11MA NOTES ON MEASUREMENT
Converting from one metric unit to another
Perhaps the most common metric conversions are between mm, cm, m and km.
1 cm = 10 mm
1 m = 100 cm = 1000 mm
1 km = 1000 m = 100 000 cm = 1 000 000 mm
kilo means 1000, so 1 km = 1000 m
centi means 1/100th, so 1 cm =0.01 m
milli means 1/1000th, so 1 mm = 1/1000th m = 0.001m
Converting units of mass and capacity
Once you are confident with conversions between mm, cm, m and km, converting units of mass and
capacity is eaier.
For example:
1 km = 1000 m, so 1 kg = 1000 g
1 m = 100 cm, so 1 l (litre) = 100 cl (centilitres)
1 m = 1000 mm, so 1 g = 1000 mg (milligrams)
Converting between metric and imperial units
Here are some examples of metric and imperial measures of length, mass and capacity:
Metric
Imperial
Length
mm, cm, m, km inch, foot, yard, mile
Mass
mg, g, kg
Capacity mL, cL,Ll
ounce (oz), pound (lb), stone
pint, gallon
You will be expected to know some common conversions between metric and imperial units. Some of these
are shown
below, but check with your teacher which ones you need to learn.







1 km = 5/8 mile
1 m = 39.37 inches
1 foot = 30.5 cm
1 inch = 2.54 cm
1 kg = 2.2 lb
1 gallon = 4.5 litres
1 litre = 1 3/4 pints
Perimeter = total length of all sides
Example 1: Find the perimeter of a triangle with sides measuring 5cm,
9cm and 11cm.
Solution:
P = 5cm + 9cm + 11cm = 25cm
Circumference = distance around the outside of a circle
Formula:
Example 2:
𝐶 = 𝜋𝑑 or
The diameter of a circle is 3cm. What is the circumference?
𝐶 = 𝜋𝑑
𝐶 = 𝜋×3
𝐶 = 9.42𝑐𝑚
Solution:
Example 3:
𝐶 = 𝜋2𝑟
The radius of a circle is 2cm. What is the circumference?
Solution:
𝐶
𝐶
𝐶
𝐶
= 𝜋2𝑟
= 𝜋2(2)
= 𝜋×4
= 12.57𝑐𝑚
Basic Area formula:
Square
= a2
Rectangle
= ab
Parallelogram
= bh
Trapezoid
=
ℎ
2
(𝑏1 + 𝑏2)
Triangle
Circle
𝜋𝑟 2
Or
𝒅
= 𝝅( )𝟐
𝟐
Surface area
Is the total area of all the faces of the 3D object
Cuboid – 6 faces
add together the areas of the 6 faces
Cylinder – e.g. can of baked beans
Surface Area
= 2 ends + Curved surface
= 2xpi. r2 + 2pi.r.h
= 2pi.r(r+h)
Cone
Base + curved surface
=pi.r2 + pi.r.l
Hint: think about what it looks like
unfolded, what would the net look like?
=pi.r(r+l)
Sphere
4pi.r2
Volume
To find the area of 3D shapes
Volume = area of an end x length
Cube
V = X3
Pyramid
V=1/3 area of base x height
Cuboid
V= l x w x h
Cone
V= 1/3 area of base x height
V = 1/3 pi.r2.h
Cylinder
V= area of circle x length
V= pi.r2.h
Sphere
V= 4/3pi.r3
Capacity
A solids volume is measured in cubic units i.e. cm3, m3...
Liquid volume is known as capacity and generally measured in mL, L or kL (note: use capital L so not to
confuse with number 1)
1cm3= 1mL
1 000cm3 = 1L
1 000 000 cm3 or 1m3 = 1 kL (kilolitre)
There is a link from capacity to the mass (weight) of a 3D object
1mL = 1g
1L = 1kg
So if we know the volume we can find the weight
Summary
Solid volume
1 cm3
1 000 cm3
1 000 000 cm3
or 1 m3
Liquid capacity
1mL
1L
1kL
Mass/weight
1g
1kg
1000kg
or 1tonne
Density
The ratio of mass to volume
e.g. a cup of sugar weighs more than a cup of marshmallows as they have different densities
The relationship can be expressed by this triangle:
Mass = Volume x Density
M
V D
𝑀𝑎𝑠𝑠
Volume = 𝐷𝑒𝑛𝑠𝑖𝑡𝑦
𝑀𝑎𝑠𝑠
Density = 𝑉𝑜𝑙𝑢𝑚𝑒
Density if often compared to that of water as water has density of exactly 1
1cm3 of water = 1mL = 1g = density of 1g/cm3
Note: Density is expressed as a weight per unit
g/cm3 or t/m3
EXAMPLE
Concrete has a density of 2.3g/cm3
Find the weight of this brick
V = 12 x 10 x 24
V = 2880cm3
Weight
10cm
24cm
= volume x density
= 2880 x 2.3
= 6624g ≈ 6.6kg
Rates
Compares two quantities measured in different units. It is used to describe how one quantity changes in
comparison to another e.g. km/h or wage per hour
Note: the word “per” indicates we are dealing with a rate
Graphically it is the slope of a line
EXAMPLES
#1. Miss Brien went for a 28km run on the weekend and it took her 2hours 38 mins, how long on average
did each km take her?
Change all to minutes first
2x60+38=158mins total
Divide by total km
158/28= 5.64mins
Can’t have .64 of a minute, convert that to seconds
60x0.64= 38.4mins round to 39seconds
So on average it took 5mins 39seconds per km
#2. Miss Brien estimates she will run at 6min per km in her marathon next weekend, how long will she take
to finish the 42.2km marathon?
Need to convert to hours
(however can’t have .22 of an hour)
Convert the .22 to minutes
Convert the .2 mins to seconds
So should take 4 hours 13mins 12seconds!
6x42.2=253.2mins
253.2/60= 4.22 hours
0.22x60=13.2mins
60x.2=12
Distance/Velocity/Time
Distance – km or m
Speed – km/hr or m/s
Time – hours, mins, seconds
This triangle shows the different relationships
Distance = Speed x Time
D
S T
Speed =
Time =
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑇𝑖𝑚𝑒
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑆𝑝𝑒𝑒𝑑
EXAMPLE
#1. A runner covered 28km in 2hrs 38mins at what speed are they travelling at?
Convert time to mins
2x60+38 = 158mins
Speed = distance over time
S=28/158
S= 0.1772 km per minute (need to convert back to hours)
S= 0.1772x60=10.63km/hr
#2. A cyclist travels at a constant speed of 34km/hr for 48mins, how far would they have travelled?
48/60= 0.8 (48mins is 0.8 of an hour)
Distance = speedxtime
D=38x0.8
D=30.4km
Or
D=38x48
D=1824 (per minute)
Need to divide by 60 to get per hour
1824/60=30.4
Converting m/s to km/hr
x 1000
÷ 3600
Km/hr
m/s
÷ 1000
x 3600
Time distance Graphs
Shows distance from a fixed point over a time period
 Time is on the horizontal (x-axis)
 Distance is on the vertical (y-axis)
The slope/gradient of the line represents the speed travelled
e.g. Your day at school
Horizontal line indicates NO movement
i.e. when you are at school
Distance from
home (km)
8:45am
3pm
Time
Significant figures
The significant figures (sig figs,or s.f.) of a number are those digits that carry meaning contributing to its
accuracy. This includes all digits except:

Leading and trailing zeros are only placeholders to indicate the scale of the number.
E.g.
o
0.00000456
3 s.f.
o
23 000
2 s.f.
o
34.3
3 s.f.
o
102.38
5 s.f.
o
84.50
4 s.f. trailing after d.p do count if written e.g. money
Rounding
Rounding is often done on purpose to obtain a value that is easier to write and handle than the original.
Whenever rounding is done you need to state the accuracy of it i.e. “3 s.f.” or “2 d.p. “
Rounding should generally be done so the answer has the same number of s.f. as the number in the
calculation
When is something is measured it is not always exact. It is only accurate to half of the unit shown by the
last s.f.
Limits of accuracy
There is a lower and upper limit of accuracy for any measurement
e.g.
34
lower limit = 33.5
Upper limit =34.5
19.8
Lower limit = 19.75
Upper limit = 19.85
1200
Lower limit = 1150
Upper limit = 1250
33
34
35
19.7
19.8
19.9
1100
1200
1300
What we really want to find is:
The smallest number that will round up to out number = lower limit
The biggest number that rounds down to our number = upper limit
Trick:
lower limit
“minus one off the last s.f.
then add 5 to the end “
Upper limit
“add 5 after last number”