Measurement

Written on 20 July 2020.

Units of Measurement

Sections:

0, Chapter 1, Introduction,
Introduction

After completing this unit you should be able to
- use and apply units of measure for length, mass and volume
- convert between metric measures
- understand how to calculate upper and lower bounds for measurements
- estimate areas
- approximate between SI units and basic Imperial units.
You have four sections to work through.
1. Units and Measuring
2. Upper and Lower Bounds
3. Estimating Areas
4. Conversion of Units
0, Chapter 2, Units and Measuring,
Units and Measuring

Different units of measurement can be used to measure the same quantities. It is important to use sensible units. For example you would not think of the distance between London and Glasgow in terms of centimetres but rather kilometres.

Some important units are listed below.

Use the slider to explore worked examples.
- Worked Example
  
  What would be the best units to use when measuring,
  
  the distance between London and Plymouth
  
  the width of a watch-face
  
  the mass of a person
  
  the mass of a letter
  
  the mass of a lorry
  
  the volume of medicine in a spoon
  
  the volume of water in a swimming pool
- Worked Example
  
  How many mm are there in \(3.72\) m?
  
  How many cm are there in \(4.23\) m?
  
  How many m are there in \(102.5\) km?
  
  How many kg are there in \(4.32\) tonnes?
1, Chapter 2, Task 1, Fitness Check,
Exercises

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Which units do you think would be the most suitable to use when measuring:

a. the distance between two towns

b. the length of a sheet of paper

c. the mass of a sheet of paper

d. the mass of a sack of cement

e. the volume of water in a cup

f. the volume of water in a large tank

Exercise 2

What value does each arrow point to?

a.

b.

c.
0, Chapter 3, Upper and Lower Bounds,
Upper and Lower Bounds

When measurements are made, they can only be obtained to a limited degree of accuracy. For example if the length of a line is given as \(11\) mm, this means that it is \(11\) mm to the nearest mm. This means, for example, that the original measurement could have been \(11.3\), as \(11.3\) would round to \(11\); but the original measurement could also have had a value like \(10.6\), as \(10.6\) would also round to \(11\).

There are many values that the original measurement could have taken.
In fact, if \(l\) is the length of \(11\) mm then it must lie in the range \(10.5\leq l\lt 11.5\),
This just says that the length can be anything between \(10.5\) mm and \(11.5\) mm

Remember

The lower bound is the smallest number which will round up to the given number, upper bound is the smallest number which will not round down to the given number.

For example, \(9\) mm, measured to the nearest mm, means that
\(8.5\) mm \(\leq\) actual length \(\lt 9.5\) mm
Here \(8.5\) mm is the lower bound and \(9.5\) mm is the upper bound

Examples

For each length below we show the range of values within which the length must be.
1. \( 18 \) cm ⇒ \(17.5\) cm \(\leq l \lt 18.5 \) cm
2. \(12.7\) cm ⇒ \(12.65\) cm \(\leq l \lt 12.75\) cm
3. \(11.06\) cm ⇒ \(11.055\) cm \(\leq l \lt 11.065 \) cm
1, Chapter 3, Task 1, Worked Example,
Worked Example
- Worked Example
  
  If the length of the sides of the rectangle below are given to the nearest cm, find:
  
  a) The minimum possible perimeter.
  
  b) The range of possible areas
- Worked Example
  
  The quantities \(x\) and \(y\) are given to 1 significant figure as
  \(x = 20\) and \(y= 40\)
  Find the minimum possible value of each expression below.
  
  \(xy\)
  
  \(\frac{x}{y}\)
1, Chapter 3, Task 2, Fitness Check,
Exercises

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

For each quantity below, state the range of values within which it must lie.

a. \(x = 4.7\)

b. \(r = 11.68\)

c. \(w = 218\)

d. \(v = 20.0\)

Exercise 2

The quantities \(x\) and \(y\) are given to \(1\) significant figure as \(x = 30\) and \(y = 60\).
Find the maximum possible value of the expressions below:

a. \(y − x\)

b. \(x + y\)

c. \(\frac{y}{x}\)

Exercise 3

If the length of the sides of the rectangle are given to the nearest cm, find:

a. the maximum possible area

b. the range of possible perimeters.
0, Chapter 4, Areas Concept,
Areas Concept

A square with sides of \(1\) cm has an area of \(1\) cm²

The area of shapes can be either counted or estimated to see how many of these squares fit inside the shape.

Use the slider to explore worked examples.
- Worked Example
  
  Find the area of the shaded shape.
- Worked Example
  
  Find the area of the shaded shape.
- Worked Example
  
  Estimate the area of the shaded shape
1, Chapter 4, Task 2, Fitness Check,
Exercises

Exercise 1

Find the area of the shaded shape.

Exercise 2

Find the area of the shaded shape.

Exercise 3

Estimate the area of the shape shaded in the diagram.
0, Chapter 5, Conversion of Units,
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 are;
length (inch, foot, yard, mile)
mass (ounce, pound, stone)
volume (pint, gallon)

SI (metric) units are;
length (mm, cm, m, km)
mass (g, kg)
volume (ml, litre)

Conversion of units

The following list shows conversions within imperial units of measurement

\( 1\) foot \(= 12 \) inches

\( 1 \) yard \(= 3 \) feet

\( 1 \) mile \(= 1760 \) yards

\( 1 \) pound (lb) \(= 16 \) ounces

\( 1 \) stone \(= 14 \) pounds

\( 1 \) gallon \(= 8 \) pints

You can convert between metric and imperial units using the following facts.

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.
1, Chapter 5, Task 1, Worked Example,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  John is measured. His height is 5 feet and 8 inches.
  
  Find his height in:
  
  inches
  
  centimetres
  
  metres
- Worked Example
  
  A family travels \(65\) miles on holiday. Convert this distance to km.
- Worked Example
  
  A line is \(80\) cm long. Convert this length to inches
- Worked Example
  
  John weighs \(8\) stone and \(5\) pounds. Find John's weight in:
  
  pounds
  
  kg
1, Chapter 5, Task 2, Fitness Check,
Exercises

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Convert each quantity to the units given.

a. \(3\) inches to cm

b. \(18\) stone to pounds

c. \(6\) lbs to ounces

d. \(6\) feet \(3\) inches to inches

e. \(15\) kg to lbs

f. \(3\) yards to inches

g. \(3\) feet to cm

h. \(5\) gallons to litres

i. \(45\) kg to lbs

j. \(9\) litres to pints

k. \(45\) gallons to litres

l. \(8\) litres to pints

Exercise 2

Jimar is \(6\) feet \(2\) inches tall and weighs \(11\) stone \(5\) pounds.
Sam is \(180\) cm tall and weighs \(68\) kg.
Who is the taller and who is the heavier?
Find your answer by converting Jimar’s measurements into metric measurements.

Exercise 3

A car travels on average \(10\) km for every litre of fuel.
The car is driven from Chichester to Woking, a distance of \(41\) miles.

a. How far does the car travel in km?

b. How many litres of fuel are used?

c. How many gallons of fuel are used?
0, Chapter 6, Summary,
Summary

SI Units

length (mm, cm, m, km)

mass (g, kg)

volume (ml, litre)

Upper and Lower Bounds

For example, \(9\) mm, measured to the nearest mm, means that
\(8.5\) mm \(\leq\) actual length \(\lt9.5\) mm
Here \(8.5\) mm is the lower bound and \(9.5\) mm is the upper bound.

Imperial Units

length (inch, foot, yard, mile)

mass (ounce, pound, stone)

volume (pint, gallon)

Conversion of Units

\(1\) km \(=1000\) m

\(1\) m \(=100\) cm \(=1000\) mm

\(1\) cm \(=10\) mm

\(1\) tonne \(=1000\) kg

\(1\) kg \(=1000\) g

\(1\) litre \(=1000\) ml

\(1\) m\(^3=1000\) litres \(=1000\) mm

\(1\) cm\(^3=1\) ml

\(5\) miles is approximately \(8\) km

\(1\) foot is approximately \(30\) cm

Interactive Exercises:

Interactive Exercises - Units of Measurement, https://www.cimt.org.uk/sif/measurement/m3/interactive.htm
Units and Measuring, https://www.cimt.org.uk/sif/measurement/m3/interactive/s1.html
Upper and Lower Bounds, https://www.cimt.org.uk/sif/measurement/m3/interactive/s2.html
Estimating Areas, https://www.cimt.org.uk/sif/measurement/m3/interactive/s3.html
Conversion of Units, https://www.cimt.org.uk/sif/measurement/m3/interactive/s4.html

YouTube URL: https://www.youtube.com/watch?v=jaexoIy3nXs&list=PL8ce1n6mRlCtS0rO_C7CNwtjhyqDVkr8g&index=18

File Attachments: media/course-resources/Units-Of-Measurement_Presentation.pptxmedia/course-resources/Units-Of-Measurement_Text.pdfmedia/course-resources/Units-Of-Measurement_Answers.pdfmedia/course-resources/Units-Of-Measurement_Essential-Information.pdfmedia/course-resources/Units-Of-Measurement_Learning-Objectives.pdf

Written on 20 July 2020.

Solids

Sections:

0, Chapter 1, Introduction,
Introduction

Here we learn about 3 dimensional shapes.

After completing this unit you should be able to
- use a 2D net to construct a 3D shape
- name the common 3D shapes
- design a net for a given 3D shape
- determine the plan and elevations (front and side) for a given 3D shape
- use isometric paper to illustrate 3D shapes.
You have four sections to work through.
1. Making Solids Using Nets
2. Constructing Nets
3. Plans and Elevations
4. Using Isometric Paper
0, Chapter 2 , Making Solids Using Nets,
Making Solids Using Nets

In this unit, you will be dealing with 3-dimensional shapes such as:

We will be working with the concept of nets. A net is a flat (1 dimensional) shape which when folded and glued makes a hollow 3 dimensinal shape.
The conept will become clearer when we look at examples. We begin by looking at the net of a prism.
1, Chapter 2.1, Prisms,
Prisms

Prisms are a common 3 dimensional shape. They can be formed by taking any two dimensional shape and extending (‘extruding’) it to create a 3 dimensional shape with a cross section of the original 2D shape.Think of a toblerone.

The prism can be named after the 2D shape, for example, the prism below could be named a triangular prism.
1, Chapter 2.1, Task 1, Worked Example 1,
Worked Example
- Worked Example
  
  Here is a net below. We show which solid is made when the net shown is folded and glued?
1, Chapter 2.1, Task 2, Exercise 1,
Exercise

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Copy and cut out larger versions of the following net. What is the name of the solid you make?

Exercise 2

Copy and cut out larger versions of the following net. What is the name of the solid you make?
0, Chapter 3, Constructing Nets,
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.

Use the slider to explore worked examples.
- Worked Example
  
  Draw the net for the cuboid shown in the diagram.
- Worked Example
  
  Draw the net for this square-based pyramid.
1, Chapter 3, Task 1, Exercise 2,
Exercise

Here are some questions to check your progress; there are more practice questions if needed.

Exercise

Draw an accurate net for the cuboid below.

Exercise

Draw an accurate net for the prism below.

Note: this is just one possible answer
0, Chapter 4, Plans and Elevations,
Plans and Elevations

The plan of a solid is the view from above, as in the diagram below.

The elevation of a solid is the view from a side.

We often use the terms front elevation and side elevation.

Looking from the side gives the Side elevation

Looking from the front gives the Front elevation

Looking from above gives the Plan
1, Chapter 4, Task 1, Worked Example 2,
Worked Example
- Worked Example
  
  Draw the front elevation, side elevation and plan of this shape, taking the L face to be the ‘front’ of the shape.
1, Chapter 4, Task 2, Exercise 3,
Exercise

Here is a question, to check your progress; there are more practice questions if needed.

Draw the front elevation, side elevation and plan of this shape, taking the triangular face to be the ‘front’ of the shape.
0, Chapter 5, Isometric Paper,
Isometric Paper

Isometric paper is paper with spots arranged in a grid and is used to produce three dimensiona views of solids. The spots on isometric paper are arranged at the corners of equilateral triangles.

Here is an example of a section of Isometric paper
1, Chapter 5, Task 1, Worked Example 3,
Worked Example
- Worked Example
  
  The diagrams below show the plan and elevations of a solid object.
  
  Plan
  
  Front Elevation
  
  Side elevation
  
  Draw the object on isometric paper.
1, Chapter 5, Task 2, Exercise 4,
Exercise

Exercise 1

Here is a question, to check your progress; there are more practice questions if needed.
Draw a cube with sides of 2 cm on isometric paper.

(note that you may draw the sides in a different order, but the end result will be the same)
0, Chapter 6, Summary,
Summary

Interactive Exercises:

Solids Interactive Exercises, https://www.cimt.org.uk/sif/measurement/m2/interactive.htm
Making Solids Using Nets, https://www.cimt.org.uk/sif/measurement/m2/interactive/s1.html
Constructing Nets, https://www.cimt.org.uk/sif/measurement/m2/interactive/s2.html
Plans and Elevations, https://www.cimt.org.uk/sif/measurement/m2/interactive/s3.html
Using Isometric Paper, https://www.cimt.org.uk/sif/measurement/m2/interactive/s4.html

YouTube URL: https://youtu.be/GqEXIWUFs8Y

File Attachments: media/course-resources/Solids_PowerPoint-Presentation.pptxmedia/course-resources/Solids_Text.pdfmedia/course-resources/Solids_Answers.pdfmedia/course-resources/Solids_Essential-Informations.pdfmedia/course-resources/Solids_Learning-Objectives.pdf

Written on 21 July 2020.

Area

Sections:

0, Chapter 1, Introduction,
Introduction

The work in this unit is focused on the very important topic of area. After completing this unit you should be able to:
- use formulae to calculate the areas of squares, rectangles and triangles
- understand and calculate the area and circumference of a circle
- use formulae to calculate the areas of parallelograms, trapeziums and kites
- calculate the surface areas of solids including cuboids, cylinders and spheres
- find the arc length and area of a sector of a circle.
You have five sections to work through.
1. Squares, Rectangles and Triangles
2. Area and Circumference of Circles
3. Sector Areas and Arc Lengths
4. Areas of Parallelograms, Trapeziums, Kites and Rhombuses
5. Surface Area
0, Chapter 2, Squares, Rectangles and Triangles,
Squares, Rectangles and Triangles

Squares, Rectangles and Triangles

Squares, Rectangles and Triangles
In this section we remind you of the formula for the area of a square, rectangle and a triangle and and show some examples of calculations.

Area of Square

For a square, the area is given by \(x\times x = x^2\) and the perimeter by \(4x\), where \(x\) is the length of a side.

Remember

The perimeter of a shape is the distance all the way round its outside.
1, Chapter 2.1, Area of Rectangle,
Area of Rectangle

Area of rectangle

For a rectangle, the area is given by \(l\times w\) (which we write as \(lw\) ) and the perimeter by \(2(l+w)\), where \(l\) is the length and \(w\) the width.
1, Chapter 2.2, Area of Triangle,
Area of Triangle

Area of triangle

For a triangle, the area is given by \(\frac{1}{2} bh\) and the perimeter by \(a+b+c\), where \(b\) is the length of the base, \(h\) the height and \(a\) and \(c\) are the lengths of the other two sides.
1, Chapter 2, Task 1, Worked Example 1,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Find the area of the following triangle:
- Worked Example
  
  Find the perimeter and area of each shape below.
  
  When working out the area of mixed shapes step one is break up the area into the basic shapes for which we have a formula - try a few examples and you will quickly get the idea
  
  The perimeter is found by adding the lengths of all the sides.
1, Chapter 2, Task 2, Exercise 1,
Exercises

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Find the area of the triangle.

Exercise 2

Find the area and perimeter of the shape.

Note that the area can be found, for example, by splitting the shape into three rectangles as below.

Exercise 3

Find the area of the shape.
0, Chapter 3, Area and Circumference of Circles,
Area and Circumference of Circles

The lengths on a circle are shown on the diagram below, where r is the radius and d is the diameter.
Also, we have the length around the outside of the circle which is called the circumference, C.

Remember

The circumference of a circle can be calculated using \(C = 2 \pi r\) or \(C = \pi d\)
where r is the radius and d the diameter of the circle. \(\pi\) represents the number 3.14
The area of a circle is found using \(A = \pi r^{2}\) or \(A = \frac{\pi d^{2}}{4}\)
1, Chapter 3, Task 1, Worked Example 2,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Find the circumference and area of this circle.
- Worked Example
  
  Find the radius of a circle if:
  
  its circumference is \(32\;cm\),
  
  its area is \(14.3\;cm^{2}\)
- Worked Example
  
  Find the area of the door shown in the diagram. The top part of the door is a semicircle.
1, Chapter 3, Task 2, Exercise 2,
Exercises

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Find the circumference and area of this circle

Exercise 2

Find the circumference and area of this circle.

Note: The diameter is given in the diagram and you need to half this to get the radius.

Exercise 3

Find the radius of a circle if its area is \(69.4 cm^2\)

Exercise 4

Find the length of one complete circuit of the track.
0, Chapter 4, Sectors and Arc Length,
Sectors and Arc Length

A sector is a slice of a circle, like a slice of a cake.
Part of the circumference of a circle is called an arc, as shown in the diagram.

Remember

If the angle at the centre of the circle is \(\theta\) then arc length \(l\) is given by: \(l=\frac{\theta}{360}\times 2\pi r\)
1, Chapter 4, Task 1, Example 1,
Example

For example, the length \(L\) below can be found with the formula \(L=\frac {\theta}{360} \times 2 \pi r\),
using \(5\;cm\) for \(r\) and \(60^\circ\) for \(\theta\), the angle.

\(L=\frac{\theta}{360} \times 2\pi r =\frac{60}{360} \times 2 \times 3.14 \times 2= 2.09\;cm\)

Note that the \(\frac {60}{360}\) at the start of the formula simplifies to \(\frac{1}{6}\) in this example; the formula works by finding the fraction of the circumference of the circle needed for the arc.
1, Chapter 4.1, Sector Area,
Sector Area

Remember

The area of the sector, as shown in the diagram below, can be found using the formula
\(Area= \frac {\theta}{360} \times \pi r^{2}\)

For example, the area of the sector in the diagram below can be found using \(6\;cm\) for \(r\) and \(40^\circ\) for \(\theta\), the angle.

\(Area= \frac {\theta}{360} \times \pi r^{2}=\frac {40}{360} \times 3.14 \times 6^{2} = 12.56\;cm^{2}\)
1, Chapter 4.1, Task 1, Exercise 3,
Exercises

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Find the length \(L\) shown in the diagram below.

Exercise 2

Find the area of the sector shown below.
0, Chapter 5, Areas of Parallelograms, Trapeziums, Kites and Rhombuses,
Areas of Parallelograms, Trapeziums, Kites and Rhombuses

Parallelograms, Trapeziums, Kites and Rhombuses are all four sided shapes, known as quadrilaterals.

Their areas can be found using the following formulae:

Area formulae

Parallelogram \(A=bh\)
Kite \(A=\frac {1}{2}ab\)
Trapezium \(A=\frac{1}{2} \left(a+b\right) h\)

The area of a rhombus, a diamond shape, can be found using either the formula for a kite or the formula for a parallelogram.
1, Chapter 5, Task 1, Worked Example 4,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Find the area of this kite.
- Worked Example
  
  Find the area of this trapezium.
1, Chapter 5, Task 2, Exercise 4,
Exercises

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

The diagram shows the end wall of a wooden shed. Find the area of this end of the shed.

Exercise 2

Find the area of the shape below.
0, Chapter 6, Surface Area,
Surface Area

Top Tip

To remind yourself about NETS, this topic is covered in the Lesson Geometry - Solids

Three dimensional shapes, like the cuboid shown below, have a surface area which can be found by adding up the areas of all the faces of the shape.

This diagram shows the net for a cuboid.
To find the surface area the area of each of the 6 rectangles must be found and then added to give the total surface area.
1, Chapter 6.1, Cuboid,
Cuboid

Surface Area: Formula for Cuboid

If \(x\), \(y\) and \(z\) are the lengths of the sides of the cuboid, then the area of the rectangles in the net are as shown on the right.

The total surface area of the cuboid is then given by:
\(A = xy + yz + xz + xy + yz + xz\)

This can be written more simply as
\(A=2xy+2yz+ 2xz\)

This is the formula for the surface area of a cuboid.
1, Chapter 6.2, Cylinder,
Cylinder

Surface Area: Cylinder

To find the surface area of a cylinder, consider how a cylinder can be broken up into three parts, the top, bottom and curved surface, as in the diagram on the bottom.

The areas of the top and bottom are the same: they are circles, and each is given by \(\pi r^2\).

The curved surface is a rectangle when rolled out: the next slide shows you how to find the area of this part of the surface.
1, Chapter 6.3, Formula for Cylinder,
Formula for Cylinder

Surface Area of Cylinder: Formula

For the curved surface, the length of one side is the same as the circumference of the circles, 2πr,
and the other side is simply the height of the cylinder, \(h\). So the area is \(2 \pi rh\)

The total surface area of the cylinder is found by adding the area of two circles to the area of this rectangle
\(2 \pi r^{2} + \pi r^{2} + 2\pi rh\)

This can be written more simply as

Surface area of a cylinder

\(A=2 \pi r^{2} + 2 \pi rh\)
1, Chapter 6.4, Sphere,
Sphere

Surface Area: Sphere

Another important result is the surface area of a sphere.

Area of sphere

\(A=4 \pi r^{2}\)
1, Chapter 6.4, Task 1, Worked Example 5,
Worked Example

Use the slider to explore worked examples.
- Worked Example
  
  Find the surface area of this cuboid
- Worked Example
  
  Cans are made out of aluminium sheets, and are cylinders of radius \(3\;cm\) and height \(10\;cm\).
  
  Find the area of aluminium needed to make one can.
- Worked Example
  
  A ball has radius \(4\;cm\). What is its surface area, to the nearest \(cm^{2}\) ?
1, Chapter 6.4, Task 2, Exercise 5,
Exercises

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Find the surface area of the cuboid below.

Exercise 2

Find the surface area of the cylinder opposite.
0, Chapter 7, Summary,
Summary

You will find it useful to be fully familiar with the facts listed here.

Interactive Exercises:

Area Interactive Exercises, https://www.cimt.org.uk/sif/measurement/m1/interactive.htm
Squares, Rectangles and Triangles, https://www.cimt.org.uk/sif/measurement/m1/interactive/s1.html
Area and Circumference of Circles, https://www.cimt.org.uk/sif/measurement/m1/interactive/s2.html
Sector Areas and Arc Lengths, https://www.cimt.org.uk/sif/measurement/m1/interactive/s3.html
Areas of Parallelograms, Trapeziums, Kites and Rhombuses, https://www.cimt.org.uk/sif/measurement/m1/interactive/s4.html
Surface Area, https://www.cimt.org.uk/sif/measurement/m1/interactive/s5.html

YouTube URL: https://youtu.be/4s3jjB5oTj8

File Attachments: media/course-resources/Area_Learning-Objectives.pdfmedia/course-resources/Area_PowerPoint-Presentation.pptxmedia/course-resources/Area_Text.pdfmedia/course-resources/Area_Answers.pdfmedia/course-resources/Area_Essential-Informations_copy_1.pdf

Written on 22 July 2020.

Volume

Sections:

0, Chapter 1, Introduction,
Introduction

This unit deals with the important topic of volume. After completing it you should be able to
- use the volume formulae for cubes, cuboids, cylinders and prisms
- calculate the density of an object, given its mass and volume
- use the volume formulae for pyramids, cones and spheres
- apply the volume formulas to solve real life problems
- apply dimensional analysis to develop formulae.
You have four sections to work through.
1. Volumes of Cubes, Cuboids, Cylinders and Prisms
2. Mass, Volume and Density
3. Volumes of Pyramids, Cones and Spheres
4. Dimensions
0, Chapter 2, Volumes of Cubes, Cuboids, Cylinders & Prisms,
Volumes of Cubes, Cuboids, Cylinders and Prisms

The volume of a shape measures the three dimensional (3D) amount of space it takes up.

Different shapes have different ways to find the volume.

Remember

The volume of solid objects is measured in cubic units. The most important thing to remember when calculating volume is that all the dimensions must be in the same units.

Volumes of Cubes

Recall that a cube has all edges the same length. The volume of a cube is found by multiplying the length of any edge by itself twice.

Or as a formula:

\(V=a^{3}\)

where \(a\) is the length of each side of the cube.

Volume Cuboids

Volume of a cuboid = (length × breadth × height) cubic units. = (l × b × h) cubic units.

Or as a formula:

\(V=abc\)

where \(a\), \(b\) and \(c\) are the lengths shown in the diagram.

Worked Example

The diagram shows a truck.

Find the volume of the load-carrying part of the truck.

Volume of Cylinders

The volume of a cylinder is given by the area of the circular base multiplied by its length (or height.)

Or as a formula:

The volume of a cylinder is given by

\(V=\pi r^{2} h\)

where \(r\) is the radius of the cylinder and \(h\) is its height.

Worked Example

The cylindrical body of a fire extinguisher has the dimensions shown in the diagram.

Find the maximum volume of water the extinguisher could hold.

Volume of Triangular Prisms

The volume of tiangular prism is given by the area of the triangular base multiplied by its length(or height)

Or as a formula:

The volume of a triangular prism is given by

\(V=A\;l\)

where \(A\) is the area of the end and \(l\) the length of the prism

or

\(V=\frac {1}{2} bhl\)

where \(b\) is the base of the triangle, \(h\) is the height of the triangle and \(l\) is the length of the prism.

Worked Example

A traffic calming road hump (sleeping policeman) is made of concrete and has the dimensions shown in the diagram.

Find the volume of concrete needed to make one road hump.
1, Chapter 2.4, Task 2, Fitness Check,
Fitness Check

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Find the volumes of the solids shown below.

a.

b.
1. \(V = 8.96\;m^{3}\)
2. \(V = 2.4\;m^{3}\)
Exercise 2

Find the volumes of the solids shown below.

a.

b.
1. \(V=\pi \times 4^{2} \times 20 = 1004.8\;mm^{3}\)
2. \(V=Al = 3 \times 15 = 45\; cm^{3}\)
0, Chapter 3, Mass, Volume and Density,
Mass, Volume and Density

Densityis a term used to describe the mass of a unit of volume of a substance.

For example, if the density of a metal is 2000 kg/m\(^{3}\), then 1 m\(^{3}\) of the substance has a mass of 2000 kg.

Mass, volume and density are related by the following equations.

Mass, Volume and Density

\(Mass=Volume\times Density\)

\(Volume=\frac{Mass}{Density}\)

\(Density=\frac{Mass}{Volume}\)

Note that the density of water is 1 gram / cm\(^{3}\) (or 1 g cm\(^{-3}\)) or 1000 kg / m\(^{3}\) (or 1000 kg m\(^{-3}\)).

Top Tip

You may find it helpful to use the following triangle, especially if you are already familiar with them,
it summarises the three equations on the previous slide in one diagram.

You cover up the quantity you wish to find. For example, to find Volume, cover up Volume
on the diagram and the triangle tells you to work out Mass ‘over’ Density (Mass ÷ Density).
1, Chapter 3, Task 2, Fitness Check,
Fitness Check

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

The diagram shows a concrete block of mass 6 kg.
1. Find the volume of the block.
2. What is the density of the concrete?
1. \(V = 8000\;cm^{3}\)
2. \(Density = 75\;g /cm^{3}\)
Exercise 2

A barrel is a cylinder with radius 40 cm and height 80 cm.

It is full of water.
1. Find the volume of the block.
2. What is the density of the concrete?
1. \(V = 402 124\;cm^{3}\) (to the nearest whole \(cm^{3}\))
  (or \(401\;920\;cm^{3}\) if you use \(p\) rounded as \(3.14\))
2. \(Mass = 402\;124\;g = 402.124\;kg\)
0, Chapter 4, Volumes of Pyramids, Cones and Spheres,
Volumes of Pyramids, Cones and Spheres

The volumes of a pyramid, a cone and a sphere are found using the following formulae.

This illustration shows you the relationship between the volumes of a cone and a cylinder that have the same sized base. Can you spot the relationship?
1, Chapter 4, Task 2, Fitness Check,
Fitness Check

Here are some questions to check your progress; there are more practice questions if needed.

Exercise 1

Find the volume of a cone with radius of 7 cm and height 5 cm, to one decimal place.

Exercise 2

Find the volume of a pyramid of square base, 10 m side lengths, and height 15 m.
0, Chapter 5, Dimensions,
Dimensions

Most measurements are based on three fundamental quantities, mass, length and time.

The letters \(M\), \(L\) and \(T\) are used to represent the dimensions of these quantities.

Mass can simply be represented with an \(M\). Square brackets are used to take the dimension of the quantity:

[mass] = [m] = M

Similarly \(L\) represents length, so, for example, if \(x\) is a length, then:

\([x]= L\)

also

\(\left[x^{2}\right] = L^{2}\)
\(\left[x^{3}\right] = L^{3}\)

If \(t\) is a time, then

\(\left[t\right] = T\)

We can also use the principle that area is found by multiplying two lengths together and volume is found by multiplying three lengths together, to represent the dimensions of area and volume:

\(\left[Area\right] = L^{2}\) or Dimensions of Area = \(L^{2}\)

Similarly,

\(\left[Volume\right] = L^{3}\) or Dimensions of Volume = \(L^{3}\)

These ideas can be used to find the dimensions of an algebraic expression involving measurements.
1, Chapter 5, Task 2, Fitness Check,
Fitness Check

Here are some questions to check your progress.

If \(x\), \(y\) and \(z\) all represent lengths, consider each expression and decide if it could be a length, an area, or a volume.
- a. \(xy + z^{2}\)
- b. \(\frac{xy}{z}\)
- c. \(\frac{2x^{2}y}{z}\)
- d. \(zy^{2} + 2x^{3} + 3\pi y^{3}\)
- a. Area
- b. Length
- c. Area
- d. Volume
0, Chapter 6, Summary,
Summary

Essential information from this unit.

Cube \( V = a^{3} \)

Cuboid \( V = abc \)

Cylinder \( V = \pi ^{2}h \)

Prism \( V = Al \)

Pryamid \( V = \frac{1}{3}ah \)

Cone \( V = \frac{1}{3} \pi r^{2}h \)

Sphere \( V = \frac{4}{3} \pi r^{3} \)

where a, b and c represent side lengths; A, area; h, height; r, radius and V, volume.

\(dencity = \frac{mass}{volume}\)

Interactive Exercises:

Interactive Exercises - Volume, https://www.cimt.org.uk/sif/measurement/m4/interactive.htm
Volumes of Cubes, Cuboids, Cylinders and Prisms, https://www.cimt.org.uk/sif/measurement/m4/interactive/s1.html
Mass, Volume and Density, https://www.cimt.org.uk/sif/measurement/m4/interactive/s2.html
Volumes of Pyramids, Cones and Spheres, https://www.cimt.org.uk/sif/measurement/m4/interactive/s3.html
Dimensions, https://www.cimt.org.uk/sif/measurement/m4/interactive/s4.html

YouTube URL: https://www.youtube.com/watch?v=RSKFteb13h0&list=PL8ce1n6mRlCtS0rO_C7CNwtjhyqDVkr8g&index=16

External Links:

3-dimensional shapes: Cylinders, https://www.bbc.co.uk/bitesize/guides/zqjhy4j/revision/5, images/Images/bitesize-logo.png, If you have enjoyed this unit on volumes then try these BBC Bitesize units on volumes.

File Attachments: media/course-resources/SIF_project_Volume.pptx/media/course-resources/volume-learning-objectives.pdf

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