# Measuring Capacity

## Introduction to Measuring Capacity

While we were discussing lengths and weights earlier, the materials which were being measured were all solids. When we discuss liquids or gases, it is not possible to describe their dimension in length, and not accurate to describe their dimension in weight. Moreover, the same weight of a liquid (say 1kg) can occupy very different spaces. That is why we need to talk about the volume of these fluids (liquids and gases). When you are cooking with a recipe which needs milk to be added, you can’t say that five cms of milk need to be added, you need a unit to describe the volume of milk that needs to be added. Therefore, we need units to measure capacity or volume.

## The Big Idea: Capacity is a three-dimensional quantity

When we measure the length of an object, it is a single dimensional dimension. When we talk about the area, it involves two dimensions. But when we talk of capacity, we refer to the volume which means that we are dealing in three dimensions. So, if a cuboidal vessel is being considered, then there is length, breadth and height, whereas is a round object is being considered then their volume or capacity is a cubic function of the radius.

### Units of capacity

The most common unit used to measure capacity is the litre. It is equal to 1000 millilitre. When we compare it to the dimension of length, then 1 litre is equal to 0.001 m^{3}, or we can say that 1 m^{3} is equal to 1000 litres. Both in real life situations as well as in mathematical calculations, these three are the units most often used. Let us see the formulas to calculate capacity or volume of some of the most common shapes.

### Capacity of cube

When the container whose capacity we wish to measure is a cube, it means that it has uniform sides. Its length, breadth and height are all equal. If we represent this equal value as ‘\({a}\)’, then the cube is said to have side ‘\({a}\)’. The volume of this cube can then be calculated as:

\(\begin{align}{\text{Volume of cube}} = a \times a \times a = {a^3}\end{align}\)

### The capacity of a rectangular prism

If the vessel does not have equal sides but is a prism with different values for length, breadth and height, then the same formula can be applied. Let us represent the length as ‘\({l}\)’, breadth as ‘\({b}\)’ and height as ‘\({h}\)’. Then, the

\(\begin{align}{\text{Volume of prism}} = l \times b \times h = lbh\end{align}\)

You will notice that this too is a multiplication of 3 dimensions of length just like a cube, but only the sides are all different, whereas a cube has equal sides.

### The capacity of pyramid and cone

The difference between a cone and a pyramid is that the base of a cone is circular whereas the base of a pyramid is a square. But when you invert a hollow cone or a hollow pyramid, those two shapes are often used for storing liquids, and so we need to know how to calculate the volumes of these two shapes.

\(\begin{align} &\text{The volume of a cone} = \frac{1}{3} \times \pi \times r \times r \times h = \frac{1}{3} \times \pi \times {r^2} \times h\text{ }\\& \left( \text{where }h \text{ is the height of the cone, } r\text{ is the radius of }\\ \text{the circular base, and } \pi \text{ is a constant} \right)\end{align}\)

\(\begin{align}\text{The volume of a pyramid} = \frac{1}{3} \times \text{area of base} \times \text{height} = \frac{1}{3} \times a \times h\text{ }\\ \left( \text{where }h\text{ is the height of the pyramid, and } \\ a\text{ is the side of square base} \right)\end{align}\)

### Capacity of cylinder

A cylinder is a tube-like structure with circular faces of the same radius at either end joined by the planar circular surface. It is the most common shape of vessels used for storing fluids. If you consider r as the radius of the circular base (and top) and h as the height of the cylinder, then the volume of the cylinder is expressed as follows:

\(\begin{align}{\text{Volume of cylinder}} = \pi \times r \times r \times h = \pi {r^2}h\end{align}\)

### Comparison of cylinder and cone

If you see the formulas for volumes of cylinder and cone above, you will realise that if there is a cone with the same radius and height as a cylinder, its volume is exactly a third of the volume of the cylinder.