Surface Area and Volume
Cubes and Cuboids
Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
A cuboid has 6 faces.
The top and the bottom of the
cuboid have the same area.
Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
A cuboid has 6 faces.
The front and the back of the
cuboid have the same area.
Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
A cuboid has 6 faces.
The left hand side and the right
hand side of the cuboid have
the same area.
Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
Can you work out the
5 cm
surface area of this cubiod?
8 cm
The area of the top = 8 × 5
= 40 cm2
7 cm
The area of the front = 7 × 5
= 35 cm2
The area of the side = 7 × 8
= 56 cm2
Surface area of a cuboid
To find the surface area of a shape, we calculate the
total area of all of the faces.
8 cm
5 cm
So the total surface area =
2 × 40 cm2
7 cm
Top and bottom
+ 2 × 35 cm2 Front and back
+ 2 × 56 cm2 Left and right side
= 80 + 70 + 112 = 262 cm2
Formula for the surface area of a cuboid
We can find the formula for the surface area of a cuboid
as follows.
Surface area of a cuboid =
l
h
w
2 × lw
Top and bottom
+ 2 × hw
Front and back
+ 2 × lh
Left and right side
= 2lw + 2hw + 2lh
Surface area of a cube
How can we find the surface area of a cube of length x?
All six faces of a cube have the
same area.
The area of each face is x × x = x2
Therefore,
x
Surface area of a cube = 6x2
Checkered cuboid problem
This cuboid is made from alternate purple and green
centimetre cubes.
What is its surface area?
Surface area
=2×3×4+2×3×5+2×4×5
= 24 + 30 + 40
= 94 cm2
How much of the
surface area is green?
48 cm2
Surface area of a prism
What is the surface area of this L-shaped prism?
3 cm
3 cm
4 cm
6 cm
To find the surface area of
this shape we need to add
together the area of the two
L-shapes and the area of the
6 rectangles that make up
the surface of the shape.
Total surface area
5 cm
= 2 × 22 + 18 + 9 + 12 + 6
+ 6 + 15
= 110 cm2
Using nets to find surface area
It can be helpful to use the net of a 3-D shape to calculate its
surface area.
Here is the net of a 3 cm by 5 cm by 6 cm cubiod.
6 cm
3 cm
18 cm2
3 cm
5 cm 15 cm2
30 cm2
15 cm2
3 cm
18 cm2
3 cm
6 cm
30 cm2
Write down the
area of each
face.
Then add the
areas together
to find the
surface area.
Surface Area = 126 cm2
Making cuboids
The following cuboid is made out of interlocking cubes.
How many cubes does it contain?
Making cuboids
We can work this out by dividing the cuboid into layers.
The number of cubes in each layer
can be found by multiplying the
number of cubes along the length
by the number of cubes along the
width.
3 × 4 = 12 cubes in each layer
There are three layers altogether
so the total number of cubes in the
cuboid = 3 × 12 = 36 cubes
Making cuboids
The amount of space that a three-dimensional object takes
up is called its volume.
Volume is measured in cubic units.
For example, we can use mm3, cm3, m3 or km3.
The 3 tells us that there are three dimensions, length, width
and height.
Liquid volume or capacity is measured in ml, l, pints or
gallons.
Volume of a cuboid
We can find the volume of a cuboid by multiplying the area of
the base by the height.
The area of the base
= length × width
So,
height, h
Volume of a cuboid
= length × width × height
= lwh
width, w
length, l
Volume of a cuboid
What is the volume of this cuboid?
Volume of cuboid
= length × width × height
5 cm
= 5 × 8 × 13
8 cm
13 cm
= 520 cm3
Volume and displacement
Volume and displacement
By dropping cubes and cuboids into a measuring cylinder
half filled with water we can see the connection between the
volume of the shape and the volume of the water displaced.
1 ml of water has a volume of 1 cm3
For example, if an object is dropped into a measuring
cylinder and displaces 5 ml of water then the volume of the
object is 5 cm3.
What is the volume of 1 litre of water?
1 litre of water has a volume of 1000 cm3.
Volume of a prism made from cuboids
What is the volume of this L-shaped prism?
3 cm
We can think of the shape as
two cuboids joined together.
3 cm
4 cm
Volume of the green cuboid
= 6 × 3 × 3 = 54 cm3
6 cm
Volume of the blue cuboid
= 3 × 2 × 2 = 12 cm3
Total volume
5 cm
= 54 + 12 = 66 cm3
Volume of a prism
Remember, a prism is a 3-D shape with the same
cross-section throughout its length.
3 cm
We can think of this prism as lots
of L-shaped surfaces running
along the length of the shape.
Volume of a prism
= area of cross-section × length
If the cross-section has an area
of 22 cm2 and the length is 3 cm,
Volume of L-shaped prism = 22 × 3 = 66 cm3
Volume of a prism
What is the volume of this prism?
12 m
7m
4m
3m
5m
Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 = 72 m2
Volume of prism = 5 × 72 = 360 m3