In this chapter you will:
solve problems involving perimeters and areas know and use the formulae for the
circumference and area of a circle
draw the nets, elevations and plans for a variety of 3D shapes
work out the volume of cubiods, prisms and cylinders.
The Tower of Pisa is a circular bell tower. Construction began in the 1170s, and the tower started leaning almost immediately because of a poor foundation and loose soil. It is 56.7 metres tall, with a diameter at the base of 15.5 metres, and there are 297 steps to the top. The tower continues to sink about 1 mm each year.
Objectives Before you start
You need to know:
how to measure or calculate the perimeters of rectangles and triangles
how to use the formula for the area of a rectangle
what a circle, semicircle and quarter circle are, and be able to name the parts of a circle and related terms
how to draw circles and arcs to a given radius.
10
AREA AND VOLUME 1
10.1
Area of triangles, parallelograms and trapeziums
1. The diagram shows a rectangle.
The length of the rectangle is 9 cm.
9 cm
9 cm The perimeter of the rectangle is 28 cm.
Work out the width and the area of the rectangle.
Get Ready
You know and can use the formula for the area of a triangle.
You know and can use the formula for the area of a parallelogram.
You know and can use the formula for the area of a trapezium.
Objectives
Zoologists at game reserves need to know the areas of different sections of their reserve, so that they know how many animals it can accommodate.
Why do this?
The area of a 2D shape is a measure of the amount of space inside the shape.
Key Point
Area of a triangle
The diagram below shows triangle ABC. A rectangle has been drawn around the triangle. The inside of the rectangle has been split into four triangles.
The length of the rectangle is the
base of the triangle and the width
of the rectangle is the perpendicular
height of the triangle.
Area of the rectangle base height
So to fi nd the area of a triangle, work out a half of its base its height.
Key Points C D 1 2 3 4 A B
Triangles 1 and 2 are congruent so area triangle 1 area triangle 2. Also area triangle 3 area triangle 4.
base b height h
A B
This means that the area of triangle ABC is half the area of the rectangle.
147 10.1 Area of triangles, parallelograms and trapeziums
Area of a triangle _ 1 2 base height
height (h)
base (b) A _ 1 2 bh
Work out the area of the triangle.
Area __ 1 2 7 4cm² 14cm²
Example 1
4 cm
7 cm
Area of a triangle 1 __ 2 base height 7 4 28 1 __ 2 28 14
Do not forget to put the units of the answer.
Examiner’s Tip
The height of a triangle is its vertical or perpendicular height.
Key Points
Area of a parallelogram
Here are two congruent triangles.The triangles can be put together to form a parallelogram. The two triangles have equal areas so the area of the
parallelogram is twice the area of one of the triangles. Area of one triangle _ 1 2 base height
Area of parallelogram 2 _ 1 2 base height base height Area of a parallelogram base height
height (h)
base (b) A bh
Work out the area of the parallelogram. Area 8 9mm²
72mm²
Example 2
9 mm
8 mm
Area of a parallelogram base height. As the lengths are in millimetres, the units of the area are mm².
1 Work out the areas of these triangles and parallelograms. a 10 cm 8 cm b 4 m 9 m c d 6 mm 9 mm e f 12 cm 9 cm 2 Copy and complete this table.
Shape Base Height Area
Triangle 6 cm 5 cm
Triangle 5 cm 10 cm
Triangle 8 cm 24 cm2
Parallelogram 8 cm 4 cm
Parallelogram 7 cm 56 cm2
3 a A rectangle has a length of 7 cm and an area of 35 cm². Work out the width of the rectangle.
b A square has an area of 144 cm². Work out the length of side of the square.
Exercise 10A 7cm 5 cm
D
12 cm 5 cmQuestions in this chapter are targeted at the grades indicated.
AO3
Area of a trapezium
Here is a trapezium. The trapezium is split into two triangles by a diagonal. Area of trapezium area of yellow triangle area of pink triangle.
Key Points Area of trapezium 1 _ 2 ah _ 1 2 bh 1 _ 2 (a b)h h b a h b Area � bh12 h a Area � ah12
149 10.1 Area of triangles, parallelograms and trapeziums
Area of a trapezium _ 1 2 sum of parallel sides distance between them. A _ 1 2 (a b)h
h a
b
Work out the area of the trapezium.
Area __ 1 2 (7 13) 11 1 __ 2 20 11 10 11 110cm2 Example 3 7 cm 13 cm 11 cm Area of a trapezium 1
__ 2 sum of parallel sides distance between them. Work out the brackets fi rst.
Examiner’s Tip
Remember that unless the question tells you to take measurements from a diagram you should not do so as diagrams are not accurately drawn.
1 Work out the area of each of these trapeziums.
a b c d Exercise 10B 8 cm 3 cm 11 cm 8 m 18 m 6 m 6 cm 9 cm 10 cm 13 cm 15 cm 7 cm
C
10.2
Problems involving perimeter and area
You can fi nd the area and perimeter of a more complicatedshape made from simpler shapes.
You can solve problems involving perimeters and areas.
1. Write down the formula for the area of:
a a rectangle b a square c a triangle d a parallelogram e a trapezium.
Get Ready Objectives
A lot of houses seen from the side are a pentagon shape, so a painter would need to work out the area of a pentagon to get the right amount of paint.
Why do this?
The perimeter or area of a compound shape can be found by splitting the shape into its simpler parts.
Key Point
Work out the area of this pentagon.
Area of rectangle A 8 10 80cm2 Area of triangle B __ 1 2 8 5 20cm2 Area of pentagon 80 20 100cm2 Example 4 15 cm 10 cm 8 cm 15 cm 5 cm B A 10 cm 8 cm
Split the pentagon into a rectangle A and a triangle B.
The height of the triangle is 15 10 5 cm. The base of the triangle is 8 cm.
The rectangle has length 8 cm and width 10 cm.
Area of a rectangle length width Area of a triangle __ 1 2 base height Area of pentagon area of A area of B
A02 A03
151 10.2 Problems involving perimeter and area
A rectangular wall is 450 cm long and 300 cm high. The wall is to be tiled. The tiles are squares of side 50 cm. How many tiles are needed?
Method 1
Number of tiles needed for the length 450 ____ 50 9 Number of tiles needed for the height 300 ____
50 6 Number of tiles needed 9 6
54 Method 2 Area of wall 450 300 cm² 135 000cm² Area of a tile 50 50 2500cm² Number of tiles 135_________ 000 2500 54 Example 5 300 cm 450 cm 50 cm 50 cm tile
wall No diagram is given with this question so it is a good idea to draw one.
So there are 6 rows of tiles, each with 9 tiles. One way to answer questions like this is to work out how many tiles are needed for the length and how many are needed for the height.
Number of tiles number of tiles in each row number of rows.
The other way to answer this question is to divide the area of the wall by the area of a tile. But remember that you should not use a calculator and the arithmetic is easier in the fi rst method.
A02 A03
1 The diagram shows the fl oor plan of a room. 5 m
5 m
9 m
3 m a Work out the perimeter of the fl oor.
Give the units of your answer.
b Work out the area of the fl oor. Give the units of your answer.
2 Karl wants to make a rectangular lawn in his garden. He wants the lawn to be 30 m by 10 m.
Karl buys rectangular strips of turf 5 m long and 1 m wide. Work out how many strips of turf Karl needs to buy.
3 A wall is a 300 cm by 250 cm rectangle. The wall is to be tiled.
The tiles are squares of side 50 cm. Work out how many tiles are needed.
4 A rectangle is 9 cm by 4 cm. A square has the same area as the rectangle.
Work out the length of side of the square.
Exercise 10C
D
AO2 AO2 AO3 AO2 AO3 AO2 AO35 Keith is going to wallpaper his living room and his bedroom.
Here are the fl oor plans of these rooms.
living room 4 m 4 m bedroom
8 m 2 m
5 m 5 m
a Work out the area of the fl oor in:
i Keith’s living room
ii Keith’s bedroom.
b Work out the perimeter of the fl oor in Keith’s living room.
To work out the number of rolls of wallpaper he needs, Keith uses this chart. Keith is going to use standard rolls of wallpaper.
Standard rolls of wallpaper are approx 10 m long How many rolls for the walls
Distance around the room including doors & windows
Wall height 10 m – 33 ft 12 m – 39 ft 14 m – 46 ft 16 m – 52 ft 18 m – 59 ft 20 m – 66 ft 22 m – 72 ft 24 m – 79 ft 2 – 2.3 m 7 – 76 5 5 6 7 8 9 10 11 2.3 – 2.4 m 76 – 8 5 6 7 8 9 10 10 11 2.4 – 2.6 m 8 – 86 5 6 7 9 10 11 12 13 2.6 – 2.7 m 86 – 9 5 6 7 9 10 11 12 13 2.7 – 2.9 m 9 – 96 6 7 8 9 10 12 12 14
The height of the walls in Keith’s living room is 2.5 m.
c Find how many rolls of wallpaper Keith needs for his living room. The height of the walls in Keith’s bedroom is 2.6 m.
d Find the number of rolls of wallpaper Keith needs for his bedroom.
6 Here is a quadrilateral.
7 cm
24 cm
20 cm 15 cm
a Work out the perimeter of the quadrilateral.
b Work out the area of the quadrilateral.
D
C
AO2 AO3 AO2 AO3 AO2153 10.3 Circumference and area of a circle
10.3
Circumference and area of a circle
You can work out the circumference of a circle. You can work out the area of a circle.
You can solve problems involving circles, including semicircles and quarter circles.
Objectives
To fi t a new tyre on the wheel of your bike, you may need to know the circumference of the wheel to fi nd the correct size.
Why do this?
1. Draw a circle of radius 5 cm. For this circle, draw and label clearly:
a a radius b a diameter c a chord d a sector e an arc f a segment g a tangent. Get Ready
For all circles circumference of circle ___________________ diameter of circle (pi). This value cannot be found exactly.
To 3 decimal places, 3.142.
circumference of circle diameter of circle
C 2r C � d C d d C Key Points Watch Out!
It is important not to confuse the diameter with the radius. Watch Out!
Examiner’s Tip
Calculator exam papers have the following instruction about , ‘If your calculator does not have a button, take the value of to be 3.142 unless the question instructs otherwise.’
7 Work out the area of the yellow shaded region in this diagram.
6 cm 9 cm 8 cm
12 cm 8 A kite has diagonals of length 10 cm and 20 cm.
Work out the area of the kite.
C
AO2 AO3AO2 AO3
Work out the circumference of a circle with:
a diameter 8.7 cm
b radius 3.1 m.
Give your answers correct to 3 signifi cant fi gures.
a C 8.7 27.3318… Circumference 27.3 cm b C 2 3.1 19.47787… Circumference 19.5 m
The circumference of a circle is 84.3 cm. Work out the radius of the circle. Give your answer correct to 3 signifi cant fi gures.
84.3 2 r 2 r r 84.3 (2)
13.4167…
Radius 13.4 cm
Example 6
Use the butt on or 3.142.
Write down at least 4 fi gures of the calculator display. Give the answer correct to 3 signifi cant fi gures. The units are the same as the diameter (cm).
Use C d with d 8.7 cm.
The diameter can be worked out from d 2r
so d 2 3.1 6.2 and then use C d. Or use C 2r with r 3.1 m.
The units are the same as the radius (m).
Watch Out!
Be careful when dividing by 2 on a calculator. It is best to use brackets.
Watch Out!
Use C 2r with C 84.3 cm as the radius is given in the question. Divide both sides by 2 and write down at least 4 fi gures of the calculator display.
Give the answer correct to 3 signifi cant fi gures. The units are the same as the circumference (cm).
Example 7
Examiner’s Tip
Remember that the circumference is approximately 3 times the diameter or 6 times the radius.
In this exercise, if your calculator does not have a button, take the value of to be 3.142. Give answers correct to 3 signifi cant fi gures unless a question says differently.
1 Work out the circumference of a circle with diameter:
a 7 cm b 12.9 mm c 5.6 cm d 40 cm e 21.9 m
2 The radius of a basketball net hoop is 23 cm.
a Work out the circumference of a basketball net hoop. A netball hoop has a radius of 19 cm.
b Work out how much longer is the circumference of a basketball net hoop than the circumference of a netball hoop.
Exercise 10D
155 10.3 Circumference and area of a circle
3 The circumference of a CD is 37.7 cm. Work out the radius of the CD. 4 The diameter of the front wheel of Michael’s bicycle is 668 mm.
a Work out the circumference of the wheel. Give your answer in cm correct to the nearest cm. Michael rides his bicycle.
b Work out the distance cycled when the wheel makes 1000 complete turns. Give your answer in km correct to 2 signifi cant fi gures.
The distance Michael rides his bicycle is 6 km.
c Work out the number of complete turns made by this wheel.
5 The length of the minute hand of a watch is 1.2 cm.
a Work out the distance moved by the point end of the hand in 1 hour.
b Work out the distance moved by the point end of the hand in: i 6 hours ii 20 minutes.
6 A circular table has a radius of 65 cm. a Work out the circumference of the table. The circumference of a circular tablecloth is 5 m.
The tablecloth is put symmetrically on the table so that the distance from the table to the edge of the tablecloth is the same all around the table.
b Work out the distance from the table to the edge of the tablecloth.
7 The diagram shows a shape made from a
semicircle, a rectangle and an equilateral triangle.
18 cm
10 cm The rectangle has length 18 cm and width 10 cm.
Work out the perimeter of the shape.
C
AO2 AO3 AO2 AO3B
Area of a circle
To fi nd the area of a circle means to fi nd the area enclosed by the circle. Here is a circle that has been divided into four equal
wedges or sectors. The sectors are then arranged as shown to form a parallelogram-like shape.
r r
�
r
The length shown as r is half the circumference, 2r, of the circle.
r
�
r
The area of the circle is the same as the area of the shape.
Here is what happens when the circle is divided into more sectors.
Key Points
AO2
AO2 AO3
The shape looks more like a parallelogram and as the number of sectors increases the parallelogram becomes more like a rectangle.
r � r
The width of this rectangle is equal to half of the circumference of the original circle and the height of the rectangle is equal to the radius of the circle.
Area of circle area of rectangle r r r2
Taking A as the area of a circle and r as the radius of the circle, A r2 That is Area radius radius
Work out the area of a circle with: a a radius of 9 cm b a diameter of 12.8 m. Give your answers correct to 3 signifi cant fi gures.
a A 92 254.4690… Area 254 cm2 b Radius 12.8 2 m 6.4 m A 6.42 128.6796… Area 129 m2
Work out the radius of a circle with area 46 cm2. 46 r2 r2 46 14.64225… r √____________14.64225… 3.8265… Radius 3.83 cm Example 8 Use A r2 with r 9 cm.
Write down at least 4 fi gures of the calculator display.
Give the answer correct to 3 signifi cant fi gures.
As the units of the radius are cm, the units of the area are cm2.
Divide the diameter by 2 to get the radius.
Write down at least 4 fi gures of the calculator display.
Give the answer correct to 3 signifi cant fi gures.
As the units of the radius are m, the units of the area are m2.
Take the square root to fi nd the value of r.
Use A r2 with A 46 cm2.
Work out the value of r2 by dividing both sides by .
Example 9
Examiner’s Tip
When the diameter of a circle is given, to work out the area of the circle fi rst fi nd the radius by dividing the diameter by 2.
157 10.3 Circumference and area of a circle
In this exercise, if your calculator does not have a button, take the value of to be 3.142. Give answers correct to 3 signifi cant fi gures unless the question says differently.
1 Work out the area of a circle with radius:
a 8 cm b 12.7 cm c 28.5 mm d 9.72 cm e 12.6 m
2 Work out the area of a circle with diameter:
a 24 cm b 8.3 cm c 0.95 m d 58.4 mm e 18.26 cm
3 The diagram shows a pond surrounded by a path. a Work out the area of the blue region of the pond.
3.5 m 2.5 m b Work out the area of the path.
c The path is made of shingle that costs £1.95 per square metre of path. Work out the cost of the shingle to make the path.
4 The diagram represents the plan of a sports fi eld. The fi eld is a rectangle with semicircular ends.
The rectangle has length 100 m and width 70 m. The semicircles have diameter 70 m.
a Work out the area of the fi eld.
70 m
100 m
The fi eld is to be covered in fertiliser that costs 23p per square metre.
b Use your answer to part a to work out the cost of the fertiliser for the fi eld.
5 A circle of diameter 8 cm is cut from a piece of yellow card.
The card is in the shape of a square of side 11 cm.
8 cm
11 cm 11 cm
The card shown yellow in the diagram is thrown away. Work out the area of the card thrown away.
6 A, B and C are three circles. Circle A has radius 5 cm and circle B has radius 12 cm. Circle C is such that
area of circle C area of circle A area of circle B. Work out the radius of circle C.
7 The diagram shows a star made by removing four identical
quarter circles from the corners of a square of side 30 cm.
30 cm
30 cm
Work out the area of the star.
Exercise 10E
D
C
B
AO2 AO3 AO2 AO3 AO2 AO3 AO2 AO3 AO2 AO3 M10_EMHA_SB_GCSE_0839_U10.indd 157 18/10/11 14:52:2610.4
Drawing 3D shapes
You can recognise and draw the net of a 3D shape.Objective
A manufacturer of chocolate boxes would have to consider the nets of different sizes of boxes in order to see how best to package their product.
Why do this?
1. Sketch these shapes.
a a triangular prism b a square-based pyramid
c a cylinder d a triangular-based pyramid Get Ready
Isometric paper will help you to make scale drawings of three-dimensional objects.
Isometric paper must be the right way up i.e. vertical lines down the page and no horizontal lines. A net of a 3D shape is a 2D shape that can be folded to make the 3D shape.
A 3D shape can have more than one net.
This cube has sides of length 2.
This cuboid has height 4, length 3 and width 2.
This prism has a triangular face.
Shapes can be joined together
Key Points
Draw two different nets for this cuboid.
Example 10
Watch Out!
A 3D shape may have many different nets. The shape of the net will depend on where the 3D shape has been split apart.
Watch Out!
There are six diff erent nets that will make this cuboid.
2 cm 5 cm 3 cm 2 cm 3 cm 2 cm 5 cm 2 cm 3 cm 5 cm 3 cm 3 cm 2 cm
159 front elevation side elevation plan
10.5 Elevations and plans
1 Use isometric paper to draw a cuboid with height 2 cm, width 4 cm and length 3 cm. 2 Sketch six different nets that will make a cube.
3 Here are the nets of some 3D shapes. Identify the shapes.
a b c d
4 Draw an accurate net for each of these. a 3 cm 5 cm 3 cm b 4 cm 6 cm 3 cm 5 cm 2 cm Exercise 10F
10.5
Elevations and plans
You can draw elevations and plans of 3D shapes.
Objective
Architectural proposals will usually contain plans and elevations of the proposed building, to give people an idea of what the building will look like from each side.
Why do this?
1. What would the shapes in question 4, above, look like if drawn from above, the side and the front. Get Ready
The front elevation is the view from the front. The side elevation is the view from the side. The plan is the view from above.
Key Points
plan
front elevation
Draw the front elevation, side elevation and plan of this 3D shape.
Sketch the shape represented by the front and side elevations and plan.
Example 11
Example 12
There are six cubes in this shape but you can see only fi ve of them. There must be a cube under the top one.
Draw the elevations and plan like this: 1. Plan at the top.
2. Front elevation under the plan. 3. Side elevation (view from the right)
to the right of the front elevation.
side elevation plan
front elevation
plan
front elevation side elevation front
1 Draw the elevations and plans of these shapes. a front b c 2 m 3 m 5 m d 5 cm 6 cm 2 cm e 6 cm 3 cm 3 cm 4 cm 5 cm f 2 cm 5 cm g 5 cm 4 cm Exercise 10G
D
161 10.6 Volume of a cubiod
10.6
Volume of a cuboid
You can work out the volume of a cuboid and shapes made from cuboids.
Objective
If you were fi lling a swimming pool you might fi rst have to consider its volume in order to work out how much water you would need.
Why do this?
1. Work out the volumes of these cuboids. Give the units with your answers. a 6 m 4 m 8 m b 8 cm 12 cm 6 cm Get Ready
2 Sketch the shapes represented by these elevations and plans. a side elevation plan front elevation b side elevation plan front elevation c side elevation plan front elevation
D
This shape is made from two cuboids. Work out the total volume of the shape.
Total volume 108 12 120 m3
Example 13
To work out the total volume of the shape add the volumes of the cuboids. 2 m 2 m 9 m 4 m 3 m 3 m
Work out the volume of each cuboid. Use volume of cuboid l w h.
9 m
Volume � 9 � 3 � 4 � 108 m3
4 m 3 m
For the larger cuboid
l 9 m, w 3 m
and h 4 m. 2 m
2 m 3 m
Volume � 2 � 3 � 2 � 12 m3
For the smaller cuboid
l 2 m, w 3 m and
10.7
Volume of a prism
You can work out the volume of a prism.Objective
Sandwiches are often sold in packs that are triangular prisms, so you can work out how much sandwich you are getting.
Why do this?
1. Work out the volume of these shapes. a 2a a a a a a b 2a a a a a a
c Find the volume of half shape b. Get Ready
Volume of prism area of cross-section length
cross-section
length
Key Point
1 These shapes are made from cuboids. Work out the volumes of the shapes. a 5 cm 6 cm 2 cm 7 cm 2 cm 3 cm 6 cm b 9 cm 2 cm 4 cm 9 cm 8 cm c 3 cm 110 mm 4 cm 2 cm 9 cm
2 Here is a net of a cuboid. Work out the volume of the cuboid.
18 cm 10 cm 14 cm Exercise 10H
D
AO3163 10.7 Volume of a prism
The area of the cross-section of this prism is 25 cm2. The length of the prism is 10 cm. Work out the volume of the prism.
Volume 25 10 250 cm3
Work out the volume of this prism.
Area of cross-section __ 1 2 3 4 6 cm2 Volume of prism 6 6.5 39 cm3
Example 14
Example 15
Use volume of prism area of cross-section length. Here, the area of cross-section 25 cm2 and
the length 10 cm.
Give the unit with your answer. The unit of area is cm2, the length is in cm so the unit of volume is cm3.
25 cm2
10 cm
The cross-section of the prism is a triangle. Remember: area of a triangle 1 __ 2 base height. Here the base 3 cm and height 4 cm.
Use volume of prism area of cross-section length. Here the area of cross-section 6 cm2 and length 6.5 cm.
4 cm
3 cm 5 cm
6.5 cm
1 Work out the volumes of these prisms. a 12 cm2 6.5 cm b 75 mm2 30 mm c 1.75 m 0.6 m 0.95 m d 6 cm 3 cm 8 cm 6 cm
2 Work out the volumes of these prisms. a 9 cm 6 cm 5 cm 5 cm b 28 cm 12 cm 15 cm 35 cm Exercise 10I
C
c 3.3 cm 4.6 cm 5.9 cm 3.5 cm 2.7 cm d 45 cm 45 cm 25 cm 25 cm 60 cm
3 The area of the cross-section of a prism is 45 cm2. The volume of the prism is 405 cm3. Work out the length of the prism.
4 Here is a prism. Show that the volume of the prism is 8x3 cm3.
x 2x
2x 3x
5 The diagram shows a triangular prism.
The volume of the prism is 45y3 cm3.
h 4y
5y
Find an expression for h in terms of y.
C
B
AO2Volume of cylinder area of cross-section length
r2h
r h
where r is the radius and h is the height.
Key Point
10.8
Volume of a cylinder
You can work out the volume of a cylinder.
Objective
You could work out the volume of liquid that your mug can hold if you wanted to boil only that exact amount of water, to save energy.
Why do this?
1. Find the area of these circles:
a radius 3 cm b diameter 5 cm c radius 10 cm.
165 10.8 Volume of a cylinder
Work out the volume of this cylinder.
Give your answer in terms of and to 3 signifi cant fi gures.
Area of cross-section 62 36 Volume of cylinder 3.142 6 6 12 1357.344 cm3 1360 cm3 (3 s.f.) Example 16
The cross-section of the cylinder is a circle with radius 6 cm. Remember: area of circle radius2.
Take as 3.142.
Give your fi nal answer correct to 3 signifi cant fi gures. Use volume of cylinder area of cross-section length. Do not round your answer at this stage.
Write down all the digits on your calculator display.
12 cm 6 cm
1 Work out the volumes of these cylinders.
Give your answers correct to 3 signifi cant fi gures.
a 5 cm 4 cm b 240 mm 300 mm c 5 cm 30 mm d 12 cm 79 cm
2 Work out the volumes of these cylinders. Give your answers in terms of . a 6 cm 10 cm b c 20 cm 6.5 cm
3 An aircraft hangar has a semicircular cross-section of diameter 20 m.
The length of the hangar is 32 m.
20 m
32 m
Work out the volume of the hangar. Give your answer in terms of .
Exercise 10J
C
AO3
0.45 m 0.5 m
4 An annulus has an external diameter of 7.8 cm, an internal diameter of 6.2 cm and a length of 6.5 cm.
Work out the volume of the annulus. Give your answer correct to 1 decimal place.
6.5 cm
6.2 cm 7.8 cm
5 A gold coin has a height of 2.5 mm and a volume of 2000 mm3. Work out the diameter of the gold coin. Give your answer correct to 2 decimal places.
6 An oil drum has a radius of 0.9 m and a height of 1.4 m. The oil drum is completely fi lled with oil.
Work out the volume of the oil in the oil drum. Give your answer correct to 3 signifi cant fi gures.
Area of a triangle _ 1 2 base height. A _ 1 2 bh
Area of a parallelogram base height. A bh
Area of a trapezium _ 1 2 sum of parallel sides distance between them. A _ 1 2 (a b)h
The perimeter or area of a compound shape can be found by splitting the shape into its simpler parts. For all circles, circumference of circle ___________________ diameter of circle (pi).
To 3 decimal places, 3.142.
Circumference of a circle d 2r where d is the diameter of the circle, and r is the radius of the circle. Area of a circle r2 where r is the radius of the circle.
The net of a 3D shape is a 2D shape that can be folded to make the 3D shape. A 3D shape can have more than one net.
The front elevation is the view from the front. plan
front elevation
side elevation
The side elevation is the view from the side. The plan is the view from above.
Volume of prism area of cross-section length.
cross section
B
AO3 AO3 AO2 AO3Chapter review
167 Volume of cylinder area of cross-section length
r2h
where r is the radius and h is the height.
r h
1 The diagram shows some nets and some solid shapes.
An arrow has been drawn from one net to its solid shape. Draw an arrow from each of the other nets to its solid shape.
Nov 08
Review exercise
2 Find the volume of this prism. Diagram NOT accurately drawn
represents 1 cm3
June 08 3 Work out the area of the shape.
12 cm 5 cm 7 cm 9 cm Diagram NOT accurately drawn Nov 2008 4 The diagram shows a solid object made of 6 identical cubes.
front
a On a centimetre grid, draw the side elevation of the solid object from the direction of the arrow.
b On a centimetre grid, draw the plan of the solid object. June 07 5 The diagram shows a cuboid.
10 cm
height
6 cm
The cuboid has: a volume of 300 cm3 a length of 10 cm a width of 6 cm.
Work out the height of the cuboid.
Nov 06 6 Boxes are packed into cartons.
box 4 cm carton 20 cm 6 cm 10 cm 30 cm 60 cm Diagram NOT accurately drawn A box measures 4 cm by 6 cm by 10 cm. A carton measures 20 cm by 30 cm by 60 cm. The carton is completely fi lled with boxes. Work out the number of boxes that will completely fi ll one carton.
Nov 07
Exam Question Report
95% of students answered this question poorly because they did not know what the different types of plans and elevations are.
169
Chapter review
7 Jane makes chocolates.
Each box she puts them in has: volume 1000 cm3
length 20 cm width 1000 cm.
a Work out the height of a box. Jane makes 350 chocolates. Each box will hold 18 chocolates.
b Work out: i how many boxes Jane can fi ll completely
ii how many chocolates will be left over.
8 Here is a net of a cuboid. Work out:
9 cm 3.2 cm
4.5 cm
a the surface area
b the volume of the cuboid.
9 The diagram shows a triangular prism.
6 cm 4.5 cm
9 cm 7.5 cm
a Draw the elevations and plan for the prism.
b Work out the surface area of the prism. Give the units with your answer.
10 Shelim is replacing the skirting boards and
coving in his living room. Skirting board can be bought in:
� 1 m
FIREPLACE
4 m lengths at £30.50 3 m lengths at £18.75 2 m lengths at £14.00. Coving can be bought in: 3 m lengths at £27.50 2.4 m lengths at £22.00.
Coving can be joined together, but skirting board must not be pieced together as the joins will be noticeable.
Find the cost of his materials for both jobs, minimising the waste.
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D
AO2 AO3 AO3C
AO2 AO311 Amy has saved £600 to spend on carpeting
her front room. There are four types she likes: Natural Twist at £14.50 per m2
2 m
5 m 5 m
1 m Medium Blend at £17.60 per m2
Heavy Weave at £19.00 per m2 Luxury Pile at £24.90 per m2.
She also needs to buy underlay, which is available in two types:
Cushion at £2.00 per m2 Super Cushion at £4.00 per m2. Fitting is £50 extra.
What can she afford to buy?
12 A landscape contractor charges:
£40 per square metre for levelling the ground and laying
grass house 10 m 12 m 2 m 10 m paving paving stones
£15 per square metre for levelling the ground and sowing grass seed.
Calculate the cost of both paving and seeding the garden shown on the right.
13 A ring-shaped fl owerbed is to be created around a circular lawn of radius 2.55 m.
2.55 m
Roses costing £4.20 are to be planted approximately every 50 cm around this fl owerbed. How much money will be needed for roses?
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C
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Chapter review
14 The diagram shows a garden that includes a lawn,
a vegetable patch, a circular pond and a fl owerbed.
All measurements are shown in metres. VEGETABLE PATCH
4 m 4 m 1 m pond, radius 1 m flower bed 1 m 1.8 m
The lawn is going to be relaid with turf costing £4.60 per square metre.
How much will this cost?
15 You are planning a party for 30 children.
You buy some concentrated orange squash and some plastic cups.
£1.2 5
10 cm
4.5 cm
Each plastic cup will have 150 ml of drink in it. (150 ml 150 cm3)
a Check that the plastic cup shown can hold 150 ml of drink. Use the formula: volume 5 h d __ 4 2
Each of the 30 children at the party will have a maximum of three drinks of orange squash. Each plastic cup is to be fi lled with 150 ml of drink.
The squash needs to be diluted as shown on the bottle label.
A bottle of concentrated orange squash contains 0.8 litres of squash and costs £1.25.
b How many bottles of concentrated orange squash do you need for the party?
c How much will they cost in total?
16
8 cm
20 cm 6 cm
5 cm
The cross-section of the prism in the diagram is a trapezium. The lengths of the parallel sides of the trapezium are 8 cm and 6 cm. The distance between the parallel sides of the trapezium is 5 cm. The length of the prism is 20 cm.
a Work out the volume of the prism. The prism is made out of gold. Gold has a density of 19.3 g/cm3.
b Work out the mass of the prism. Give your answer in kilograms.
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17 A swimming pool has a cross-sectional area in the
shape of a trapezium, as shown in the diagram. Water is pumped in at 2 m3 per minute.
25 m
10 m 1 m
3 m Using the dimensions shown in the diagram, fi nd how long
it takes to fi ll the pool?
18 A running track consists of two 60 m straights and two semicircular bends of diameter 60 m.
60 m 60 m
a Find the length of one lap of this running track.
b The owners of the track wish to stage athletics meetings and need it to be exactly 400 m long. This can be done by just altering the straights or just widening the bends.
Calculate what adjustments would need to be made.
19 Discs of diameter 2 cm are cut from a metal strip that is 2 cm by 100 cm. 100 cm
2 cm
What is the minimum amount of waste material?
20 A cylindrical oil tank has a radius 60 cm and a length of 180 cm.
It is made from reinforced steel and is full of oil.
60 cm
180 cm
The oil has a density of 4.3 g/cm3.
The reinforced steel has a mass of 2.8 g/cm2. Find the total mass of the tank and the oil in kg.
21 The solid shape, shown in the diagram, is made by cutting a hole all the way through a wooden cube.
The cube has edges of length 7 cm.
The hole has a square cross-section of side 2 cm.
2 cm
7 cm
7 cm 2 cm
7 cm a Work out the volume of wood in the solid shape.
The mass of the solid shape is 189 grams.
b Work out the density of the wood.
March 2009, adapted
