**Introduction**

The volume of a three-dimensional figure represents the amount of space that a solid takes up overall. A three-dimensional figure is anything that possesses length, breadth, and thickness. The volume of a hollow three-dimensional object is the sum of the volume of the empty space inside the body minus the volume filled by the body. Let's now discuss capacity. As is well known, we frequently encounter hollow items in our daily lives. The most material that a thing can carry is referred to as its capacity (about volume).

**Volume vs Capacity**

The total amount of space a substance occupies is its **volume**; hence, the volume of a solid object is the area of space it occupies. On the other hand, **capacity** refers to the amount of anything that a container can hold. Simply said, capacity refers to the container's volume.

**Difference Between Volume and Capacity in Tabular Form**

Basis of Comparison | Volume | Capacity |

Meaning | The total amount of space a thing occupies is implied by its volume. | The term "capacity" describes an object's ability to hold a substance, such as a solid, liquid, or gas. |

What is it? | It is the precise quantity of something that fills a particular place. | It is the maximum amount of substance that a thing might possibly hold. |

Measurement | measured using units of volume, such as cubic meters and cubic centimeters. | based on metric measurements such as liters, gallons, etc. |

Object | Volume exists in both solid and hollow objects. | Only hollow things have capacity. |

**What is Volume?**

In three dimensions, everything takes up some space. Here, the area's volume is being measured. The area included within a three-dimensional object's limits is referred to as its volume. This is sometimes referred to as the ability of the target.

According to the volume of the object, you can calculate how much is needed to fill it; for example, how much water is needed to fill a bottle, aquarium, or watering can.

We split the area into equal square units to determine the area of any two-dimensional form. Like this, we shall split the volume of solid objects into equal cubical units while calculating it.

**History of Volume**

With the advancement of modern mathematics, the idea of volume in mathematics has a long and rich history that spans various ancient cultures. The measurement of the three-dimensional space occupied by an object or region is the main focus of the study of volume.

- Ancient Egypt: The pyramids and other architectural marvels created by the ancient Egyptians are well-known today. Although they lacked a rigorous mathematical framework, they did have experience with volume measurement. The Egyptians employed a technique known as "hekat," which included estimating the volume of asymmetrical vessels by measuring the number of hekat units of grain each one could store.
- Mesopotamia in antiquity: The Babylonians in particular made substantial contributions to mathematics. They created complex mathematical methods, such as the calculation of volumes. To calculate the volume of straightforward structures like rectangular prisms and cylindrical shapes, the Babylonians employed approximations and formulas.
- The study of geometry grew in popularity in ancient Greece, where the idea of volume also formalized. Mathematicians like Archimedes and Euclid made important contributions. In his renowned work "Elements," Euclid created the groundwork for geometrical principles and provided formulas for determining the volumes of various solids.
- Particularly well-known for his work on volume is Archimedes. He made important strides in the estimation of volume for irregular shapes and found the buoyancy laws. His "Method of Exhaustion," which he used to determine the volume of a sphere and the surface area and volume of a cone, is his most well-known contribution.
- Islamic Golden Age: Mathematical research advanced significantly throughout the Islamic Golden Age, which lasted from the eighth to the fourteenth century. Islamic mathematicians like Al-Khwarizmi, Ibn al-Haytham, and Al-Jayyani improved on the Greeks' work and established the idea of volume. The knowledge of geometric solids like spheres, pyramids, and cylinders was aided by them.
- Renaissance and Modern Periods: Volume calculations continued to progress during the Renaissance, when mathematical studies saw a comeback. Galileo Galilei and other mathematicians like Leonardo da Vinci contributed to our understanding of volume, particularly as it relates to solids of revolution.
- Calculus was created in the 17th century by mathematicians like Gottfried Wilhelm Leibniz and Isaac Newton, who revolutionized mathematical analysis and gave us sophisticated tools for estimating the volumes of curved and irregular objects.
- The idea of volume has been expanded upon and generalized in modern mathematics. Mathematicians have been able to define volume for more abstract spaces, such as fractals and infinite-dimensional spaces, thanks to the development of measure theory in the 19th and 20th centuries.

**The Volume of 3-D Shapes**

Everything in our environment has the inherent ability to occupy space. These actual things can be directly compared to the fundamental 3-D shapes. Let's take a closer look at these solid shapes' volumes.

**Volume of Sphere**

A sphere is the simplest and most common type of three-dimensional shape. We frequently see spheres in the form of balls, globes, decorative lights, oranges, etc. A sphere's radius is the simplest parameter to measure. The radius of the sphere is used to determine its volume.

*Volume of Sphere: - 4/3 **πr^2, where r=radius*

**Volume of Cube**

The cube is the following basic and typical three-dimensional shape. Each side of the cube is the same length, which distinguishes it from other objects. Dice, Rubik's cubes, sugar cubes, gift boxes, and other items in the shape of a cube are some examples of commonplace cube-shaped objects. The length of a cube's side is used to determine its volume.

*Volume of Cube: - a^3, where a= length of each side of cube*

**Volume of Cuboid**

Another name for the cuboid shape is the rectangular prism. The lengths of the sides in a cuboid will differ. The sides of a cuboid are denoted by the following notation.

Height = l

width = b

height = h

The volume of a cuboid is calculated using each of these measurements. Books, shoe boxes, bricks, mattresses, etc. are typical cuboids.

*Volume of Cuboid: - l*b*h*

**Volume of Cylinder**

A cylinder is another example of a three-dimensional object with two circular bases and a height between them. The cylindrical items we use daily include water bottles, buckets, candles, cans, etc. The base's radius and the cylinder's height are measured to determine its volume.

*Volume of Cylinder: *- πr^2h, *where r = radius of base and h = height of cylinder.*

**Volume of Cone**

A cone is a frequent three-dimensional shape in the world. A cone can be anything from an ice cream cone to a party hat, a funnel, or a Christmas tree. An iconic three-dimensional geometric shape called a cone has a flat and curved surface that points upward.

*Volume of cone = 1/3 πr^2h, where r = the cone's base radius and h = height from base to top*

**Volume Measurement**

Because it is a commonly used (unit)3 and can be written in different ways such as cubic centimeters, cubic inches, cubic feet, cubic meters, etc., volume is calculated for three-dimensional objects. The volume is expressed in cubic centimeters (cm3) if the length or radius is measured in centimeters. The volume is expressed in cubic meters (m3) if the measurements are in meters.

When measuring the volume of liquids (for example, to determine how much water can be contained in a cylindrical bottle), we must convert values â€‹â€‹from cm3 or m3 to liters. Below is the formula for converting volume from liters to centimeters:

1 liter = 1000 cm3

1 liter = 1000 ml

1000 cm3 = 1000 ml

Hence,1 ml = 1 cm^3

**How do you Determine Volume?**

The following procedures must be followed to determine the volume of any solid shape:

- To find the parameter values for a given volume, you must substitute in the appropriate formula. For instance, you may determine the volume using the diameter, slope height, radius (r), and height (h) parameters.
- Each parameter must have the same number of units, which is crucial. Your data won't be correct if that happens.
- Substitute the values â€‹â€‹in the volume formula of the corresponding number.
- Finish your answer in writing cubic meters.

**Units of Volume**

Since volume is a measurement of the amount of three-dimensional space filled by a shape or surface, the cubic meter (m^3) is the S.I. unit of volume. However, the liter is the unit of volume that is most frequently used. Other units used to measure large and small amounts include milliliters (ml), pints, gallons, and others.

**Facts about Volume**

- Geometry's most basic notion, volume indicates how much three-dimensional space is filled by an object or region.
- Depending on the measurement system being used, the volume unit will vary. The cubic meter (m3) serves as the standard unit of volume in the metric system.
- Based on a solid's shape, multiple formulas can be used to determine its volume. For instance, V = s3 represents the volume of a cube, where s stands for the length of the cube's sides.
- A rectangular prism's volume is determined by multiplying its length, width, and height together: V = lwh.
- The volume of an object submerged in a fluid is equal to the volume of the fluid that the object has displaced, according to Archimedes' principle.
- V = πr2h, where r is the radius of the base and h is the height, equals the volume of a cylinder.
- The volume of a sphere is denoted as V = (4/3) πr3, where r is the radius.
- The formula V = (1/3) πr2h, where r is the radius of the base and h is the height, can be used to determine the volume of a cone.
- The cone’s volume is calculated as V = (1/3) πr²h, where r represents the radius of the bottom and h represents the height of the one.
- To locate the quantity of a pyramid, discover one-third of the base region and then multiply it by using the peak. This will decide the volume of the pyramid.
- Another way to discover the prism’s volume or pyramid is by dividing the area of the base by its height. This calculation will provide the quantity of the prism or pyramid.
- When handling more than one separate gadget that does not overlap, the whole quantity is obtained by summing up the individual volumes of these gadgets.
- To determine the extent of irregularly fashioned items, diverse strategies can be used, together with dissection or an approximate technique that includes breaking down the object into smaller recognized shapes and determining their volumes.
- Volume isn't always the simplest relevant to strong items however can also be used to explain abstract mathematical domains such as fractals and curved surfaces.
- By calculating the integration of a function that represents the cross-sectional area of a solid over a specified time interval, the volume can be determined using integral calculus.
- The volume of a torus, which is a doughnut-shaped object, can be calculated using the formula V = 2π²Rr², where R represents the distance from the center of the tube to the center of the torus, and r denotes the radius of the tube.
- V = 2π2Rr2, where R is the distance from the tube's center to the torus's center and r is the tube's radius, gives the volume of a torus (a doughnut-shaped object).
- A rectangular pyramid's volume is equal to one-sixth of the sum of its base and height.
- By using the scalar triple product of its three edge vectors, a parallelepiped—a three-dimensional shape with six parallelogram faces—can have its volume calculated.
- Combining the volumes of two cones allows one to determine the volume of a frustum, which is a segment of a cone with the top removed.
- You can calculate the volume of a composite solid, which consists of several different, distinct pieces, by adding the volumes of each one separately.
- Numerous disciplines, including architecture, engineering, physics, fluid dynamics, and computer graphics, have practical uses for the study of volume.

**What is Capacity?**

Have you ever noticed that there is a maximum amount of water that you can put into a bottle or a pan? The maximum amount for each drink and chocolate syrup is specified on the container. The capacity of a pan, bottle, or other container refers to the maximum amount it can hold. The maximum amount a container can hold is known in mathematics as capacity.

**Types Of Capacity Measurement**

- Metric measurement
- Customary measurement

**Metric Measurement**

Liters and milliliters are the accepted units for measuring capacity.

Milliliter (ml): The letters ml stand for milliliter. A milliliter is used to measure incredibly tiny amounts of liquids, such as cough syrup.

L: liter, the letter l stands for liter. A liter is a unit of measurement for liquids like milk, juice, and water.

Milliliter (ml) and liter (l), two units, are related:

A liter is equivalent to 1,000 milliliters.

1L = 1000 ml

**Customary Measurement**

Gallons, quarts, pints, cups, and fluid ounces are all examples of common measurements. English or customary units are used here; this is US Standard Unit.

**Fluid Ounce**: To measure the volume of liquids, one uses the fluid ounce unit. It is frequently shortened to fl. oz.- 16 tablespoons make up one
**cup**, the volume measuring unit. About 237 ml makes a US cup. Typically, it is employed for consuming beverages like coffee or tea. - Two cups make up one
**pint**. About 473 ml makes a US pint. We use pints to measure the juice jars' capacity. **Quart**: Two pints make into a quart (qt). The number of quarts represents the milk carton's volume.**Gallon**: The greatest unit of liquid measurement is the gallon (gal). It corresponds to 4 quarts. In gallons, fuel tank capacity is specified.

**Facts about Capacity**

- The utmost amount that a container, system, or other entity can accommodate is referred to as its capacity.
- It can be measured in several different ways, including liters, gallons, cubic meters, and weight units like kilograms and pounds.
- The highest output or production rate that a system, machine, or production line can produce in a specific amount of time is referred to as capacity in manufacturing and engineering.
- Identifying the production capacity that an organization needs to meet its demands and goals is the process of capacity planning.
- How much of the available capacity is being used or utilized is determined by capacity utilization.
- When the demand exceeds the available capacity or when capacity utilization falls below a predetermined level, overcapacity results.
- When the demand exceeds the capacity, it means that the system is operating below its full capacity, which can lead to bottlenecks in production or service delivery.
- In transportation, capacity refers back to the maximum wide variety of passengers or objects that an automobile, vessel, or infrastructure can delivery inside a selected length.
- 9. Storage capability, however, refers to the number of records that can be saved in specific garage mediums like hard drives, solid-state drives, or cloud storage solutions.
- 10. In computing, capability often refers back to the processing pace and computational capabilities of computer systems or additives like the RAM or CPU.
- Eleven. In human resources, capability pertains to the competencies, talents, and knowledge possessed using people or organizations to perform unique responsibilities or activities.
- 12. Effective capacity represents the best attainable potential underneath the best conditions, without considering external elements like maintenance, downtime, or different boundaries.
- Thirteen. Effective ability represents the best possible potential underneath ideal conditions, without considering outside factors like preservation, downtime, or other obstacles.
- Lastly, the concept of capacity is closely intertwined with scalability, which refers to the ability of an organization or system to handle increasing workloads or demands.
- The capacity of a battery or energy storage device determines the amount of energy it can store and subsequently release when needed.
- Capacity planning in project management entails determining the resources and expertise needed to finish a project and ensuring they are accessible when required.
- The idea of capacity is important in the field of healthcare as well because it describes how many patients can be housed or treated in a facility.
- Capacity utilization is a crucial economic indicator of the strength and effectiveness of a country's economy or industry.
- Investing in new infrastructure, streamlining processes, or outsourcing are just a few techniques that can be used to reduce capacity constraints, which can restrict the growth or expansion of enterprises.
- To ensure efficient resource allocation, locate bottlenecks, and make educated decisions about upcoming investments and upgrades, capacity must be continuously monitored and evaluated.

**Main Difference Between Volume and Capacity in Points**

In relation to the distinction between volume and capacity, the following details are crucial:

- Volume is the amount of space that matter occupies. The capacity of an object refers to the most amount of a substance it can hold.
- A substance's volume is a measurement of the amount of space it occupies. As opposed to this, an object's capacity is just the quantity of interior space that can be filled.
- The units used to measure volume are always cubic, such as the cubic centimeter and cubic meter. On the other hand, capacity is calculated using metric units such as milliliters, liters, gallons, pounds, etc.
- A solid thing just has volume, whereas a hollow object has both volume and capacity.

**Conclusion**

You may have learned after a thorough explanation of the two issues that volume refers to the amount of space occupied by the matter and capacity refers to the amount of space available for the matter to occupy.

When referring to containers, beakers, or any other hollow object, the word capacity is utilized. Furthermore, a container's volume can be altered by varying the amount of material it contains, while a container's capacity cannot be altered.