# Volume

## What is volume?

The volume of a body is the numerical value that measures the amount of space occupied by it. The height, width and depth determine the volume, the larger they are, the greater the occupied space.

The concept of volume is very important, since the world is three-dimensional and all objects have width, height and depth, therefore they have volume. People use it frequently, for example when estimating whether the piece of furniture they want to buy fits in their living room or whether it fits into a certain dress size.

In certain areas of science and engineering, such as when working with fluids, be they liquids or gases, the volume occupied takes on a great importance: when filling containers and pumping liquids such as water, or in the design of a ship to make sure it doesn’t sink. All this makes it essential to determine it for a multitude of processes.

There are formulas to calculate the volume of geometric bodies of regular shapes, such as prisms, spheres, cylinders and cones, for example, based on some of their dimensions. And there are also ways to find out the volume of irregular objects, as will be described a little later.

## Volume formulas in geometric figures

For the most popular geometric objects there are formulas that allow calculating their volume:

• #### Cube

V = ℓ 3

Where V represents the volume and ℓ is the edge (side) of the cube.

• #### Parallelepiped

A parallelepiped is a rectangular box with width “a”, length ℓ and height “h”. Its volume is given by the product of its three dimensions:

V = a ∙ ℓ ∙ h

• #### Sphere

The volume of the sphere depends on its radius r:

$V=\frac{4}{3}\pi&space;r^{3}$

• #### Straight circular cylinder

The volume of the right circular cylinder is the product of the area of ​​its base and its height “h”. Since the base is a disk of radius “r”, whose area is A = π · r 2 , the volume remains:

V = πr 2 ∙ h

• #### Cone

The volume of the cone is one third of the product of the area of ​​the circular base A and the height h. Since A = πr 2 , then:

$V=\frac{1}{3}\pi&space;r^{2}h$

• #### Pyramid

For a pyramid whose base area is A and has a height “h”, the volume is given by:

$V=\frac{1}{3}A\cdot&space;h$

If the pyramid has a square base with side “a”, as in the figure, the area A of the base is a 2 and the volume of the pyramid is:

V = (1/3) ⋅a 2 ⋅h

• #### Prism

The volume of the prism is the product of the area of ​​the base A and the height “h”:

V = A ∙ h

### Volume units

In the International System of Units SI, the unit for volume is the cubic meter or m 3 , while in the Anglo-Saxon system it is the cubic foot or ft 3 (from feet , which in English means “foot”).

There are many other units, according to the size of the space occupied. For example, cubic kilometers km 3 for larger volumes or cubic millimeters mm 3 for small volumes. There are also units for local use.

Mention should also be made of the units of capacity, closely related to those of volume, which are preferably used for liquids. The central unit of capacity is the liter, abbreviated L, which is equal to one dm 3 (cubic decimeter).

Other units that are worth mentioning are the gallon, the cubic inch, the cup and the drop, the latter widely used to dose medicines.

## How is volume measured?

The volume of a body, like any other measurement, is carried out by comparing it with a suitable standard, in this case a unit of volume.

The unit of volume is defined as that of the cube whose edge is 1 unit. This unit can be meter, centimeter, foot, inch, or anything else. So the volume of the object corresponds to the number of cubic units occupied by the figure and is always a positive quantity.

### Volume of a geometric body

When it comes to a geometric body such as those already mentioned, the volume is calculated through the appropriate formula, measuring the dimensions indicated by the formula.

For example, if you want to know the volume of a sphere, you need to measure its diameter and thereby know its radius, which is half. If it is a rectangular box, the width, height and depth of the box are measured.

Then the requested values ​​are inserted into the formula, taking care that all the units are the same, the required operations are carried out and voila, you have the volume of the object.

### Volume of an irregular body

Irregular solids do not have a geometric shape, like a stone or pebble. Even so, its volume can be found with the help of a graduated container filled with water, using the liquid displacement method.

First, the volume occupied by the water is determined and then the irregular object is completely submerged, measuring the new volume, which is greater than the original. The volume of the irregular object is the difference between this volume and that of the water alone.

For this method to work, the object must not be made of some substance that dissolves in water easily, it must remain completely submerged, and of course, you must have a graduated container of the necessary size to fully house it.

## Volume examples

The approximate volume of some known objects is:

• Earth: 1.08321 × 10 12  km³
• Amazon River: 225,000 m 3 / s (The volume per unit of time is called “flow”)
• The Great Pyramid of Giza: 2,600,000 m³
• A soccer ball: 5600 cm 3
• A backpack: 50 dm 3

## Volume and mass

Volume and mass are not synonymous, the first is linked to the dimensions of the object and the second to the amount of matter it contains.

There can be a lot of matter in a small object, or very little in a large object, which depends on the density of the material, which is the ratio of the mass to the volume of an object:

$Densidad=\frac{Masa}{Volumen}$

## Solved exercises

### Exercise 1

Calculate the volume of a rectangular box whose dimensions are 34 cm × 22 cm × 8 cm.

• #### Solution

The volume of a rectangular box is simply the product of its three dimensions:

V = 34 cm × 22 cm × 8 cm = 5984 cm 3

### Exercise 2

The base of a quadrangular pyramid has an area of ​​16 cm 2 and its height is 6 cm. Calculate the volume of said pyramid.

• #### Solution

The formula given above is used for the volume of a pyramid, known as the area of ​​its base:

$V=\frac{1}{3}A\cdot&space;h$

And the numerical values ​​are substituted:

V = (1/3) × 16 cm 2 × 6 cm = 32 cm 3