Volume
Definition
Definition Volume
The volume of a shape is the amount of space it takes up in three dimensions. We measure it in cubic units, like cubic centimeters or cubic meters.
To find the volume, imagine filling the shape with small cubes, like building blocks. Count how many cubes fit inside the shape.
Example
Find the volume:
Units of Volume
We can measure volume using different units like blocks or small boxes. But these units are not the same for everyone. Your friend might have a bigger block than yours, making it hard to compare. To solve this, mathematicians created universal units like the cubic centimeter and cubic meter, so everyone can measure and compare volumes the same way.
Definition Units of Volume
- Cubic Millimeter \(\left(\mathrm{mm}^3\right)\): A very small unit of volume, about the size of a tiny grain of sand.
- Cubic Centimeter \(\left(\mathrm{cm}^3\right)\): A small unit of volume, about the size of a small ice cube.
- Cubic Meter \(\left(\mathrm{m}^3\right)\): A larger unit of volume, about the space it takes for a washing machine.
Conversion of Volume Units
Let’s explore how volume units are related by looking at a small cube. Below is a cube that measures 1 cm by 1 cm by 1 cm, which is 1 cm³. We divide it into smaller cubes, each 1 mm by 1 mm by 1 mm (1 mm³). Let’s count how many 1 mm³ cubes fit inside 1 cm³.
$$\begin{aligned}1 \, \text{cm}^3 &= 1 \, \text{cm} \times 1 \, \text{cm} \times 1 \, \text{cm} \\ &= 10 \, \text{mm} \times 10 \, \text{mm} \times 10 \, \text{mm} \quad (1 \, \text{cm} = 10 \, \text{mm}) \\ &= 1\,000 \, \text{mm}^3\end{aligned}$$
$$\begin{aligned}1 \, \text{cm}^3 &= 1 \, \text{cm} \times 1 \, \text{cm} \times 1 \, \text{cm} \\ &= 10 \, \text{mm} \times 10 \, \text{mm} \times 10 \, \text{mm} \quad (1 \, \text{cm} = 10 \, \text{mm}) \\ &= 1\,000 \, \text{mm}^3\end{aligned}$$
Proposition Conversion of Volume Units
- \(1 \, \text{m}^3 = 1\,000\,000 \, \text{cm}^3\)
- \(1 \, \text{cm}^3 = 1\,000 \, \text{mm}^3\)
Method Converting Using Multiplication or Division
- Use multiplication to go from a larger unit to a smaller one (like cubic meters to cubic centimeters).
- Use division to go from a smaller unit to a larger one (like cubic centimeters to cubic meters).
Method Converting Using a Table
To convert between units of volume, we can use a conversion table. For example, to convert 10.5 cubic meters to cubic centimeters:
- Write the units in the table: m³, cm³, mm³.
- Place the number in the column of the unit you start with.
- Fill in zeros in the columns to the right until you reach the unit you want to convert to.
- Read the number in the column of the target unit.
So, 10.5 m³ = \(10\,500\,000\) cm³.
Volume of a Rectangular Cuboid
To calculate the volume of a rectangular cuboid, we can count the cubes inside, but that takes a long time. Let’s look at how the volume changes when we make the box taller, one step at a time, to find an easier way. We’ll also see how counting layer by layer can help us understand the pattern.
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Proposition Volume of a Rectangular Cuboid
The volume of a rectangular prism is multiplying its length, width, and height:$$\text{Volume} = \textcolor{colordef}{\text{length}} \times \textcolor{colorprop}{\text{width}} \times \textcolor{olive}{\text{height}}$$
Example
Find the volume:
$$\begin{aligned}\text{Volume} &= \textcolor{colordef}{\text{length}} \times \textcolor{colorprop}{\text{width}} \times \textcolor{olive}{\text{height}}\\ &= 3\times 2 \times 4 \\ &= 24 \, \text{cm}^3\end{aligned}$$
Capacity
We often need to measure liquids like water, milk, or juice. Instead of using cubic centimeters, there’s an easier way to talk about these amounts: we use a unit called a liter. For example, instead of saying “1000 cubic centimeters,” we can simply say “1 liter”. This makes it much easier to understand how much liquid we have!
Definition Liter
A liter is a unit we use to measure the volume of liquids.
- 1 liter is the volume of a cube that measures \(10 \, \text{cm}\) on each side. $$ 1 \, \text{L} = 1000 \, \text{cm}^3 \text{ and } 1\,000 \, \text{L} = 1 \, \text{m}^3 $$
- We write it with the symbol L (a capital “L”).
- A smaller unit, the centiliter (cL), is often used for smaller volumes: $$ 1 \, \text{L} = 100 \, \text{cL} $$
- An even smaller unit, the milliliter (mL), is used for very small volumes: $$ 1 \, \text{L} = 1\,000 \, \text{mL} \quad \text{and} \quad 1 \, \text{cL} = 10 \, \text{mL} $$
Example
- A big water bottle holds about 1 liter of water, which is 100 centiliters:
- A small soda can holds about 0.33 liters of soda, which is 33 centiliters:
