Measurement


Area - Introduction


The area of a shape is a measure of how many square units it contains. The units employed would be typically

  • • square meters - m2
  • • square centimeters - cm2
  • • square millimeters - mm2

For larger areas, the hectare is used - an area equivalent to the area of a square with sides of 100 meters.

A hectare would have an area of

(100 m)2 = 100 m × 100 m = 10 000 m2

This means that a square kilometer contains 100 hectares

so an area of 679 hectares = 6.79 km2

Note that, in the case of area, conversion to different units is more complicated than with 'straightforward' length. For example, a square kilometer is 1 000 000 times as large as a square meter ( NOT 1 000 times as large ).

Consider squares for simplicity - a square kilometer will have sides 1 000 times as long as a square meter, but by multiplying 1 000 by 1 000 you can see that a square kilometer is indeed 1 000 000 ( 1 million ) times as large as a square meter.

Quick Quiz

How many square centimeters are there in

  1. 1.    2 m2
  2. 2.    6.4 m2

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Perimeters


Calculation of perimeters is fairly straightforward. The two main things to watch out for are

  • • Include every side in your addition. For example, normally you are only given two dimensions to a rectangle but you need to add four sides together.

  • • Make sure all your dimensions are in the same units before adding.

So the perimeter of the rectangle shown is

9 + 9 + 4 + 4 = 26 cm
or alternatively

(2 × 9) + (2 × 4) = 26 cm

Quick Quiz

What are the lengths of the perimeters of the following figures ?



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Area of Rectangles


The area of a rectangle is calculated straightforwardly as

length x width

For the rectangle on the left, the area would be given by

l x w

For a square this would just be the same two numbers multiplied together - you can see the derivation of the phrase 'three squared' to describe an expression like 32.

This concept of multiplying two lengths generalizes to all area formulae. So an expression like

4 p r2

where two 'lengths' (i.e. r) are multiplied together could be the formula for the area of a shape (p is a constant, as is 4, which have same value for all shapes - and therefore do not correspond to a 'length').

In fact, this expression does correspond to a real-life area - the surface area of a sphere.

Quick Quiz

  1. What is the area of this rectangle ?
  2. What is the area of this Danish flag (all dimensions are in centimeters)
  3. This diagram gives the range of dimensions allowed for a tag rugby pitch. What is the smallest area that such a pitch can be ?

  1. Which of these could   be the formula for an area

    • abc
    • ab2
    • ab
    • b2
    • 3c ÷ b
    • 3ca

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The Area of More Complicated 'Rectangular' Shapes


The technique for more complicated shapes is to try and break them up into rectangles whose area can be calculated, and then summing the areas of these separate rectangles.

For an L-Shape, as shown on the left this can be split into two separate rectangles. You have a couple of options - here we will form a rectangle from the 'bar' at the top, and another rectangle from what is left.

So the top rectangle has an area of

7 × 1 = 7 cm 2

and the lower rectangle will have an area of

4 × 2 = 8 cm 2

So the total area will be

7 + 8 = 15 cm 2

Quick Quiz

What is the area of these shapes ?

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Area of a Triangle




Stated in a nutshell :-

The area of a triangle is given by half the base multiplied by the altitude (or height).


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Circles


Circumference = 2 p r

Area = p r2


Differentiate between the two formulas by remembering that at a simple level, the area of a rectangle is found by multiplying two lengths together. This concept generalizes such that an expression for an area will consist of two lengths (or variables) multiplied together.

So of the two expressions above

p r2

must be the expression for the area of the circle.

For completeness, when you come to find the volume of a sphere, you might expect that the formula will contain three lengths (or variable terms) multiplied together. And this is indeed so : The formula for the volume of a sphere is

\[ \frac{4}{3}\pi r^3 \]

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Volumes


This is 'one step further on' from the calculation of areas.

In areas, we multiplied two lengths together. For volumes, we multiply three lengths together.

The volume is given by

4 × 2 × 5

= 40 cm2

For a cube this would just be the same three numbers multiplied together - you can see the derivation of the phrase 'three cubed' to describe an expression like 33.

This concept of multiplying three lengths generalizes to all volume formulae. So an expression like

\[ \frac{4}{3}\pi r^3 \]

where three 'lengths' (i.e. r) are multiplied together could be the formula for the volume of a shape (p is a constant, as is $\frac{4}{3}$, which have same value for all shapes - and therefore do not correspond to a 'length').

In fact, this expression does correspond to a real-life volume - the volume of a sphere.

Quick Quiz

  1. 1.    The height of this aquarium is 1.4 meters, its length is 3.5 meters and its width is 2 meters. What is its volume?
  2. 2.    This case has length 80 cms, height 50 cms and depth 150 mms. What is its volume?

  1. 3.    Which of these could   be the formula for a volume

    • abc
    • ab2
    • ab
    • b2
    • 3c ÷ b
    • 3ca

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The Number of Items within a Given Container


Proceeding by example -

As already mentioned, a liter is a decimeter cubed (dm3). Or put another way, if a liter was contained in a cubic-shaped pack then its sides would all be 1 dm (or 10 centimeters).

Now what if I was to ask how many cubic-shaped liter packs could fit into a container with dimensions

1 m × 2 m × 3.5 m


The first thing to note is that they will fit exactly into such a container, because 1 dm. (or 10 centimeters, or 0.1 meters) is an exact multiple of each of the dimensions.

I then consider

How many times will 0.1 meters go into 1 m ?    10
How many times will 0.1 meters go into 2 m ?   20
How many times will 0.1 meters go into 3.5 m ?   35


Therefore the total no of liter packs that will fit into the container is

10 × 20 × 35 = 7 000

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The Volume of More Complex Shapes

Still to be written

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Surface Area of Three-Dimensional Objects


For 'simple' shapes this is fairly straightforward. Normally you would find the area of each side individually, so you would make use of your knowledge of the area of two-dimensional figures, and then sum all your areas together.

Just make sure that you include all the sides of the figure, and note that sometimes the symmetry of the object can reduce the calculations involved.

For example, to find the surface area of a cuboid house-brick could involve finding the area of six shapes (the six sides) separately and summing them to find the total surface area. However, note that because of the symmetry of the brick, we only need to calculate the area of three sides, and then double the area of each side.

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Questions from Past Papers


Ali needs an 18-inch zip to replace a broken one in a cushion. The shop only sells them in centimeters. Which zip is the closest to the one Ali needs given that 1 inch is approximately equal to 2.5cm?

  • A    7cm
  • B    36cm
  • C    45cm
  • D    54cm




This is the floor plan of a room. What is the area of the floor ?


  • A    24.5 m2
  • B    25.5 m2
  • C    29.5 m2
  • D    31.5 m2




The diagram below shows a carton, seen from the top, full of tins of shoe polish. The carton contains 5 layers of tins. Each tin is 8 centimeters in diameter and 2 centimeters high.


How many tins of shoe polish are in 55 cartons ?

  • A    825
  • B    1055
  • C    1100
  • D    5500

The tins fit tightly into the carton. What are the measurements of the inside of the carton ?

  • A    8 cm by 8 cm by 2 cm
  • B    8 cm by 8 cm by 10 cm
  • C    40 cm by 32 cm by 2 cm
  • D    40 cm by 32 cm by 10 cm

The shoe polish sells for 39p per tin or 3 tins for £1. What is the least amount of money that could be collected from the sale of one carton of shoe polish tins ?

  • A    £ 33.00
  • B    £ 33.39
  • C    £ 39.00
  • D    £ 390.00




A carpet company has some floor plans. Which floor plan represents an area of 109m2 ?




The diagram below shows the lawn and a patio in a garden


Which of these is a correct way to find the area of the lawn ?

  • A    (6 × 8) + (7 × 10)
  • B    (7 × 8) + (3 × 2)
  • C    (8 × 7) + (6 × 3)
  • D    (10 × 6) +(8 × 7)

The area of the patio is

  • A    4 m2
  • B    6 m2
  • C    9 m2
  • D    18 m2




The crate below is to be packed with boxes. The boxes are all the same size as each other.


What is the largest number of boxes that can fit inside the crate ?

  • A    19 boxes
  • B    27 boxes
  • C    33 boxes
  • D    57 boxes




The diagram shows a piece of card. Its area has to be $\frac{1}{8}$ m2. It is 50 cm long.


1.   What must the width of the card be

  • A    6 $\frac{1}{4}$ cm
  • B    16 cm
  • C    25 cm
  • D    40 cm

2.   The card is cut from a sheet whose area is one square meter. The sheet weighs 180 grams. What is the weight of the card ?

  • A    2.20 g
  • B    14.4 g
  • C    22.0 g
  • D    22.5 g




Stage 3

A swimming pool is 21.0 meters long by 15.6 meters wide. It has a level bottom (floor). The initial quantity of water in the pool is 90 000 gallons.

[1 gallon is approximately 4.55 liters,   1000 liters = 1 m³]


a.   The amount of chlorine required to maintain the quantity of water in the pool is 80 parts per million (80 ppm).

How many liters of chlorine are required for the pool?

b.   How many more gallons of water and chlorine mixture are required to provide a depth of 1.5 meters in the pool?


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