Table of Contents

## Definition of Volume

Volume is the quantity of three-dimensional space occupied by a liquid, solid, or gas. Common units used to express volume include liters, cubic meters, gallons, milliliters, teaspoons, and ounces, though many other units exist.

- The volumes of some typical objects can vary enormously.
**For example**, the human body has a volume of roughly 26 gallons (0.1 cubic meter or 100 liters). A grown elephant’s volume is about 1,500 gallons (6 cubic meters), and the volume of Earth is about 1021 cubic meters. By contrast, the volume of a hydrogen atom is far smaller, about 10-30 cubic meters. - Volume of most physical objects is a function of two other factors, temperature and pressure.
- In general, the volume of an object increases with an increase in temperature and decreases with an increase in pressure. Some exceptions exist to this general rule.
**For example**, when water is heated from a temperature of 32°F (0°C) to 39° F (4°C), it decreases in volume. Above 39°F (4°C). However, further heating of water results in an increase in volume that is more characteristic of matter. - Volume is often quantified numerically using the SI derived unit, the cubic meter.

### Volume of Container

The volume of a container is generally understood to be the capacity of the container, i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.

Three-dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes, can be easily calculated using arithmetic formulas.

If a formula exists for the shape’s boundary, volumes of complicated shapes can be calculated with integral calculus. One-dimensional figures (such as lines) and two-dimensional shapes (like squares) are assigned zero volume in the three-dimensional space.

In differential geometry, volume is expressed by means of the volume form and is an important global Riemannian invariant.

In thermodynamics, volume is a fundamental unit and is a conjugate variable to pressure.

- Volume is the three-dimensional space occupied by a substance or enclosed by a surface.
- The International System of Units (SI) standard unit of volume is the cubic meter (m
^{3}). - The metric system uses the liter (L) as a volume unit. One liter is the same volume as a 10-centimeter cube.
- The volume of a liquid is measured with a measuring container, such as a measuring cup or graduated cylinder.
- Volume of a gas depends on the volume of its container: gases expand to fill whatever space is available to them.
- The volume of a regularly shaped solid can be calculated from its dimensions. For example, the volume of a rectangular solid is the product of its length, width, and height.
- The volume of an irregularly shaped solid can be measured by the displacement method. You can read below how this method works.

## Examples of Volume

- As a volume example, a student might use a graduated cylinder to measure volume of a chemical solution in milliliters.
- You could buy a quart of milk.
- Gases are commonly sold in units of volume, such as cubic centimeters, cm
^{3}, or cubic liters.

## Units of Volume

Mathematically, the volume would seem to be a simple extension of the area concept, but it is actually more complicated. The volume of simple figures with integral sides is found by determining the number of unit cubes that fit into the figure.

However, when this idea is extended to include all possible positive real numbers, paradoxes of volume occur. It theoretically is possible to take a solid figure apart into a few pieces and reassemble it so that it has a different volume.

- Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length.

For example, a cubic centimeter (cm^{3}) is the volume of a cube whose sides are one centimeter (1 cm) in length.

- The units in which volume is measured depend on a variety of factors, such as the system of measurement being used and the type of material being measured. Volume in the British system of measurement may be measured in barrels, bushels, drams, gills, pecks, teaspoons, or other units. Each of these units may have more than one meaning, depending on the material being measured.

**For example**, the precise size of a barrel ranges anywhere from 31 to 42 gallons, depending on federal and state statutes. Themore standard units used in the British system, however, are the cubic inch or cubic foot and the gallon.

### In SI Unit

In the International System of Units (SI), the standard unit of volume is the cubic meter (m^{3}). The metric system also includes the liter (L) as a unit of volume, where one liter is the volume of a 10-centimetre cube.

Thus

1 liter = (10 cm)^{3} = 1000 cubic centimeters = 0.001 cubic meters,

so

1 cubic meter = 1000 liters.

Small amounts of liquid are often measured in milliliters, where

1 milliliter = 0.001 liters = 1 cubic centimeter.

In the same way, large amounts can be measured in megaliters, where

1 million liters = 1000 cubic meters = 1 megaliter.

Various other traditional units of volume are also in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot. These all are the units of volume.

#### Two other volume units are commonly encountered in the chemical laboratory — the litre (l) and the millilitre (ml — one-thousandth of a litre).

The litre was originally defined as the volume of one kilogram of pure water at the temperature of its maximum density (3.98°C), but in 1964 the definition was changed. The litre is now exactly one-thousandth of a cubic meter, that is, 1 dm^{3}. A millilitre is, therefore, exactly 1 cm^{3}. Because the new definition of litre altered its volume slightly.

It is recommended that the results of highly accurate measurements be reported in the SI units cubic decimeters or cubic centimetres rather than in litres or millilitres.

However, for most situations discussed in this online textbook, the units cubic decimeters and litre and cubic centimetres and millilitres may be used interchangeably.

Thus when recording a volume obtained from laboratory glassware calibrated in millilitres, you can just as well write 24.7 cm^{3} as 24.7 ml.

## The Volume Of Solids

- The volume of solids is relatively less affected by pressure and temperature changes than is that of liquids or gases.

**For example**, heating a litre of iron from 32° F (0° C) to 212° F (100° C) causes an increase in the volume of less than 1%, and heating a litre of water through the same temperature range causes an increase in the volume of less than 5%. However, heating a litre of air from 32° F (0° C) to 212° F (100° C) causes an increase in the volume of nearly 140%.

- The volume of a solid object can be determined in one of two general ways, depending on whether or not a mathematical formula can be written for the object.

**For example**, the volume of a cube can be determined if one knows the length of one side (s). In such a case, V=s^{3}, or the volume of the cube, is equal to the cube of the length of any one side (all sides being equal in length).

The volume of a cylinder, on the other hand, is equal to the product of the area of the base multiplied by the altitude of the cylinder. For a right circular cylinder, the volume is equal to the product of the radius of the circular base (r) squared multiplied by the height (h) of the cone and by pi (π), or V = πr^{2}h.

- Many solid objects have irregular shapes for which no mathematical formula exists. One way to find the volume of such objects is to sub-divide them into recognizable shapes for which formulas do exist (such as many small cubes) and then approximate the total volume by summing the volumes of individual subdivisions.

## The Volume Of Liquids And Gases

Measuring the volume of a liquid is relatively straight forward. Since liquids take the shape of the container in which they are placed, a liquid whose volume is to be found can be poured into a graduated container, that is, a container on which some scale has been etched.

Graduated cylinders of various sizes, ranging from 10 ml to 1l, are commonly available in science laboratories for measuring the volumes of liquids. Other devices, such as pipettes and burettes, are available to measure exact volumes, especially small ones.

## Calculation of Volume

There are several different ways to calculate the volume of an object. Because every object has different properties – such as mass, shape, and displacement – which relate back to its volume.

For a simple shape, like a cube or a sphere, you can find its volume by first determining its overall measurements of length or diameter. You can also find the volume by figuring an object’s displacement.

Here are three different methods for finding volume. Depending on the object you are trying to measure, you will find that one method or another is preferable

### Solving by Space

All physical objects occupy space, and you can find the volume for some of them by measuring their physical dimensions. This is the easiest way to calculate the volume of objects with simple shapes, like cones, rectangular prisms, spheres, and cylinders.

**eg** a honeydew melon is close enough in shape to a sphere that you can use the sphere equation to calculate its volume and still get a fairly accurate answer.

#### Example

Calculate the volume by multiplying the measured length and width of the space together, then multiply the result by the height of the room.

From the example, 10 *25 feet = 250 square feet, and 5 * 10 feet = 50 square feet.

### Solving by Density and Mass

Density is defined as an object’s mass per a given unit of volume. So, if you know the object’s density, and you’re able to weigh it, you can determine its volume with the equation:

Volume = weight / density

#### Example

You find a large rock with a roughly spherical shape and want to know its composition. Finding its density can help you determine that.

**Solution**

Calculate the volume of the rock by measuring its diameter and dividing by 2 to find its radius (r).

The volume of a sphere is 4/3πr^{3}, so if the rock has a radius of 10 inches, its volume is 418.67 cubic inches.

Convert to cubic feet by multiplying by 0.00057. The result is 0.239 cubic feet.

If the rock weighs 40 pounds, its density is 40 lb/0.239 ft^{3} = 167. 36 lb/ft^{3}. This is very close to the density of granite, so there’s a good chance the rock is solid granite.

### Solving by Displacement

This is another way of measuring the physical space that an object occupies. If the object has an abnormal shape, you might be unable to measure its physical dimensions accurately.

Instead, what you can do is measure the volume which is displaced when the object is immersed in a liquid or a gas. This is a very common method for measuring volume, and when done correctly, it is highly accurate.

**eg**. if you want to know the volume of a piece of ginger root, you can fill a beaker or a measuring cup with a known volume of water – let’s say one cup. Next, add the ginger. Make sure that it is submerged underwater.

Then, measure the new volume at the water line. The new volume will always be more than the starting volume. Subtract the starting volume (one cup) from this new volume, and you will have the volume of the ginger.

#### Examples

- Pour enough water from your cup into the graduated cylinder to reach a height that will cover the sample. Read and record the volume.
- Slightly tilt the graduated cylinder and carefully place the sample into the water.
- Place the graduated cylinder upright on the table and look at the level of the water. If the sample floats, use a pencil to gently push the top of the sample just under the surface of the water. Record the number of milliliters for this final water level.
- Find the amount of water displaced by subtracting the initial level of the water from the final level. This volume equals the volume of the cylinder in cm
^{3}. - A student closely examines the water level in a graduated cylinder after placed a plastic cylinder in the water.
- Record this volume in the chart on the activity sheet.
- Remove the sample by pouring the water back into your cup and taking the sample out of your graduated cylinder.

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