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Cuboids
Shape and Space
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 surface area of this cuboid?
Surface area of a cuboid
7 cm
8 cm5 cm
The area of the top = 8 × 5
= 40 cm2
The area of the front = 7 × 5
= 35 cm2
The area of the side = 7 × 8
= 56 cm2
To find the surface area of a shape, we calculate the total area of all of the faces.
So the total surface area =
Surface area of a cuboid
7 cm
8 cm5 cm
2 × 40 cm2
+ 2 × 35 cm2
+ 2 × 56 cm2
Top and bottom
Front and back
Left and right side
= 80 + 70 + 112 = 262 cm2
We can find the formula for the surface area of a cuboid as follows.
Surface area of a cuboid =
Formula for the surface area of a cuboid
h
lw
2 × lw Top and bottom
+ 2 × hw Front and back
+ 2 × lh Left and right side
= 2lw + 2hw + 2lh
How can we find the surface area of a cube of length x?
Surface area of a cube
x
All six faces of a cube have the same area.
The area of each face is x × x = x2
Therefore,
Surface area of a cube = 6x2
This cuboid is made from alternate purple and green centimetre cubes.
Chequered cuboid problem
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
What is the surface area of this L-shaped prism?
Surface area of a prism
6 cm
5 cm
3 cm
4 cm
3 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
= 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15= 110 cm2
5 cm
6 cm
3 cm6 cm
3 cm3 cm
3 cm
It can be helpful to use the net of a 3-D shape to calculate its surface area.
Using nets to find surface area
Here is the net of a 3 cm by 5 cm by 6 cm cuboid
Write down the area of each face.
15 cm2 15 cm2
18 cm2
30 cm2 30 cm2
18 cm2
Then add the areas together to find the surface area.
Surface Area = 126 cm2
The following cuboid is made out of interlocking cubes.
Making cuboids
How many cubes does it contain?
We can work this out by dividing the cuboid into layers.
Making cuboids
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
The amount of space that a three-dimensional object takes up is called its volume.
Making cuboids
For example, we can use mm3, cm3, m3 or km3.
The 3 tells us that there are three dimensions, length, width and height.
Volume is measured in cubic units.
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.
Volume of a cuboid
= length × width × height
= lwh
height, h
length, lwidth, w
The area of the base
= length × width
So,
Volume of a cuboid
What is the volume of this cuboid?
Volume of cuboid
= length × width × height
= 5 × 8 × 13
= 520 cm3
5 cm
8 cm 13 cm
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.
What is the volume of this L-shaped prism?
Volume of a prism made from cuboids
6 cm
5 cm
3 cm
4 cm
3 cmWe can think of the shape as two cuboids joined together.
Volume of the green cuboid
= 6 × 3 × 3 = 54 cm3
Volume of the blue cuboid
= 3 × 2 × 2 = 12 cm3
Total volume
= 54 + 12 = 66 cm3
Remember, a prism is a 3-D shape with the same cross-section throughout its length.
Volume of a prism
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
3 cm
Volume of a prism
Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 =
Volume of prism = 5 × 72 = 360 m3
3 m
4 m
12 m
7 m
5 m
72 m2
What is the volume of this prism?
