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UNITS
You should be familiar with the following units:
Length: mm, cm, m, km Area: mm^2 , cm^2 , m^2 , ha, km^2 Volume: mm^3 , cm^3 , m^3 Capacity: ml, cl, l Mass: g, kg, t
To convert from smaller to larger units we divide by the conversion factor. To convert from larger to smaller units we multiply by the conversion factor.
LENGTH
The perimeter of a figure is the measurement of the distance around its boundary.
For a polygon the perimeter is the sum of the lengths of all sides.
AREA
The area of a figure is the amount of surface within its boundaries.
You should be able to use these formulae for area:
Rectangles
Area = length £ width
Circles and sectors
An arc is any continuous part of the circle. The length of an arc is called its arclength. Every arc has a corresponding sector , which is the portion of the circle subtended by the same angle μ±^ as the arc.
For a circle: Circumference C = ¼d = 2¼r Area A = ¼r^2
For a sector of angle μ±:
Arclength s =
360
£ 2 ¼r Area A =
360
£ ¼r^2
SURFACE AREA
Solids with plane faces
The surface area of a three dimensional figure with plane faces is the sum of the areas of the faces.
To assist in your calculations, you can draw a net of the solid, correctly labelling the dimensions.
Solids with curved surfaces
You should be able to use these formulae for surface area:
Hollow cylinder Hollow can
A = 2¼rh A = 2¼rh + ¼r^2
Solid cylinder Hollow cone
A = 2¼rh + 2¼r^2 A = ¼rl
Solid cone Sphere
A = ¼rl + ¼r^2 A = 4¼r^2
km m cm mm
km^2 ha m^2 cm^2 mm^2
length
width
Triangles
Area = 12 (base £ height) height
base
Parallelograms Area = base £ height (^) height
base
r q°
arc
sector
Trapezia
Area = 12 (a + b) £ h (^) h
b
a
TOPIC 6: MENSURATION
r
r
hollow
hollow
h
r
hollow
solid
h
r
solid
solid
h l
r
l
r
Exam Preparation & Practice Guide^33 Cambridge IGCSE International Mathematics (0607) Extended
VOLUME
The volume of a solid is the amount of space it occupies.
You should be able to use these formulae for volume: Solids of uniform cross-section
Volume of uniform solid = area of end £ height
Pyramids and cones
Volume of a pyramid or cone = 13 (area of base £ height)
Spheres
Volume of a sphere = 43 ¼r^3
You can find the volumes of compound solids by separating the solid into sections like those above.
CAPACITY
The capacity of a container is the quantity of fluid or gas required to fill it.
Connecting volume to capacity
1 ml ´ 1 cm^3 1 litre ´ 1000 cm^3 1 kl = 1000 litres ´ 1 m^3
MASS
The mass of an object is the amount of matter in it.
SKILL PRACTICE
1 Convert: a 72 mm to cm b 5 : 8 m to mm c 9 : 75 km to m d 28 000 000 cm to km.
2 Kevin counts the light poles on the footpath as he walks to school. Kevin walks 2 : 4 km, and counts 80 light poles. How far is it between each light pole?
3 Find the perimeter of: a b c
4 Convert: a 44 mm^2 to cm^2 b 0 : 059 ha to cm^2 c 21 : 85 ha to km^2 d 0 :000 006 2 km^2 to mm^2 e 360 m^2 to cm^2 f 39 500 m^2 to ha.
5 A rectangle is 3 : 2 m by 2 : 4 m and has the same perimeter as a square. Find the length of the sides of the square.
6 The base area of a box of stickers is 85 cm^2. How many of these boxes will fit in one layer of a pallet of area 1 : 36 m^2?
7 Find a formula for the perimeter P of: a b c
8 A circle has area 36 : 4 m^2. Find: a its radius b its circumference.
9 Find the area of the following: a b
10 Find the surface area of: a a cube with sides 16 cm b a cuboid 36 mm £ 48 mm £ 21 mm.
11 Convert: a 3 : 71 b 58 215
12 Calculate the length of guard rail needed to construct a safety fence for the following viewing platform:
m^3 cm^3 mm^3
height
end
r
base
height
base
height
height
end
height
end
kl litres cl ml
t kg g mg
12 cm
15 cm
3 5. m
2 m
1 5. m
2 5. m
z (^) b
a
p
q
9 m
6 m
0 9. m 1 m 1 m
2 m 1 6. m
3 cm
5 cm
8 cm
litres into cl ml into litres.
Cambridge IGCSE International Mathematics (0607) Extended^34 Exam Preparation & Practice Guide
30 A cylindrical drinking flask has radius 3 : 42 cm and height 16 : 33 cm. Find its capacity.
31 How many cylindrical cookies with diameter 5 cm and thickness 1 cm could be made from a rectangular block of dough 20 cm £ 15 cm £ 8 cm?
32 Three sizes of tile are used to form the 3 : 25 m £ 2 : 25 m floor of a bathroom using the pattern shown. The large tiles are 10 cm £ 10 cm. What proportion of the area is covered by the smallest tiles?
33 Pauline has a wooden block with the dimensions shown. She paints a 1 cm wide border around the edge of every face. Find:
a the total surface area of the block b the painted area c the unpainted area.
34 Find a formula for the area A of the following regions:
a
b c
d
35 A solid cone has diameter 15 mm and slant height 34 mm. Find its surface area.
36 Emma has just bought 60 timber posts to help build a fence. Each post is a cylinder 1 : 8 m long with diameter 16 cm. The total mass of Emma’s posts is 1 : 08 tonnes. Find: a the mass of each post in kilograms b the volume of each post in m^3
37 Find a formula for the surface area A of the following solids: a b
38 Eliza has a bucket with the dimensions shown. She fills it with water, but there is a hole in the bucket, so the water drips out at a rate of 1 : 2 ml/min. How much water remains in the bucket when Eliza returns 3 hours later?
39 Find formulae for the volume V of the following objects: a b
c
40 Des buys a 500 g wedge of his favourite cheese. The wedge is a right angle and is 6 : 1 cm high. Its volume is 460 cm^3. Find the radius of the wedge.
41 A concrete bench for a bus stop is made with the dimensions shown. Show that the volume of concrete used is given by the formula V = a^2 l( ¼ 8 + 8).
42 A metal door handle is formed from three cylindrical pieces. The handles are 4 cm deep and have radius 3 cm. The shaft in the middle has length 12 cm and radius 1 : 5 cm.
Find the total volume of the door handle.
43 a 55 litres of water is added to the cylindrical aquarium shown. How far from the top does the water rise? b Glass marbles of diameter 12 mm are carefully added to the aquarium. How many marbles can be added without causing the water to overflow?
10 cm
10 cm 20 cm
r
q b
c
a
l
d
y
2x
x
35 cm
55 cm
36 cm
g
h
a b (^) 2p
h
5a
3a
b
500 g
6 1. cm
a 2a
2a 2a (^) l
50 cm
40 cm
4 cm 4 cm
12 cm
radius 3 cm
radius radius 1.5cm 3 cm
Cambridge IGCSE International Mathematics (0607) Extended^36 Exam Preparation & Practice Guide
