What is volume and how is it calculated? All formulas for volumes of geometric bodies

One of the most interesting problems of geometry, the result of which is important in physics, chemistry, and other areas, is the determination of volumes. Doing math at school, children often ask themselves the thought: “Why do we need this?” The world around seems so simple and clear that certain school knowledge classified as "unnecessary". But it is necessary to face, for example, transportation and the question arises of how to calculate the volume of cargo. You say that there is nothing easier? You are wrong. Knowledge of calculation formulas, the concepts of "substance density", "bulk density of bodies" become necessary.

School knowledge - practical basis

School teachers, teaching the basics of geometry, offer us the following definition of volume: the part of the space occupied by the body. At the same time, the formulas for determining volumes have long been written down, and you can find them in reference books. Mankind learned to determine the volume of a body of the correct form long before the appearance of the treatises of Archimedes. But only this great Greek thinker introduced a technique that makes it possible to determine the volume of any figure. His conclusions became the basis of integral calculus. Volumetric figures are considered to be obtained in the process of rotating flat

Euclidean geometry with a certain accuracy allows you to determine the volume:

The difference between flat and volumetric figures does not allow answering the question of some sufferers about how to calculate the volume of a rectangle. It's about the same as finding something, I don't know what. Confusion in the geometric material is possible, with a rectangle sometimes called cuboid.

What to do if the shape of the body is not so well defined?

Determining the volume of complex geometric structures is not an easy job. It is necessary to be guided by several unshakable principles.

• Any body can be broken down into simpler parts. Volume is equal to the sum volumes of its individual parts.
• Equal-sized bodies have equal volumes, the parallel transfer of bodies does not change its volume.
• The unit of volume is the volume of a cube with an edge of unit length.

The presence of irregularly shaped bodies (remember the notorious crown of King Heron) does not become a problem. Determining the volumes of bodies is quite possible. This is the process of directly measuring the volumes of a liquid with a body immersed in it, which will be discussed below.

Various applications for volume determination

Let's return to the problem: how to calculate the volume of transported goods. What is the cargo: packaged or bulk? What are the container parameters? There are more questions than answers. The issue of the mass of the cargo will become important, since the transport differs in carrying capacity, and the routes - in the maximum weight. vehicle. Violation of the rules of transportation threatens with penalties.

Task 1. Let the cargo be rectangular containers filled with goods. Knowing the weight of the goods and the container, you can easily determine the total weight. The volume of the container is defined as the volume of a rectangular parallelepiped.

Knowing the carrying capacity of the transport, its dimensions, it is possible to calculate the possible volume of the transported cargo. The correct ratio of these parameters allows you to avoid a catastrophe, premature failure of transport.

Task 2. Cargo - bulk material: sand, crushed stone and the like. At this stage, only a great specialist can do without knowledge of physics, whose experience in cargo transportation allows you to intuitively determine the maximum volume allowed for transportation.

The scientific method involves knowledge of such a parameter as the load.

The formula V=m/ρ is used, where m is the mass of the load, ρ is the density of the material. Before calculating the volume, it is worth knowing the density of the load, which is also not at all difficult (tables, laboratory definition).

This technique also works remarkably well in determining volumes of liquid cargoes. The liter is used as the unit of measure.

Determining the volume of building forms

The issue of determining volumes plays an important role in construction. The construction of houses and other structures is a costly business, building materials require careful attention and extremely accurate calculation.

The basis of the building - the foundation - is usually a cast structure filled with concrete. Before that, you need to determine the type of foundation.

The slab foundation is a slab in the form of a rectangular parallelepiped. Columnar base - rectangular or cylindrical pillars of a certain section. By determining the volume of one column and multiplying it by the quantity, it is possible to calculate the cubic capacity of concrete for the entire foundation.

When calculating the volume of concrete for walls or floors, they do it quite simply: they determine the volume of the entire wall, multiplying the length by the width and height, then separately determine the volumes of window and door openings. The difference between the volume of the wall and the total volume of openings is the volume of concrete.

How to determine the volume of a building?

Some applied tasks require knowledge about the volume of buildings and structures. These include problems of repair, reconstruction, determination of air humidity, issues related to heat supply and ventilation.

Before answering the question of how to calculate the volume of a building, measurements are taken on its outer side: the cross-sectional area (length multiplied by the width), the height of the building from the bottom of the first floor to the attic.

The determination of the internal volumes of heated premises is carried out by internal strokes.

The device of heating systems

Modern apartments and offices cannot be imagined without a heating system. The main part of the systems are batteries and connecting pipes. How to calculate the volume of the heating system? The total volume of all heating sections, which is indicated on the radiator itself, must be added to the volume of pipes.

And at this stage the problem arises: how to calculate the volume of the pipe. Imagine that the pipe is a cylinder, the solution comes by itself: we use the cylinder formula. In heating systems, pipes are filled with water, so it is necessary to know the area of ​​\u200b\u200bthe internal section of the pipe. To do this, we determine its inner radius (R). The formula for determining the area of ​​a circle: S=πR 2 . The total length of the pipes is determined by their length in the room.

Sewerage in the house - pipe system

When laying pipes for drainage, it is also worth knowing the volume of the pipe. At this stage, an outer diameter is needed, the steps are similar to the previous ones.

Determination of the volume of metal that goes to the manufacture of pipes - also interesting task. Geometrically, a pipe is a cylinder with voids. Determining the area of ​​a ring lying in its cross section is a rather complicated task, but it can be solved. A simpler way out is to determine the external and internal volumes of the pipe, the difference between these values ​​\u200b\u200band will be the volume of the metal.

Determination of volumes in problems of physics

The famous legend about the crown of King Heron became known not only as a result of solving the problem of bringing “to clean water» thieving jewelers. The result of the complex mental activity of Archimedes is the determination of the volumes of bodies of irregular geometric shape. The main idea extracted by the philosopher is that the volume of liquid displaced by the body is equal to the volume of the body.

AT laboratory research use a measuring cylinder (beaker). The volume of liquid is determined (V 1), the body is immersed in it, secondary measurements are performed (V 2). Volume is equal to the difference secondary and primary measurements: V t \u003d V 2 - V 1.

This method of determining the volumes of bodies is used in calculating the bulk density of bulk insoluble materials. It is extremely convenient in determining the density of alloys.

You can calculate the volume of a pin using this method. It seems difficult enough to determine the volume of such a small body as a pin or a pellet. It cannot be measured with a ruler, the measuring cylinder is also large enough.

But if you use several completely identical pins (n), then you can use a graduated cylinder to determine their total volume (V t \u003d V 2 - V 1). Then divide the resulting value by the number of pins. V= V t \n.

This task becomes clear if it is necessary to cast many pellets from one large piece of lead.

Liquid volume units

The international system of units assumes the measurement of volumes in m 3. In everyday life, off-system units are more often used: liter, milliliter. When it is determined how to calculate the volume in liters, the conversion system is used: 1 m 3 \u003d 1000 liters.

Use in Everyday life other off-system measures can cause difficulties. The British use barrels, gallons, bushels, which are more familiar to them.

Translation system:

Tasks with non-standard data

Task 1. How to calculate the volume, knowing the height and area? Typically, this problem is solved by determining the amount of coating of various parts by galvanization. The surface area of ​​the part (S) is known. Layer thickness (h) - height. Volume is determined by the product of area and height: V=Sh.

Problem 2. For cubes, the problem of determining the volume may look interesting, from a mathematical point of view, if the area of ​​\u200b\u200bone face is known. It is known that the volume of a cube is: V=a 3 , where a is the length of its face. The area of ​​the side surface of the cube S=a 2 . Extracting from the area, we get the length of the face of the cube. We use the volume formula, calculate its value.

Task 3. Calculate the volume of a figure if the area is known and some parameters are given. Additional parameters include conditions for the ratio of sides, heights, base diameters, and much more.

To solve specific problems, you will need not only knowledge of the formulas for calculating volumes, but also other geometry formulas.

Determining the amount of memory

A task completely unrelated to geometry: to determine the amount of memory of electronic devices. In the modern, fairly computerized world, this problem is not superfluous. Precise devices, such as personal computers, do not tolerate approximation.

Knowing the amount of memory on a flash drive or other storage device is useful when copying or moving information.

It is important to know the amount of RAM and permanent memory of the computer. Often the user is faced with a situation where “the game is not running”, “the program hangs”. The problem is quite possible with a low amount of memory.

A byte and its derivatives (kilobyte, megabyte, terabyte) are considered.

1 kB = 1024 B

1 MB = 1024 kB

1 GB = 1024 MB

The strangeness in this system of recalculation follows from binary system information encoding.

The memory size of a storage device is its main characteristic. Comparing the amount of information transferred and the amount of memory of the drive, you can determine the possibility of its further operation.

The concept of "volume" is so broad that it is possible to fully understand its versatility only by solving applied problems, interesting and exciting.

Measure all required distances in meters. The volume of many three-dimensional figures is easy to calculate using the appropriate formulas. However, all values ​​substituted into the formulas must be measured in meters. Thus, before substituting values ​​into the formula, make sure that they are all measured in meters, or that you have converted other units of measure to meters.

• 1 mm = 0.001 m
• 1 cm = 0.01 m
• 1 km = 1000 m

The video course "Get an A" includes all the topics you need to successful delivery USE in mathematics for 60-65 points. Completely all tasks 1-13 profile exam mathematics. Also suitable for passing the Basic USE in mathematics. If you want to pass the exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!

Preparation course for the exam for grades 10-11, as well as for teachers. Everything you need to solve part 1 of the exam in mathematics (the first 12 problems) and problem 13 (trigonometry). And this is more than 70 points on the Unified State Examination, and neither a hundred-point student nor a humanist can do without them.

All the necessary theory. Quick Ways solutions, traps and secrets of the exam. All relevant tasks of part 1 from the Bank of FIPI tasks have been analyzed. The course fully complies with the requirements of the USE-2018.

The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.

Hundreds of exam tasks. Text problems and probability theory. Simple and easy to remember problem solving algorithms. Geometry. Theory, reference material, analysis of all types of USE tasks. Stereometry. Cunning tricks for solving, useful cheat sheets, development of spatial imagination. Trigonometry from scratch - to task 13. Understanding instead of cramming. Visual explanation of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Base for solution challenging tasks 2 parts of the exam.

All measurements are in mm

H— Liquid level.

Y- Reservoir height.

L- The length of the container.

X- The tank is wide.

This program performs calculations of the volume of liquid in rectangular containers of various sizes, it will also help to calculate the surface area of ​​the tank, free and total volume.

As a result of the calculation, you will find out:

• The total area of ​​the tank;
• Lateral surface area;
• bottom area;
• Free volume;
• The amount of liquid;
• Capacity volume.

Technology for calculating the amount of liquid in tanks different shapes

When the container has an irregular geometric shape (for example, in the form of a pyramid, parallelepiped, rectangle, etc.), it is necessary first of all to measure the internal linear dimensions and only after that to make calculations.

The calculation of the volume of liquid in a small rectangular container can be manually performed as follows. It is necessary to fill the entire tank with liquid to the brim. Then the volume of water in this case will be equal to the volume of the tank. Next, carefully drain all the water into separate containers. For example, in a special reservoir of the correct geometric shape or a measuring cylinder. On the measuring scale you can visually determine the volume of your tank. To calculate the amount of liquid in a rectangular container, it is best for you to use our online program, which quickly and accurately perform all calculations.

If the tank big size, and it is impossible to manually measure the amount of liquid, then you can use the gas mass formula with a known molar mass. For example, the mass of nitrogen is M = 0.028 kg / mol. These calculations are possible when the tank can be closed tightly (hermetically). Now, using a thermometer, we measure the temperature inside the tank, and the internal pressure with a manometer. Temperature must be expressed in Kelvin and pressure in Pascals. The volume of internal gas can be calculated using the following formula (V=(m∙R∙T)/(M∙P)). That is, we multiply the mass of the gas (m) by its temperature (T) and the gas constant (R). Next, the result should be divided by the gas pressure (P) and molar mass(M). The volume will be expressed in m³.

How to calculate and find out the volume of an aquarium by size yourself

Aquariums are glass vessels that fill clean water up to a certain level. Many aquarium owners have repeatedly thought about how large their tank is, how to perform calculations. The easiest and most reliable method is to use a tape measure and measure all the necessary parameters that should be entered into the appropriate cells of our calculator, and you will immediately get the finished result.

However, there is another way to determine the volume of the aquarium, which is a longer process, using liter jar, gradually filling the entire container to the appropriate level.

The third method for calculating the volume of an aquarium is a special formula. We measure the depth of the tank, height and width in centimeters. For example, we got the following parameters: depth - 50 cm, height - 60 cm and width - 100 cm. According to these dimensions, the volume of the aquarium is calculated by the formula (V \u003d X * Y * H) or 100x50x60 \u003d 3,000,000 cm³. Next, we need to convert the result to liters. For this finished value multiply by 0.001. From here it follows - 0.001x3000000 centimeters, and we get, the volume of our tank will be 300 liters. We have calculated the total capacity of the tank, then we need to calculate the actual water level.

Each aquarium is filled much lower than its actual height, in order to avoid overflow of water, in order to cover it with a lid, taking into account the screed. For example, when our aquarium is 60 centimeters high, then the glued screeds will be located 3-5 centimeters lower. With our size of 60 centimeters, a little less than 10% of the volume of the container falls on 5 cm ties. From here we can calculate the real volume of 300 liters - 10% \u003d 270 liters.

Important! A few percent should be taken away, taking into account the volume of glasses, the size of the aquarium or any other container, we remove it from the outside (excluding the thickness of the glasses).

From here, the volume of our tank will be equal to 260 liters.

Shipping cost - important question which interests many of our customers. Majority transport companies prepares a price list for its services, taking into account the volume of cargo in cubic meters - in other words, the volume of the place that the packaged cargo will occupy in the transport compartment of the aircraft, sea ​​container, cargo truck or railway wagon.

Which delivery to choose - air, railway or car?

In order to navigate the prices for delivery and choose the most optimal mode of transport when ordering goods from China, you need to know the total volume of cargo in m 3 that you want to receive. A calculator on our website will help you calculate the volume, but in order to quickly get the desired result, you must use the following data:

• type of packaging (box or cylinder);
• basic packaging parameters - length, width and height (for boxes) or height and diameter (for cylinders);
• number of packages in pieces.

By measuring the main packaging parameters with a ruler, you can calculate the volume of a box or cylinder, and then calculate the volume of the entire consignment in cubic meters. The figures obtained will help you compare the prices for delivery by one or another transport and choose the appropriate option.

Why do you need a volume calculator?

One of the main qualities of a modern businessman is the ability to quickly accept important decisions and timely respond to changes in market trends. Our volume calculator helps you save time on calculations and get the numbers you need in just a few minutes.

Using the volume calculator is convenient and very simple: for calculations, enter the necessary numbers in the appropriate fields, and then feel free to click on the "Calculate" button. The volume calculator in m 3 gives the finished result, regardless of the units in which you entered the tare parameters - in centimeters or meters. The system automatically converts the data into the required format and displays the final result in cubic meters.

Knowing the volume of containers and the total volume of cargo, you can correctly choose the appropriate type of transport and place the goods in it as compactly as possible without overpaying for empty space. Use the online packaging volume calculator to quickly calculate the volume of a box or tubes, as well as the entire batch of goods. The second calculator will help you find out the estimated cost of shipping cargo from China various types transport, taking into account its volume in m 3.

How to calculate the volume of a box?

In order to calculate the volume of a box, it is necessary to measure its length, height and width. If you have a sample of packaging for your products, use a ruler to measure. Information about the parameters of the box can also be obtained from the supplier. There are two ways to calculate the volume of a box in cubic meters: using our online volume calculator in m 3 or using the formula yourself. Let's consider both options.

• In order for the volume calculator to correctly calculate the volume of the box, select the option "Box volume". Measure the box, focusing on the image next to the calculator (or copy the information about its parameters from the seller's website), and enter the numbers into the volume calculation calculator. You can also specify the number of boxes and shipping costs for cubic meter. Click on the "Calculate" button - in the table below you will see the final result: the volume of the box in m 3. If you provided data on the number of boxes and the cost of delivery for calculations, then the plate will also show the total volume of cargo and the estimated amount of delivery.
• You can independently calculate the volume of packaging using the formula that is studied in mathematics lessons at school: V = a * b * h. Here V is the volume, a is the length, b is the width and h is the height (note: all data obtained during measurements must be converted from centimeters to meters). Just multiply these numbers and you will get the required volume of the box in cubic meters.

How to calculate the volume of a cylinder (pipe)?

Your goods will be packed in a cylindrical container, and you want to know the volume of cargo? The calculator will easily cope with this task. For calculations, you will need parameters such as the height of the container and its diameter. Use a ruler to measure, as in the case of a box, or ask the supplier for packaging parameters. Next, our calculator for calculating the volume will be used:

• mark the type of container (cylinder/pipe);
• enter the packaging parameters in the appropriate lines;
• indicate the number of pipes (if you know it);
• click the "Calculate" button.

Done: the calculator calculated the volume of cargo in a second! The table with the results shows the volume in cubic meters of both one pipe and the total volume of your cargo (if the number of packages was indicated).

For independent calculations and consolidation of knowledge gained at school, use the formula V=π*r 2 *h. As we remember, V denotes volume, π is the number "pi" equal to 3.14, r 2 is the radius of the pipe squared, and h is its height. Multiplying all the numbers, you get the volume of the container of a cylindrical shape. Do not forget: after measuring the radius of the pipe and its height, convert centimeters to meters - and then you will get the correct result in m 3.

How to calculate the volume of cargo in different containers?

It's good when the entire load has the same dimensions - the online volume calculator solves such problems in a matter of seconds. And how to calculate the volume of cargo if it is packed in containers of various shapes - large and small boxes and cylinders?

There is nothing complicated here, the main thing is to know the exact parameters of each type of container and its quantity. Our volume calculator in m 3 will help you quickly calculate the volume of cargo packed in containers of the same shape and size, after which all you have to do is add up all the numbers and get the total volume of your cargo.

How to calculate shipping cost?

Knowing the total volume of cargo in cubic meters, you can easily navigate the cost of delivery from China by various means of transport. To do this, use the calculation results provided by our Volume Calculator. In a special form located under the calculator, enter the received numbers in the "Volume" field. Select the delivery option (by sea, by air, by road, etc.), enter the points of departure and destination, and fill in other fields, then click the "Calculate" button. The system will automatically calculate the cost of shipping your cargo for the selected shipping option.