# Volume

# Volume

The volume of liquids and gases

Volume is the amount of space occupied by an object or a material. Volume is said to be a derived unit, since the volume of an object can be known from other measurements. In order to find the volume of a rectangular box, for example, one only needs to know the length, width, and depth of the box. Then the volume can be calculated from the formula, V = l× w× d. Specifically, if the length (l) is 30 centimeters, width (w) is 20 centimeters, and depth (d) is 10 centimeters, then the volume is: V = 30 centimeters x 20 centimeters x 10 centimeters = 6,000 cubic centimeters.

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 10^{21} cubic meters. By contrast, the volume of a hydro-genatom is far smaller, about 10^{-30} cubic meters.

Greek mathematicians, especially Archimedes (287–212 BC), were among the first scientists to derive formulas for the volumes of common shapes. However, it was not until the invention of the calculus by English physicist and mathematician Sir Isaac Newton (1642–1727) and German mathematician Gottfried Wilhelm Leibniz (1646–1716) in the seventeenth century that the volume for an arbitrary shape could be calculated. One can show, for example, that for an object of a certain volume, a sphere has the smallest ratio of surface area to volume. This example explains why a cat curls up into a ball when it sleeps— it is trying to make itself like a sphere, so its surface area is as small as possible in order to minimize heat loss through its skin.

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.

## Units of volume

The term unit volume refers to the volume of one something: one quart, one milliliter, or one cubic inch, for example. Every measuring system that exists defines a unit volume for that system. Then, when one speaks about the volume of an object in that system, what he or she means is how many times that unit volume is contained within the object. If the volume of a glass of water is said to be 35.6 cubic inches, for example, what is meant is that 35.6 cubic inch unit volumes could be placed into that glass.

Mathematically, volume would seem to be a simple extension of the concept of area, 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. When this idea is extended to include all possible positive real numbers, however, 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.

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. For example, 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. The more standard units used in the British system, however, are the cubic inch or cubic foot and the gallon.

Variability in the basic units also exists. For example, the quart differs in size depending on whether it is being used to measure a liquid or dry volume and whether it is a measurement made in the British or customary U.S. system. As an example, 1 customary liquid quart is equivalent to 57.75 cubic inches, while 1 customary dry quart is equivalent to 67.201 cubic inches. In contrast, 1 British quart is equivalent to 69.354 cubic inches.

The basic unit of volume in the international system (often called the metric system) is the liter (abbreviated as l), although the cubic centimeter (cc or cm^{3}) and milliliter (ml) are also widely used as units for measuring volume. The fundamental relationship between units in the two systems is given by the fact that 1 U.S. liquid quart is equivalent to 0.946 L or, conversely, 1 liter is equivalent to 1.057 customary liquid quarts.

## 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 liter of iron from 32° F (0° C) to 212° F (100° C) causes an increase in volume of less than 1%, and heating a liter of water through the same temperature range causes an increase in volume of less than 5%. However, heating a liter of air from 32° F (0° C) to 212° F (100° C) causes an increase in 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. This method of approximation can become exact by using calculus. Another way is to calculate the volume by water displacement, or the displacement of some other liquid.

Suppose, for example, that one wishes to calculate the volume of an irregularly shaped piece of rock. One way to determine that volume is first to add water to some volume-measuring instrument, such as a graduated cylinder. The exact volume of water added to the cylinder is recorded. Then, the object whose volume is to be determined is also added to the cylinder. The water in the cylinder will rise by an amount equivalent to the volume of the object. Thus, the final volume read on the cylinder less the original volume is equal to the volume of the submerged object.

This method is applicable, of course, only if the object is insoluble in water. If the object is soluble in water, then another liquid, such as alcohol or cyclohexane, can be substituted for the water.

## 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 simply 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 1 l are commonly available in science laboratories for measuring the volumes of liquids. Other devices, such as pipettes and burettes, are available for measuring exact volumes, especially small volumes.

### KEY TERMS

**British (customary) system** —A collection of measuring units that has developed haphazardly over many centuries and is now used almost exclusively in the United States and for certain specialized types of measurements.

**Displacement method** —A method for determining the volume of an irregularly shaped solid object by placing it in a measured amount of water or other liquid.

**International (metric) system** —A system of measurement used by all scientists and in common practice by almost every nation of the world.

**Unit volume** —The basic size of an object against which all other volumes are measured in a system.

The volume of a liquid is only moderately affected by pressure, but it is often quite sensitive to changes in temperature. For this reason, volume measurements made at temperatures other than ambient temperature are generally so indicated when they are reported, as V = 35.89 ml (95° F; 35° C).

The volume of gases is very much influenced by temperature and pressure. Thus, any attempt to measure or report the volume of the gas must always include an indication of the pressure and temperature under which that volume was measured. Indeed, since gases expand to fill any container into which they are placed, the term volume has meaning for a gas *only* when temperature and pressure are indicated.

David E. Newton

# Volume

# Volume

Volume is the amount of space occupied by an object or a material. Volume is said to be a derived unit, since the volume of an object can be known from other measurements. In order to find the volume of a rectangular box, for example, one only needs to know the length, width, and depth of the box. Then the volume can be calculated from the formula, V = l × w × d.

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** .

## Units of volume

The term unit volume refers to the volume of one something: one quart, one milliliter, or one cubic inch, for example. Every measuring system that exists defines a unit volume for that system. Then, when one speaks about the volume of an object in that system, what he or she means is how many times that unit volume is contained within the object. If the volume of a **glass** of water is said to be 35.6 cubic inches, for example, what is meant is that 35.6 cubic inch unit volumes could be placed into that glass.

Mathematically, volume would seem to be a simple extension of the concept of area, 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. When this idea is extended to include all possible positive **real numbers** , however, 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.

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. For example, 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. The more standard units used in the British system, however, are the cubic inch or cubic foot and the gallon.

Variability in the basic units also exists. For example, the "quart" differs in size depending on whether it is being used to measure a liquid or dry volume and whether it is a measurement made in the British or customary U.S. system. As an example, 1 customary liquid quart is equivalent to 57.75 cubic inches, while 1 customary dry quart is equivalent to 67.201 cubic inches. In contrast, 1 British quart is equivalent to 69.354 cubic inches.

The basic unit of volume in the international system (often called the **metric system** ) is the liter (abbreviated as l), although the cubic centimeter (cc or cm3) and milliliter (ml) are also widely used as units for measuring volume. The fundamental relationship between units in the two systems is given by the fact that 1 U.S. liquid quart is equivalent to 0.946 L or, conversely, 1 liter is equivalent to 1.057 customary liquid quarts.

## 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 liter of **iron** from 32°F (0°C) to 212°F (100°C) causes an increase in volume of less than 1%, and heating a liter of water through the same temperature range causes an increase in volume of less than 5%. But heating a liter of air from 32°F (0°C) to 212°F (100°C) causes an increase in 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. In such a case, V = s3, 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 = πr2h.

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 sub-divisions. This method of approximation can become exact by using **calculus** . Another way is to calculate the volume by water displacement, or the displacement of some other liquid.

Suppose, for example, that one wishes to calculate the volume of an irregularly shaped piece of rock. One way to determine that volume is first to add water to some volume-measuring instrument, such as a graduated cylinder. The exact volume of water added to the cylinder is recorded. Then, the object whose volume is to be determined is also added to the cylinder. The water in the cylinder will rise by an amount equivalent to the volume of the object. Thus, the final volume read on the cylinder less the original volume is equal to the volume of the submerged object.

This method is applicable, of course, only if the object is insoluble in water. If the object is soluble in water, then another liquid, such as **alcohol** or cyclohexane, can be substituted for the water.

## 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 simply 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 1 l are commonly available in science laboratories for measuring the volumes of liquids. Other devices, such as pipettes and burettes, are available for measuring exact volumes, especially small volumes.

The volume of a liquid is only moderately affected by pressure, but it is often quite sensitive to changes in temperature. For this reason, volume measurements made at temperatures other than ambient temperature are generally so indicated when they are reported, as V = 35.89 ml (95°F; 35°C).

The volume of gases is very much influenced by temperature and pressure. Thus, any attempt to measure or report the volume of the gas must always include an indication of the pressure and temperature under which that volume was measured. Indeed, since gases expand to fill any container into which they are placed, the term volume has meaning for a gas *only* when temperature and pressure are indicated.

David E. Newton

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**British (customary) system**—A collection of measuring units that has developed haphazardly over many centuries and is now used almost exclusively in the United States and for certain specialized types of measurements.

**Displacement method**—A method for determining the volume of an irregularly shaped solid object by placing it in a measured amount of water or other liquid.

**International (metric) system**—A system of measurement used by all scientists and in common practice by almost every nation of the world.

**Unit volume**—The basic size of an object against which all other volumes are measured in a system.

# Volume

# Volume

Volume is the amount of space occupied by an object or a material. Volume is said to be a derived unit, since the volume of an object can be known from other measurements. In order to find the volume of a rectangular box, for example, one only needs to know the length, width, and depth of the box. Then the volume can be calculated from the formula, V = l · w · d.

The 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, however, further heating of water results in an increase in volume that is more characteristic of matter.

## Units of volume

The term unit volume refers to the volume of "one something": one quart, one milliliter, or one cubic inch, for example. Every measuring system that exists defines a unit volume for that system. Then, when one speaks about the volume of an object in that system, what he or she means is how many times that unit volume is contained within the object. If the volume of a glass of water is said to be 35.6 cubic inches, for example, what is meant is that 35.6 cubic inch unit volumes could be placed into that glass.

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. For example, 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. The more standard units used in the British system, however, are the cubic inch or cubic foot and the gallon.

Variability in the basic units also exists. For example, the quart differs in size depending on whether it is being used to measure a liquid or dry volume and whether it is a measurement made in the British or customary U.S. system. As an example, 1 customary liquid quart is equivalent to 57.75 cubic inches, while 1 customary dry quart is equivalent to 67.201 cubic inches. In contrast, 1 British quart is equivalent to 69.354 cubic inches.

The basic unit of volume in the metric system is the liter (abbreviated as L), although the cubic centimeter (cc or cm^{3}) and milliliter (mL) are also widely used as units for measuring volume. The fundamental relationship between units in the two systems is given by the fact that 1U.S. liquid quart is equivalent to 0.946 liter or, conversely, 1 liter is equivalent to 1.057 customary liquid quarts.

## Words to Know

**British system:** A system of measurement long used in many parts of the world but now used commonly only in the United States among the major nations of the world.

**Displacement method:** A method for determining the volume of an irregularly shaped solid object by placing it in a measured amount of water or other liquid and noting the increase in volume of the liquid.

**Metric system:** A system of measurement used by all scientists and in common practice by almost every nation of the world.

**Unit volume:** The basic size of an object against which all other volumes are measured in a system.

## The volume of solids

The volumes of solids are relatively less affected by pressure and temperature changes than are the volumes of most liquids and all gases. For example, heating a liter of iron from 0°C to 100°C causes an increase in volume of less than 1 percent. Heating a liter of water through the same temperature range causes an increase in volume of less than 5 percent. But heating a liter of air from 0°C to 100°C causes an increase in volume of nearly 140 percent.

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. 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 height of the cylinder.

Many solid objects have irregular shapes for which no mathematical formula exists. One way to find the volume of such objects is to subdivide 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 sub-divisions. This method of approximation can become exact by using calculus.

Another way is to calculate the volume by water displacement, or the displacement of some other liquid. Suppose, for example, that one wishes to calculate the volume of an irregularly shaped piece of rock. One way to determine that volume is first to add water to some volume-measuring instrument, such as a graduated cylinder. The exact volume of water added to the cylinder is recorded. Then, the object whose volume is to be determined is also added to the cylinder. The water in the cylinder will rise by an amount equivalent to the volume of the object. Thus, the final volume read on the cylinder less the original volume is equal to the volume of the submerged object.

This method is applicable, of course, only if the object is insoluble in water. If the object is soluble in water, then another liquid, such as alcohol or cyclohexane, can be substituted for the water.

## The volume of liquids and gases

Measuring the volume of a liquid is relatively straightforward. Since liquids take the shape of the container in which they are placed, a liquid whose volume is to be found can simply 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 milliliters to 1 liter are commonly available in science laboratories for measuring the volumes of liquids. Other devices, such as pipettes and burettes (small measuring tubes), are available for measuring exact volumes, especially small volumes.

The volume of a liquid is only moderately affected by pressure, but it is often quite sensitive to changes in temperature. For this reason, volume measurements made at temperatures other than ambient (the surrounding) temperature are generally so indicated when they are reported, as V = 35.89 milliliters (35°C).

The volume of gases is very much influenced by temperature and pressure. Thus, any attempt to measure or report the volume of the gas must always include an indication of the pressure and temperature under which that volume was measured. Indeed, since gases expand to fill any container into which they are placed, the term volume has meaning for a gas *only* when temperature and pressure are indicated.

# volume

vol·ume / ˈvälyəm; -ˌyoōm/ • n. 1. a book forming part of a work or series. ∎ a single book or a bound collection of printed sheets. ∎ a consecutive sequence of issues of a periodical. ∎ hist. a scroll of parchment or papyrus containing written matter. 2. the amount of space that a substance or object occupies, or that is enclosed within a container, esp. when great: *the sewer could not cope with the volume of rainwater*| *a volume of air.* ∎ the amount or quantity of something, esp. when great: *changes in the volume of consumer spending.* ∎ (a volume of/volumes of) a certain, typically large amount of something: *the volumes of data handled are vast.* ∎ fullness or expansive thickness of something, esp. of a person's hair. 3. quantity or power of sound; degree of loudness: *he turned the volume up on the radio.*

# volume

**volume** (hist.) roll of parchment, etc. forming a book; tome XIV; size, bulk (†of a book) XVI, (of other things) XVII; (poet.) coil, convolution XVII. ME. *volym*, *volum(e)* — OF. *volum*, (also mod.) *volume* — L. *volūmen* roll of writing, book, coil, f. *volū-*, var. of base **wolw-* of *volvere* roll = Gr. *eilúein*, f. IE. **wel-* **wol-* turn.

So **voluminous** XVII. — late L. *volūminōsus*.

# volume

**volume** A removable unit of any data storage medium, e.g. a reel or cartridge of magnetic tape or a demountable disk.

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