,

50 = 6x – 10

60 = 6x

x = 10

Therefore, the length of the field is 10 metres. 

Question

             

The diagram shows a block of stone in the shape of a prism of length $42 \mathrm{~cm}$. The cross-section is a trapezium $A B C D$. $A B=19 \mathrm{~cm}, A D=10 \mathrm{~cm}, D C=13 \mathrm{~cm}$ and angle $A D C=90^{\circ}$.
(a) Calculate
(i) the perimeter of the rectangular face $A B F E$,

(ii) the area of the cross-section ABCD,

(iii) the volume of the block of stone.

(b) The mass of 1 cubic centimetre of the stone is 4 grams. Calculate the mass of the block. Give your answer in kilograms.

▶️Answer/Explanation

(a) (i) 122

(ii) 160

(iii) 6720 or their (a)(ii) × 42 evaluated

(b) 26.88 or their (a)(iii) × 0.004 evaluated or 26.9

Question

In the diagram, ABCD is a square of side 7 cm. BLC and DMA are equilateral triangles.
(a) Find the perimeter of the shape ABLCDM.

(b) (i) Write down the size of angle CBL.

(ii) Calculate the length of LX.

(c) (i) Calculate the area of triangle BLC.

(ii) Calculate the area of the shape ABLCDM.

▶️Answer/Explanation

(a) 42
(b) (i) $60^{\circ}$
(ii) $6.06(217 \ldots)$
(c) (i) 21.2 to $21.4 \mathrm{ft}$
(ii) 91.4 to $91.7 \mathrm{ft}$

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