Defn Volume
• The volume of a solid is the amount of space it occupies. In other words, it is the number of cubes which fit inside.
• Volume is measured in either cm3 or m3
Here is a cubic centimetre:
It is a cube which measures 1cm in all directions.1cm
1cm1cm
Defn: Prism
• A prism is a solid with two opposite faces that are the same shape, size and are parallel. I.e. it has the same cross-section for the entire length of the shape.
The following are all prisms
Cylinder
Cuboid
Triangular PrismTrapezoid Prism
Volume of Prism = length x Cross-sectional area
Cross section
Volumes Of Cuboids.Look at the cuboid below:
10cm
3cm
4cm
We must first calculate the area of the base of the cuboid:
The base is a rectangle measuring 10cm by 3cm:
3cm
10cm
10cm
3cm
4cm
3cm
10cm
Area of a rectangle = length x breadth
Area = 10 x 3
Area = 30cm2
We now know we can place 30 centimetre squares on the base of the cuboid. But we can also place 30 cubic centimetres on the base:
10cm
3cm
4cm
We have now got to find how many layers of 1cm cubes we can place in the cuboid:
We can fit in 4 layers.
Volume = 30 x 4
Volume = 120cm3
That means that we can place 120 of our cubes measuring a centimetre in all directions inside our cuboid.
10cm
3cm
4cm
We have found that the volume of the cuboid is given by:
Volume = 10 x 3 x 4 = 120cm3
This gives us our formula for the volume of a cuboid:
Volume = Length x Breadth x Height
V=LBH for short.
Do we need to see that again?How to find volume
3x6 = 18 cubes
3x6 = 18 cubes
18 cubes x 4 lots = 72 cubes
Find the volume of the following? Calculate the volumes of the cuboids below:
(1)
14cm5 cm
7cm(2)
3.4cm
3.4cm
3.4cm
(3)
8.9 m
2.7m
3.2m
490cm3
39.3cm3
76.9 m3
The ConeA Cone is a three dimensional solid with a circular base and a curved surface that gradually
narrows to a vertex.
Volume of a Cone
=
+ + =
A Pyramid is a three dimensional figure with a regular polygon as its base and all the outside faces are identical isosceles triangles meeting at a point.
Pyramids
base = quadrilateral
base = pentagon
base = heptagon
Identical isosceles triangles
Volume of a Pyramid:
V = (1/3) Area of the base x height
V = (1/3) Ah
Volume of a Pyramid = 1/3 x Volume of a Prism
Volume of Pyramids
+ + =
The Cross Sectional Area.When we calculated the volume of the cuboid :
10cm
3cm
4cm
We found the area of the base : This is the Cross Sectional Area.
The Cross section is the shape that is repeated throughout the volume.We then calculated how many layers of cross section made up the volume.This gives us a formula for calculating other volumes:
Volume = Cross Sectional Area x Length.
For the solids below identify the cross sectional area required for calculating the volume:
Circle
(2)
Right Angled Triangle.
(3)
Pentagon
(4)A2
A1
Rectangle & Semi Circle.
(1)
The Volume Of A Cylinder.Consider the cylinder below:
4cm
6cm
It has a height of 6cm .
What is the size of the radius ?2cm
Volume = cross section x heightWhat shape is the cross section?Circle
Calculate the area of the circle:A = r 2
A = 3.14 x 2 x 2A = 12.56 cm2
Calculate the volume:V = r 2 x hV = 12.56 x 6V = 75.36 cm3
The formula for the volume of a cylinder is:
V = r 2 h
r = radius h = height.
The Volume Of A Triangular Prism.
Consider the triangular prism below:
Volume = Cross Section x HeightWhat shape is the cross section ?Triangle.Calculate the area of the triangle:
5cm
8cm
5cmA = ½ x base x heightA = 0.5 x 5 x 5 A = 12.5cm2
Calculate the volume:Volume = Cross Section x Length
V = 12.5 x 8V = 100 cm3
The formula for the volume of a triangular prism is :
V = ½ b h l
B= base h = height l = length
What Goes In The Box ? 2Calculate the volume of the shapes below:
(1)
16cm
14cm
(2)
3m
4m
5m
(3)
6cm12cm
8m
2813.4cm3
30m3
288cm3
More Complex Shapes.Calculate the volume of the shape below:
20m
23m
16m
12m
Calculate the cross sectional area:
A1A2
Area = A1 + A2Area = (12 x 16) + ( ½ x (20 –12) x 16)
Area = 192 + 64
Area = 256m2
Calculate the volume:
Volume = Cross sectional area x length.
V = 256 x 23
V = 2888m3
Calculate the volume of the shape below:
12cm 18cm
10cm
Calculate the cross sectional area:
A2
A1
Area = A1 + A2Area = (12 x 10) + ( ½ x x 6 x 6 )Area = 120 +56.52Area = 176.52cm2
Calculate the volume.
Volume = cross sectional area x LengthV = 176.52 x 18 V = 3177.36cm3
Example 2.
What Is Volume ?
The volume of a solid is the amount of space inside the solid. In other words it is the number of cubes which fit inside. Consider the cylinder below:
If we were to fill the cylinder with water the volume would be the amount of water the cylinder could hold:
Measuring Volume.
Volume is measured in cubic centimetres (also called centimetre cubed).
Here is a cubic centimetre
It is a cube which measures 1cm in all directions.1cm
1cm1cm
We will now see how to calculate the volume of various shapes.
Volumes Of Cuboids.Look at the cuboid below:
10cm
3cm
4cm
We must first calculate the area of the base of the cuboid:
The base is a rectangle measuring 10cm by 3cm:
3cm
10cm
10cm
3cm
4cm
3cm
10cm
Area of a rectangle = length x breadth
Area = 10 x 3
Area = 30cm2
We now know we can place 30 centimetre squares on the base of the cuboid. But we can also place 30 cubic centimetres on the base:
10cm
3cm
4cm
We have now got to find how many layers of 1cm cubes we can place in the cuboid:
We can fit in 4 layers.
Volume = 30 x 4
Volume = 120cm3
That means that we can place 120 of our cubes measuring a centimetre in all directions inside our cuboid.
10cm
3cm
4cm
We have found that the volume of the cuboid is given by:
Volume = 10 x 3 x 4 = 120cm3
This gives us our formula for the volume of a cuboid:
Volume = Length x Breadth x Height
V=LBH for short.
What Goes In The Box ? Calculate the volumes of the cuboids below:
(1)
14cm5 cm
7cm(2)
3.4cm
3.4cm
3.4cm
(3)
8.9 m
2.7m
3.2m
490cm3
39.3cm3
76.9 m3
The Cross Sectional Area.When we calculated the volume of the cuboid :
10cm
3cm
4cm
We found the area of the base : This is the Cross Sectional Area.
The Cross section is the shape that is repeated throughout the volume.We then calculated how many layers of cross section made up the volume.This gives us a formula for calculating other volumes:
Volume = Cross Sectional Area x Length.
For the solids below identify the cross sectional area required for calculating the volume:
Circle
(2)
Right Angled Triangle.
(3)
Pentagon
(4)A2
A1
Rectangle & Semi Circle.
(1)
The Volume Of A Cylinder.Consider the cylinder below:
4cm
6cm
It has a height of 6cm .
What is the size of the radius ?2cm
Volume = cross section x heightWhat shape is the cross section?Circle
Calculate the area of the circle:A = r 2
A = 3.14 x 2 x 2A = 12.56 cm2
Calculate the volume:V = r 2 x hV = 12.56 x 6V = 75.36 cm3
The formula for the volume of a cylinder is:
V = r 2 h
r = radius h = height.
The Volume Of A Triangular Prism.
Consider the triangular prism below:
Volume = Cross Section x HeightWhat shape is the cross section ?Triangle.Calculate the area of the triangle:
5cm
8cm
5cmA = ½ x base x heightA = 0.5 x 5 x 5 A = 12.5cm2
Calculate the volume:Volume = Cross Section x Length
V = 12.5 x 8V = 100 cm3
The formula for the volume of a triangular prism is :
V = ½ b h l
B= base h = height l = length
What Goes In The Box ? 2Calculate the volume of the shapes below:
(1)
16cm
14cm
(2)
3m
4m
5m
(3)
6cm12cm
8m
2813.4cm3
30m3
288cm3
More Complex Shapes.Calculate the volume of the shape below:
20m
23m
16m
12m
Calculate the cross sectional area:
A1A2
Area = A1 + A2Area = (12 x 16) + ( ½ x (20 –12) x 16)
Area = 192 + 64
Area = 256m2
Calculate the volume:
Volume = Cross sectional area x length.
V = 256 x 23
V = 2888m3
Calculate the volume of the shape below:
12cm 18cm
10cm
Calculate the cross sectional area:
A2
A1
Area = A1 + A2Area = (12 x 10) + ( ½ x x 6 x 6 )Area = 120 +56.52Area = 176.52cm2
Calculate the volume.
Volume = cross sectional area x LengthV = 176.52 x 18 V = 3177.36cm3
Example 2.
Volume Of A Cone.Consider the cylinder and cone shown below:
The diameter (D) of the top of the cone and the cylinder are equal.
D D
The height (H) of the cone and the cylinder are equal.
H H
If you filled the cone with water and emptied it into the cylinder, how many times would you have to fill the cone to completely fill the cylinder to the top ?
3 times. This shows that the cylinder has three times the volume of a cone with the same height and radius.
The experiment on the previous slide allows us to work out the formula for the volume of a cone:
The formula for the volume of a cylinder is : V = r 2 h
We have seen that the volume of a cylinder is three times more than that of a cone with the same diameter and height .
The formula for the volume of a cone is:
hr π3
1V 2
h
r
r = radius h = height
Calculate the volume of the cones below:
hr π3
1V 2
13m
18m(2)
9663.143
1V
9m
6m(1)
hr π3
1V 2
139914.33
1V
31102.14mV 3339.12mV
The ConeA Cone is a three dimensional solid with a circular base and a curved surface that gradually
narrows to a vertex.
Volume of a Cone
=
+ + =
Find the volume of a cylinder with a radius r=1 m and height h=2 m. Find the volume of a cone with a radius r=1 m and height
h=1 m
Volume of a Cylinder = base x height = r2h
= 3.14(1)2(2)
= 6.28 m3
Exercise #1
Volume of a Cone = (1/3) r2h
= (1/3)(3.14)(1)2(2)
= 2.09 m3
Find the area of a cone with a radius r=3 m and height h=4 m.
Use the Pythagorean Theorem to find l
l 2 = r2 + h2
l 2= (3)2 + (4)2
l 2= 25
l = 5
Surface Area of a
Cone
Surface Area of a Cone
= r2 + rl
= 3.14(3)2 + 3.14(3)(5)
= 75.36 m2
r = the radius h = the height l = the slant height
A Pyramid is a three dimensional figure with a regular polygon as its base and lateral faces are identical isosceles triangles meeting at a point.
Pyramids
base = quadrilateral
base = pentagon
base = heptagon
Identical isosceles triangles
Volume of a Pyramid:
V = (1/3) Area of the base x height
V = (1/3) Ah
Volume of a Pyramid = 1/3 x Volume of a Prism
Volume of Pyramids
+ + =
Find the volume of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m
Area of base = ½ Pa
Exercise #2
h
as
Volume = 1/3 (area of base) (height)
= 1/3 ( 60m2)(8m)
= 160 m3
= ½ (5)(6)(4)
= 60 m2
Surface Area
= area of base
+ 5 (area of one lateral face)
Area of Pyramids
Find the surface area of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m
h
as
Area of a pentagon
l
What shape is the base?
= ½ Pa
= ½ (5)(6)(4)
= 60 m2
Area of Pyramids
Find the surface area of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m
h
as
Area of a triangle
= ½ base (height)
l
What shape are the lateral sides?
l 2 = h2 + a2
= 82 + 42
= 80 m2
l = 8.9 m
Attention! the height of the triangle is the slant height ”l ”
= ½ (6)(8.9)
= 26.7 m2
Area of Pyramids
Find the surface area of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m
h
as
l
Surface Area of the Pyramid
= 60 m2 + 5(26.7) m2
= 60 m2 + 133.5 m2
= 193.5 m2
A PrismCylinder
Cuboid
Triangular PrismTrapezoid Prism
Volume of Prism = length x Cross-sectional area
Cross section
Area Formulae
Area Circle = π x r2
r
Area Rectangle = Base x height
h
b
b
h
Area Triangle = ½ x Base x height
h
b
Area Trapezium = ½ x (a + b) x h
a
5cm
3cm
Cross-sectional Area = π x r2
= π x 32
= 28.2743…..cm2
Volume = length x CSA
= 5 x 28.2743….
= 141.4cm3
DO NOT ROUND!
USE CALCULATOR ‘ANS’!
= 141.3716….cm3
Volume Cylinder
Volume Cuboid
5.3cm
7.2cm
10.6cm
Cross-sectional Area = b x h
= 7.2 x 5.3= 38.16cm2
Volume = length x CSA
= 10.6 x 38.16
= 404.5cm3
DO NOT ROUND!
USE ‘ANS’!
= 404.496cm3 Sensible degree of accuracy
Volume Trapezoid PrismCross-sectional Area = ½ x(a + b) x h
= ½ x (6.3 + 1.7) x 4.9
Volume = length x CSA
= 19.6 x 8.2
= 160.7cm3
DO NOT ROUND!
USE ‘ANS’!
= 160.72cm3 Sensible degree of accuracy
1.7cm8.2cm
6.3cm
4.9cm= 19.6cm2