1. Volume

2. 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

3. Defn Polyhedron
Polyhedron: A solid three dimensional shape, with flat faces
e.g:

4. Defn Cuboid:
A cuboid, is a polyhedron with six rectangular plane faces
E.g:

5. 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.
This web site has a few examples of prisms

6. The following are all prisms
Cylinder
Cuboid
Cross section
Triangular Prism
Trapezoid Prism
Volume of Prism = length x Cross-sectional area

7. Volumes Of Cuboids. Look at the cuboid below: 4cm 3cm 10cm
We must first calculate the area of the base of the cuboid:
The base is a rectangle measuring 10cm by 3cm:
3cm
10cm

8. 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

9. 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.

10. The Volume of a prism is simply the area of one end times the length of the prism

11. 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.

12. Do we need to see that again? How to find volume
3x6 = 18 cubes
3x6 = 18 cubes
18 cubes x 4 lots = 72 cubes

14. Find the volume of the following?
Calculate the volumes of the cuboids below:
(1)
14cm
5 cm
7cm
(2)
3.4cm
490cm3
39.3cm3
(3)
8.9 m
2.7m
3.2m
76.9 m3

16. The Cone + + = Volume of a Cone =
A Cone is a three dimensional solid with a circular base and a curved surface that gradually narrows to a vertex. + + =
Volume of a Cone =

17. Identical isosceles triangles
Pyramids
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.
Identical isosceles triangles
base = quadrilateral
base = heptagon
base = pentagon

18. = + + Volume of Pyramids 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 = + +

20. 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.

21. For the solids below identify the cross sectional area required for calculating the volume:
(2)
(1)
Right Angled Triangle.
Circle
(4)
(3) A2 A1
Rectangle & Semi Circle.
Pentagon

22. The Volume Of A Cylinder.
Consider the cylinder below:
It has a height of 6cm .
4cm
6cm
What is the size of the radius ?
2cm
Volume = cross section x height
What shape is the cross section?
Circle
Calculate the area of the circle:
A =  r 2
A = 3.14 x 2 x 2
A = cm2
The formula for the volume of a cylinder is:
V =  r 2 h
r = radius h = height.
Calculate the volume:
V =  r 2 x h
V = x 6
V = cm3

23. The Volume Of A Triangular Prism.
Consider the triangular prism below:
5cm
8cm
Volume = Cross Section x Height
What shape is the cross section ?
Triangle.
Calculate the area of the triangle:
A = ½ x base x height
A = 0.5 x 5 x 5
A = 12.5cm2
Calculate the volume:
Volume = Cross Section x Length
The formula for the volume of a triangular prism is :
V = ½ b h l
B= base h = height l = length
V = 12.5 x 8
V = 100 cm3

24. What Goes In The Box ? 2 Calculate the volume of the shapes below: (2)
(1)
16cm
14cm
(3)
6cm
12cm 8m
2813.4cm3
30m3
288cm3

25. More Complex Shapes. Calculate the volume of the shape below: 20m 23m
Volume = Cross sectional area x length.
V = 256 x 23 A1 A2
V = 2888m3
Calculate the cross sectional area:
Area = A1 + A2
Area = (12 x 16) + ( ½ x (20 –12) x 16)
Area =
Area = 256m2

26. Example 2.
Calculate the volume of the shape below:
12cm
18cm
10cm A2 A1
Calculate the volume.
Volume = cross sectional area x Length
V = x 18
V = cm3
Calculate the cross sectional area:
Area = A1 + A2
Area = (12 x 10) + ( ½ x  x 6 x 6 )
Area =
Area = cm2

27. What Goes In The Box ? 3 11m (1) 4466m3 14m 22m (2) 18m 17cm

29. Other slides:

30. Volumes Of Solids. 8m 5m
7cm
5 cm
14cm
6cm
4cm
4cm
3cm
10cm

31. 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:

32. 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
We will now see how to calculate the volume of various shapes.

33. Volumes Of Cuboids. Look at the cuboid below: 4cm 3cm 10cm
We must first calculate the area of the base of the cuboid:
The base is a rectangle measuring 10cm by 3cm:
3cm
10cm

34. 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

35. 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.

36. 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.

37. What Goes In The Box ? Calculate the volumes of the cuboids below: (1)
14cm
5 cm
7cm
(2)
3.4cm
490cm3
39.3cm3
(3)
8.9 m
2.7m
3.2m
76.9 m3

38. 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.

39. For the solids below identify the cross sectional area required for calculating the volume:
(2)
(1)
Right Angled Triangle.
Circle
(4)
(3) A2 A1
Rectangle & Semi Circle.
Pentagon

40. The Volume Of A Cylinder.
Consider the cylinder below:
It has a height of 6cm .
4cm
6cm
What is the size of the radius ?
2cm
Volume = cross section x height
What shape is the cross section?
Circle
Calculate the area of the circle:
A =  r 2
A = 3.14 x 2 x 2
A = cm2
The formula for the volume of a cylinder is:
V =  r 2 h
r = radius h = height.
Calculate the volume:
V =  r 2 x h
V = x 6
V = cm3

41. The Volume Of A Triangular Prism.
Consider the triangular prism below:
5cm
8cm
Volume = Cross Section x Height
What shape is the cross section ?
Triangle.
Calculate the area of the triangle:
A = ½ x base x height
A = 0.5 x 5 x 5
A = 12.5cm2
Calculate the volume:
Volume = Cross Section x Length
The formula for the volume of a triangular prism is :
V = ½ b h l
B= base h = height l = length
V = 12.5 x 8
V = 100 cm3

42. What Goes In The Box ? 2 Calculate the volume of the shapes below: (2)
(1)
16cm
14cm
(3)
6cm
12cm 8m
2813.4cm3
30m3
288cm3

43. More Complex Shapes. Calculate the volume of the shape below: 20m 23m
Volume = Cross sectional area x length.
V = 256 x 23 A1 A2
V = 2888m3
Calculate the cross sectional area:
Area = A1 + A2
Area = (12 x 16) + ( ½ x (20 –12) x 16)
Area =
Area = 256m2

44. Example 2.
Calculate the volume of the shape below:
12cm
18cm
10cm A2 A1
Calculate the volume.
Volume = cross sectional area x Length
V = x 18
V = cm3
Calculate the cross sectional area:
Area = A1 + A2
Area = (12 x 10) + ( ½ x  x 6 x 6 )
Area =
Area = cm2

45. What Goes In The Box ? 3 11m (1) 4466m3 14m 22m (2) 18m 17cm

46. Volume Of A Cone. Consider the cylinder and cone shown below: D
The diameter (D) of the top of the cone and the cylinder are equal. H
The height (H) of the cone and the cylinder are equal.
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 ?
This shows that the cylinder has three times the volume of a cone with the same height and radius.
3 times.

47. 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: h r
r = radius h = height

48. Calculate the volume of the cones below:
(2) 9m 6m
(1)

49. Summary Of Volume Formula.h
V =  r 2 h l b h
V = l b h b l h
V = ½ b h l h r

53. The Cone + + = Volume of a Cone =
A Cone is a three dimensional solid with a circular base and a curved surface that gradually narrows to a vertex. + + =
Volume of a Cone =

54. Exercise #1 = (1/3)(3.14)(1)2(2) = 3.14(1)2(2) = 2.09 m3 = 6.28 m3
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 = pr2h
= 3.14(1)2(2)
= 6.28 m3
Volume of a Cone = (1/3) pr2h
= (1/3)(3.14)(1)2(2)
= 2.09 m3

55. Find the area of a cone with a radius r=3 m and height h=4 m.
Surface Area of a Cone
Find the area of a cone with a radius r=3 m and height h=4 m.
r = the radius h = the height l = the slant height
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
= pr2 + prl
= 3.14(3) (3)(5)
= m2

56. Identical isosceles triangles
Pyramids
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.
Identical isosceles triangles
base = quadrilateral
base = heptagon
base = pentagon

57. = + + Volume of Pyramids 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 = + +

58. Exercise #2
Find the volume of the pyramid height h = 8 m apothem a = 4 m side s = 6 m
Volume = 1/3 (area of base) (height)
= 1/3 ( 60m2)(8m)
= 160 m3 h
Area of base = ½ Pa a
= ½ (5)(6)(4)
= 60 m2 s

59. Area of Pyramids
Find the surface area of the pyramid height h = 8 m apothem a = 4 m side s = 6 m
Surface Area
= area of base
+ 5 (area of one lateral face)
What shape is the base?
Area of a pentagon h
= ½ Pa
= ½ (5)(6)(4)
= 60 m2 l a s

60. Area of Pyramids h l a s What shape are the lateral sides?
Find the surface area of the pyramid height h = 8 m apothem a = 4 m side s = 6 m
Area of a triangle = ½ base (height)
= ½ (6)(8.9)
= 26.7 m2
Attention! the height of the triangle is the slant height ”l ” h l
l 2 = h2 + a2 =
= 80 m2
l = 8.9 m a s

61. Area of Pyramids h l a s Surface Area of the Pyramid
= 60 m2 + 5(26.7) m2
= 60 m m2
= m2
Find the surface area of the pyramid height h = 8 m apothem a = 4 m side s = 6 m h l a s

64. A Prism Cross section Volume of Prism = length x Cross-sectional area
Cylinder
Cuboid
Cross section
Triangular Prism
Trapezoid Prism
Volume of Prism = length x Cross-sectional area

65. Area Formulae r h b Area Circle = π x r2 Area Rectangle
= Base x height h b h b
Area Trapezium
= ½ x (a + b) x h a b h
Area Triangle
= ½ x Base x height

66. Volume Cylinder Cross-sectional Area = π x r2 = π x 32 = 28.2743…..cm2
DO NOT ROUND!
3cm
5cm
USE CALCULATOR ‘ANS’!
Volume = length x CSA
= 5 x ….
= ….cm3
= 141.4cm3

67. Volume Cuboid Cross-sectional Area = b x h = 7.2 x 5.3 = 38.16cm2
DO NOT ROUND!
7.2cm
Volume = length x CSA
USE ‘ANS’!
= x 38.16
= cm3
Sensible degree of accuracy
= 404.5cm3

68. Volume Trapezoid Prism
Cross-sectional Area = ½ x(a + b) x h
= ½ x ( ) x 4.9
1.7cm
8.2cm
6.3cm
4.9cm
= 19.6cm2
DO NOT ROUND!
Volume = length x CSA
USE ‘ANS’!
= x 8.2
= cm3
Sensible degree of accuracy
= 160.7cm3

69. Volume Triangular Prism
Cross-sectional Area = ½ x b x h
= ½ x 8.6 x 4.1
= 17.63cm2
4.1cm
4.9cm
DO NOT ROUND!
6.2cm
8.6cm
Volume = length x CSA
USE ‘ANS’!
= x 6.2
= cm3
Sensible degree of accuracy
= 109.3cm3