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7 Mensuration
This Unit is concerned with measuring, calculating and estimating lengths, areas and volumes, as well as the construction of three-dimensional (3D) objects.
7.1 Units and Measuring
Different units can be used to measure the same quantities. It is important to use sensible units. Some important units are listed below. 1 km = 1000 m 1 m = 100 cm 1 m = 1000 mm 1 cm = 10 mm
1 tonne = 1000 kg 1 kg = 1000 g
1 litre = 1000 ml 1 m^3 = 1000 litres 1 cm^3 = 1 ml
Worked Example 1
What would be the best units to use when measuring, (a) the distance between Birmingham and Manchester, (b) the length of a matchbox, (c) the mass of a person, (d) the mass of a letter, (e) the mass of a lorry, (f) the volume of medicine in a spoon, (g) the volume of water in a swimming pool?
Solution
(a) Use km (or miles). (b) Use mm or cm. (c) Use kg. (d) Use grams. (e) Use tonnes (f) Use ml. (g) Use m 3.
Worked Example 2
(a) How many mm are there in 3.72 m? (b) How many cm are there in 4.23 m? (c) How many m are there in 102.5 km? (d) How many kg are there in 4.32 tonnes?
Solution
(a) 1 m = 1000 mm (b) 1 m = 100 cm So So 3 72. m = 3 72. × 1000 4 23. m = 4 23. × 100 = 3720 mm = 423cm (c) 1 km = 1000 m (d) 1 tonne = 1000 kg So So 1 02.5 km = 102 5. × 1000 4 .32 km = 4 32. × 1000 = 102 500 m = 4320 kg
Worked Example 3
What value does each arrow point to? (a)
(b)
(c)
Solution
(a) Here the marks are 0.1 units apart. So the arrow points to 12.6. (b) Here the marks are 0.2 units apart. So the arrow points to 11.8. (c) Here the marks are 0.4 units apart. So the arrow points to 6.8.
12 13
10 11 12
6 8 10
- Read off the value shown by the arrow on each scale (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
- A jug contains 1 litre of water. (a) If 150 ml is poured out, how much water is left? (b) A glass holds 200 ml of water. How many glasses could be filled from a full jug?
- State whether the following lengths would be best measured to the nearest m, cm or mm. (a) Your height. (b) The length of a ship. (c) The height of a hill. (d) The thickness of a book. (e) The height of a building. (f) The length of a matchstick. (g) The width of a matchstick.
10 20
10 11
100 150
40 50
40 60
18 19
5 7
3 5
8 10 12
2 4 6
20 30 40
20 60 100
100 300 500
- A cuboid has sides as shown in the diagram. Convert the lengths of these sides to mm.
- Each length below is given in mm. Give each length to the nearest cm. (a) 42 mm (b) 66 mm (c) 108 mm (d) 3 mm (e) 7 mm (f) 9.4 mm
- (a) What metric unit of length would you use to measure the length of a large coach? (b) Using the unit you gave in part (a) estimate the length of a large coach. (LON)
7.2 Estimating Areas
A square with sides of 1 cm has an area of 1 cm 2.
Worked Example 1
Find the area of the shaded shape.
Solution
The shape covers 11 squares, so its area is 11 cm 2.
2.1 cm
5.6 cm 4.2 cm
Just for Fun
Which is heavier, 1 kg of iron or 1 kg of feathers?
1 cm 1 cm Area = 1 cm^2
Just for Fun
Use 12 cocktail sticks to form 6 equilateral triangles, all of the same area. Move only 4 cocktail sticks from your figure so as to get 3 equilateral triangles, 2 of which are of the same area.
Exercises
- Find the area of each of the following shapes.
(a) (b)
(c) (d)
(e) (f)
- By counting the number of whole squares and half squares, find the area of each of the following shapes. (a) (b)
(c) (d)
(e) (f)
- The diagrams below shows the outlines of two islands, A and B. The grid squares have sides of length 1 km. Find the approximate area of each island.
- Each of the squares in this grid has an area of 1 square centimetre. Work out the area of the shaded shape.
(LON)
Investigation
Which of the following shaded figures has the greatest area? The squares are of the same length and the curved lines are all arcs of circles.
A B
7.3 Making Solids Using Nets
A net can be cut out, folded and glued to make a hollow shape. In this Unit, you will be dealing with 3-dimensional shapes such as
cuboid prism pyramid tetrahedron
Worked Example 1
What solid is made when the net shown is folded and glued?
Solution
It is important to add tabs to the net so that it can be glued. You could put tabs on every edge, but this would mean gluing tabs to tabs. The diagram opposite shows one possible position of the tabs. Before gluing, crease all the folds.
The final solid is a triangular prism.
Exercises
- Copy and cut out larger versions of the following nets. Fold and glue them to obtain cubes. Do not forget to add tabs to the nets.
(f)
(g)
- The diagram shows the net for a dice with some of the spots in place. Fill in the missing spots so that the opposite faces add up to 7. Then make the dice.
7.4 Constructing Nets
A net for a solid can be visualised by imagining that the shape is cut along its edges until it can be laid flat.
Worked Example 1
Draw the net for the cuboid shown in the diagram.
Solution
Imagine making cuts as below:
- cut along the edges AB, BC and CD to open the top like a flap.
- then cut down AE, BF, CG and DH, and press flat to give the net below.
A B
D C
E (^) F
G
H
A
B (^) C
D
A E
B (^) C
D
B C
F (^) G
H
A B
D C
E (^) F
G
H
B
C
Exercises
- Draw an accurate net for each cuboid below. (a)
(b)
(c) (d)
- Draw a net for each of the following solids. (a) (b)
2 cm 4 cm 2 cm
1 cm 4 cm
3 cm
2 cm
1 cm
4 cm 2.5 cm
2 cm
3 cm
4 cm
3 cm
5 cm
4 cm 4 cm
3 cm
3 cm
5 cm
(c) (d)
(e) (f)
- (a) Draw and cut out four equally sized equilateral triangles. (b) How many different ways can they be arranged with sides joined together? One example is shown.
(c) Which of your arrangements of triangles form a net for a tetrahedron?
- The diagrams below show the ends of two of prisms that each have length of 8 cm. Draw a net for each prism. (a) (b)
All edges 5 cm
2 cm
2 cm
2 cm
2 cm
3 cm
6 cm
4 cm
4 cm
3 cm
6 cm
4 cm
6 cm
6 cm
4 cm (^) 4 cm
3 cm
5 cm
5 cm
2 cm
2 cm
2 cm
2 cm
2 cm 2 cm
7.5 Conversion of Units
It is useful to be aware of both metric and imperial units and to be able to convert between them. Imperial Units 1
foot = 12 inches 1 yard = 3 feet
pound (lb) = 16 ounces 1 stone = 14 pounds
1 gallon = pints
Conversion Facts 1 kg is about 2.2 lbs.
1 gallon is about 4.5 litres. 1 litre is about 1.75 pints.
5 miles is about 8 km. 1 inch is about 2.5 cm. 1 foot is about 30 cm.
Worked Example 1
John is measured. His height is 5 feet and 8 inches. Find his height in: (a) inches, (b) centimetres (c) metres.
Solution
(a) There are 12 inches in one foot, so John's height = 5 × 12 + 8 = 60 + 8 = 68 inches
(b) 1 inch is about 2.5 cm, so John's height = 68 ×2 5. = 170 cm (c) 1 metre = 100 cm, so John's height = 1 7. m
Worked Example 2
A family travels 365 miles on holiday. Convert this distance to km.
Solution
As 5 miles is approximately equal to 8 km, first divide by 5 and then multiply by 8. 365 ÷ 5 = 73 73 × 8 = 584 So 365 miles is approximately the same as 584 km.
Worked Example 3
Jared weighs 8 stone and 5 pounds. Find Jared's weight in: (a) pounds, (b) kg.
Solution
(a) There are 14 pounds in 1 stone, so Jared's weight = 8 × 14 + 5 = 112 + 5 = 117 lbs (b) As 1 pound is about 0.45 kg, Jared's weight = 117 ×0 45. = 53 kg (to the nearest kg)
Worked Example 4
A line is 80 cm long. Convert this length to inches.
Solution
1 inch = 2 5. cm 80 2 5. =^32 , so the line is about 32 inches long.
