Finding volume by counting cubes
The volume of a shape measures the three dimensional (3D) amount of space it takes up.
Volume is measured in cubes.
A cubic centimetre is the volume within a cube that has sides of length \({1~cm}\), as shown below.
It has a volume of \({1~cm}^3\) (\(1~cm\) cubed).
The cuboid below contains \(12\) cubes.
Each cube has a volume of \({1~cm}^3\).
So the volume of this cuboid is \({12~cm}^3\).
Question
Find the volume of the following cuboids:
a)
b)
Answer
a) \({8~cm}^3\)
b) There are two layers of nine cubes, so the volume is \({18~cm}^3\).
Question
Find the volume of the 3D shape below.
Answer
\({11~cm^3}\)
Volume of a cuboid
To find the volume of a cuboid, multiply its length by its width by its height.
This can be written as:
\(volume = l \times w \times h\)
Example
The volume of this cereal packet is:
\({8~cm}\times{20~cm}\times{30~cm}={4,800~cm}^{3}\)
Question
What is the volume of this tin?
Answer
The volume is \(\text{48 cm}^3\)
Remember that \(volume = length \times width \times height\), so in this example the volume is \({4~cm}\times{2~cm}\times{6~cm}={48}~cm^{3}\).
Volume of a prism
Volume of a prism
We've learned that the volume of a cuboid is its length multiplied by its width multiplied by its height (\(l \times w \times h\)).
The area of the green shaded end of the cuboid (the cross section) is \(w \times h\), so you can also say that the volume of a cuboid is: \(Volume = area~of~cross~section \times length\)
Different types of prism
This formula works for all prisms:
Volume of a cylinder = \(\text{area of circle}\times\text{length}\)
Volume of a triangular prism = \(\text{area~of~triangle}\times\text{length}\)
Volume of L-shaped prism = \(\text{area~of~L-shape}\times\text{length}\)
a) What is the volume of this triangular prism?
b) What is the volume of this prism?
Answer
a) \(volume = area~of~triangle \times length\) \(=(\frac{1}{2}\times{2~cm}\times{5~cm})\times{4~cm}\)
\(= \text{20 cm}^3\)
b) The area of the cross section is \(\text{5 cm}^2\) and the length is \(\text{8 cm}\), so the volume is \({5~cm}^{2}\times{8~cm}={40~cm}^{3}\).
Remember that the volume is:
\(the~area~of~the~cross~section\times the~length\)
Test section
Question 1
What is the volume of the following 3D shape?
Answer
Each cube is \({1}~{cm}^{3}\) so the total volume is \({16}~{cm}^{3}\).
Question 2
What is the volume of the following 3D shape?
Answer
Two layers of \({9}~{cm}^{3}\) gives a total of \({18}~{cm}^{3}\)
Question 3
What is the volume of a cube that has sides \({3}~{cm}\) long?
Answer
The volume of a cube \(={length}\times{width}\times{height}\), so \({3}~{cm}\times{3}~{cm}\times{3}~{cm}={27}~{cm}^{3}\).
Question 4
What is the volume of a cuboid that has the following dimensions:
length \({9}~{cm}\)
width \({3}~{cm}\)
height \({6}~{cm}\)?
Answer
The volume of a cuboid \(={length}\times{width}\times{height}\), so \({9}~{cm}\times{13}~{cm}\times{6}~{cm}={162}~{cm}^{3}\).
Question 5
What is the volume of a cuboid if its length is \({8}~{cm}\), its width is \({14}~{cm}\) and its height is \({5}~{cm}\)?
Answer
The volume of a cuboid \(={length}\times{width}\times{height}\), so \({8}~{cm}\times{14}~{cm}\times{5}~{cm}={560}~{cm}^{3}\).
Question 6
What is the length of a cuboid if its volume is \({200}~{cm}^{3}\), its width is \({10}~{cm}\) and its height is \({5}~{cm}\)?
Answer
\({Length}={volume~of~cuboid}\div{({width}\times{height})}\), so \({200}~{cm}^{3}\div({10}~{cm}\times{5}~{cm})={4}~{cm}\).
Question 7
What is the height of a cuboid if its volume is \({504}~{cm}^{3}\), its length is \({9}~{cm}\) and its width is \({8}~{cm}\)?
Answer
When rearranging remember that \({height}={volume~of~cuboid}\div{({length}\times{width})}\), so \({504}~{cm}^{3}\div{({9}~{cm}\times{8}~{cm})}={7}~{cm}\).
Question 8
What is the volume of a prism that has a cross-section of \({16}~{cm}^{2}\) and a length of \({8}~{cm}\)?
Answer
The \(\text{volume~of~a~prism}=\text{area~of~cross-section}\times\text{length}\), so \({16}~{cm}^{2}\times{8}~{cm}={128}~{cm}^{3}\).
Question 9
What is the volume of a triangular prism if its length is \({12}~{cm}\), its height is \({7}~{cm}\) and its base is \({5}~{cm}\)?
Answer
The \(\text{volume~of~a~prism}=\text{area~of~cross-section}\times\text{length}\), so \({(\frac{1}{2}\times{7}~{cm}\times{5}~{cm})}\times{12}~{cm}={210}~{cm}^{3}\).
Question 10
What is the area of the cross-section of a prism if its volume is \({432}~{cm}^{3}\) and its length is \({9}~{cm}\)?
Answer
When rearranging remember that the \(\text{area~of~a~cross-section}=\text{volume~of~prism}\div\text{length}\), so \({432}~{cm}^{3}\div{9}~{cm}={48}~{cm}^{2}\).
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