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# Cubic Unit

Out of the many quantities that we need to measure in our day to day lives, the most common is volume. When we are preparing a dish, we need to measure the required volumes of the relevant ingredients. When we are feeding milk to a baby, we need to measure the volume mentioned on the formula box and mix the contents accordingly.

In big warehouses and industries, the space occupied by inventory is measured in volume. The capacity of an oil barrel or even a complete oil tanker is measured in volume.

In simple words, volume refers to the amount of space that an object occupies. Consider a solid ball. If we drop this ball in a barrel completely filled with water, then the amount of water displaced by the ball is the volume of that ball.

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## What is a Cubic Unit?

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Volume or capacity can be measured using various units. Usually for liquids, the unit used for measuring capacity is litre or ounce. However, for solids, it is not always possible to measure capacity in litres. Say, for example, if we have a glass tumbler, we can fill it with water to measure its capacity. (If 250 ml of water fits into the tumbler then we say that its capacity is 250 ml.) However, suppose we wanted to measure the volume or capacity of a cardboard box. What shall we do in that case? Can we fill it with water? Not really.

That's where cubic unit steps in.

Definition:

A cubic unit is a volume that is occupied by a cube that has a length of 1unit, width of 1unit and height of 1unit. Thus, to measure the volume of that box, we can fill it with such unit cubes. The number of cubes that fit into the box would be the volume of the box in cubic units. Well, how inconvenient is that? Can we actually fill all the objects that we want to measure the volume of, with such unit cubes? I don't think so. So what do we do then?

## Cubic Unit Formula

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The formula to determine the volume of an object in cubic units depends on the shape of the object. Let us consider a few common shapes.
Cube: The volume of a cube is given by the formula:

Volume = side $\times$ side$\times$ side

If the side is measured in centimetres, then the unit of the volume would be cubic centimetres. Similarly, if the side is measured in meters, then the volume of the cube would be in cubic metres.

In general, if the side of the object is measured in any unit then its volume would be cubic of that unit. Symbolically they are written like this:

cubic centimetre $\rightarrow$ $cm^3$

cubic metre $\rightarrow$ $m^3$

Let us now take a look at the formulas for other shapes as well.

Prism: The volume of any prism is given by the formula:

Volume = area of base $\times$ height

Thus for a rectangular prism, since the base is a rectangle of length L and width W, if the height is h, then the volume would be:

Volume of rectangular prism = L $\times$ W $\times$ h cubic units

Similarly for a cylinder, the area of the base is the area of the circle at the base. Suppose the radius of the circle is r. Then,

area of base = $\pi^2$

Therefore the volume of cylinder of height h would be:

Volume = area of base $\times$ height

Volume = $\pi^2\ h$ cubic units.

The 'unit'can also be inches or feet. That would make the volume in cubic inches or cubic feet.

Pyramid: The volume of a pyramid is given by the formula:

Volume = $\frac{1}{3}$ $\times$ area of base $\times$ height

Thus if we have a square based pyramid, then its volume would be:

Volume = $\frac{1}{3}$ $\times$ $a^2$ $\times$ $h$

Where, $a$ is the length of side of the base and $h$ is the height of the pyramid.

Similarly, a cone is a pyramid with a circular base. Thus its volume would be:

Volume = $\frac{1}{3}$ $\pi^2\ h$

## Units for Measuring Volume

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The unit that we use for measuring volume largely depends on the size of the object. If we were to measure the volume of a box of cookies, we'd normally use cubic centimetres or cubic inches. However, if we were to measure the volume of an oil tanker or a shipping container, we'd used cubic metres or cubic feet to measure volume. That is because, the dimensions of the oil tanker or the shipping container, are normally measured in feet or metres. Also, geographical location plays an important part here. In the US, feet and inches are more common units of measuring length. Therefore there the common unit for volume is cubic feet or cubic inches. However, in Europe, the SI system of units is more prevalent. Thus the volumes here are usually measured in cubic meters and cubic centimetres.

## Units of Conversion

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Just like units of distance, it is possible to convert units of volumes from one unit to another. We can convert from cubic metres to cubic centimetres and vice versa. Same applies to cubic inches and cubic feet. Let see how.

We know that,

1m = 100 cm

Therefore,

$1\ m^3$ = $1m\ \times\ 1m \times\ 1m$ = $100\ cm \times\ 100cm \times\ 100cm$

Thus,

$1m^3$ = $1000000\ cm^3$

This is the rule to be used to convert from cubic metres to cubic centimetres and vice versa.

Similarly for inches and feet we have:

1 $ft$ = 12 $in$

$\therefore$ 1ft $\times$ 1ft $\times$ 1ft = 12in $\times$ 12in $\times$ 12in

$\therefore$ 1 $ft^3$ = 1728 $in^3$

Let us look at some examples based on this to better understand this.

## Examples

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Example 1: The dimensions of a rectangular prism are 120 cm by 140 cm by 200 cm. Calculate its volume in cubic metres.

Solution:

Volume of prism=area of base ×height

$\therefore$ Volume = L $\times$ W $\times$ h

$\therefore$ Volume = 120 $\times$ 140 $\times$ 200 = 3360000 $cm^3$

Now, we know that:

1000000 $cm^3$ = 1 $m^3$

$\therefore$ 3360000 $cm^3$ = $\frac{3360000\ \times\ 1}{1000000\ m^3}$

$\therefore$ Volume = 3.36 $m^3$ $\leftarrow$ Answer!
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