Education, Secondary education and schools
How to calculate the volume - calculation formulas
One of the most interesting problems of geometry, the result of solving which is important in physics, in chemistry, and in other fields is the definition of volumes. Being engaged in mathematics at school, children are often asked by the thought: "Why do we need this?" The world around seems so simple and understandable that certain school knowledge is classed as "unnecessary". But it is worth to face, for example, with transportation and the question arises as to how to calculate the volume of cargo. Will you say that nothing is easier? You are mistaken. Knowledge of computational formulas, concepts of "matter density", "bulk density of bodies" become necessary.
School knowledge is the practical basis
Teachers of schools, teaching the basics of geometry, offer us such a definition of volume: part of the space occupied by the body. In this case, the formulas for the determination of volumes have long been recorded, and they can be found in reference books. To determine the volume of the body of the correct form, mankind has learned 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 reasoning became the basis of integral calculus. The figures obtained during the rotation of plane geometric figures are considered to be three-dimensional .
Euclidean geometry with a certain accuracy makes it possible to determine the volume:
Geometrical body | Calculation formula | Main settings |
Rectangular parallelepiped | V = lbh | L - length, b - width, h - height |
Cube | V = a 3 | A is the edge of the cube |
Cylinder | V = Sh | S - base area, h - height |
Sphere | V = 4πR 3/3 | R is the radius of the sphere |
What should I do if the body shape is not so clearly defined?
Determining the volume of complex geometric constructions is not an easy task. It is necessary to be guided by several unshakable principles.
- Any body can be broken down into simpler parts. The volume is equal to the sum of the volumes of its individual parts.
- Equal bodies have equal volumes, parallel transfer of bodies does not change its volume.
- The unit volume is the volume of the 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 volume of bodies by hydrostatic weighing is quite possible. This is a process of direct measurement of fluid volumes with a body immersed in it, which will be considered below.
Various applications for determining the volume
Let's return to the problem: how to calculate the volume of transported goods. What is the load: packed or loose? What are the packaging parameters? There are more questions than answers. Important is the issue of the mass of the cargo, because the transport is different in carrying capacity, and the route is the maximum weight of the vehicle. Infringement of rules of transportation threatens with penal sanctions.
Knowing the carrying capacity of transport, its dimensions, it is possible to calculate the possible volume of transported cargo. The correct ratio of these parameters allows to avoid catastrophe, premature failure of transport.
Task 2. Cargo - loose material: sand, gravel and the like. At this stage, only a class specialist can do without the knowledge of physics, whose experience in cargo transportation allows you to intuitively determine the maximum allowable volume for transportation.
The scientific method presupposes the knowledge of such a parameter as the density (bulk density) of the cargo.
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 to know the density of the cargo, which is also not at all difficult (tables, laboratory definition).
This technique also works remarkably in determining the volumes of liquid cargo. In this case, a liter is used as a unit of measurement.
Determination of the volume of building forms
The issue of determining volumes plays an important role in construction. The erection of houses and other structures is a costly business, building materials require attentive attitude and extremely precise calculation.
The foundation of the building - the foundation - is usually a cast construction filled with concrete. Before calculating the volume of concrete, it is necessary to determine the type of foundation.
Plate foundation - plate in the form of a rectangular parallelepiped. The columnar base is rectangular or cylindrical columns of a certain section. Having determined the volume of one column and multiplying it by the number, you can calculate the cubature of the concrete on the whole foundation.
Calculating the volume of concrete for walls or ceilings, it is quite simple: determine the volume of the entire wall, multiplying the length by the width and height, then separately determine the volume of window and door openings. The difference in the volume of the wall and the total volume of the openings is the volume of concrete.
How to determine the volume of a building?
Some applied tasks require knowledge of the volume of buildings and structures. These include the 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, make measurements on its outer side: the area of the section (the length is multiplied by the width), the height of the building from the bottom of the first floor to the attic.
Determination of internal volumes of heated rooms is carried out by internal strokes.
The device of heating systems
Modern apartments and offices can not 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 folded with 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 naturally: we use the formula for calculating the volume of a cylinder. In heating systems, the pipes are filled with water, so it is necessary to know the area of the internal section of the pipe. To do this, determine its inner radius (R). 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.
Sewer in the house - pipe system
When laying pipes for water disposal, it is also worth knowing the volume of the pipe. At this stage, an external diameter is required, the actions are similar to the previous ones.
Determining the amount of metal that goes into making pipes is also an interesting task. Geometrically the pipe is a cylinder with voids. To determine the area of the ring lying in its cross section is a rather complicated problem, but it can be solved. A simpler solution is to determine the outer and inner volumes of the tube, the difference between these quantities and will be the volume of the metal.
The definition of volumes in problems of physics
The famous legend of the crown of King Heron became known not only because of the solution of the task of removing "thieving jewelers" into clean water. The result of the complex intellectual activity of Archimedes is the determination of the volumes of bodies of irregular geometric shapes. The main idea, extracted by the philosopher - the volume of fluid displaced by the body is equal to the volume of the body.
In laboratory studies use a measuring cylinder (beaker). Determine the volume of the liquid (V 1 ), immerse the body in it, perform secondary measurements (V 2 ). The volume is equal to the difference between the secondary and primary measurements: V m = V 2 - V 1 .
This method of determining the volume of bodies is used to calculate the bulk density of free-flowing insoluble materials. It is extremely useful in determining the density of alloys.
The pin size can be calculated using this method. It seems difficult enough to determine the volume of such a small body as a pin or a pellet. The ruler does not measure it, the measuring cylinder is also large enough.
But if you use several exactly the same pins (n), then you can use the graduated cylinder to determine their total volume (V m = V 2 - V 1) . Then the resulting value divided by the number of pins. V = V m \ n.
This task becomes understandable if from a single large piece of lead it is necessary to cast a lot of pellets.
Units of measurement of liquid volume
The use of other off-system measures in everyday life can cause difficulties. The British use more common for them barrels, gallons, bushels.
Translation system:
English measures | Russian measures | ||
Bushel | 36.4 l | Bucket | 12 l |
Gallon | 4,5 l | Barrel | 490 l |
Barrel (dry) | 115.628 l | Shtof | 1, 23 L |
Barrel (oil) | 158, 988 L | Charka | 0, 123 l |
English barrel for bulk solids | 163.65 l | Shkolik | 0.06 L |
Tasks with non-standard data
Task 1. How to calculate the volume, knowing the height and area? Usually such a problem is solved by determining the volume of coating of various parts by galvanic means. The surface area of the part (S) is known. The thickness of the layer (h) is the height. The volume is determined by the product of area and height: V = Sh.
Problem 2. For cubes, an interesting problem, from the mathematical point of view, may be the problem of determining the volume if the area of one face is known. It is known that the volume of the cube: V = a 3 , where a is the length of its face. The area of the lateral surface of the cube is S = a 2 . Extracting the square root from the area, we get the length of the face of the cube. We use the formula of volume, we calculate its value.
Task 3. Calculate the volume of the figure, if the area is known and some parameters are given. The additional parameters include the conditions of the ratio of sides, heights, diameters of the base and much more.
To solve specific problems, you need not only the knowledge of formulas for calculating volumes, but also other formulas of geometry.
Determining the amount of memory
Completely unrelated to geometry: to determine the amount of memory of electronic devices. In a modern, sufficiently computerized world, this problem is not superfluous. Precise devices, such as personal computers, do not tolerate approximation.
Knowledge of the memory capacity of a flash drive or other drive is useful when copying, moving information.
It is important to know the amount of operative and permanent memory of the computer. Often the user is faced with a situation where "the game is not going", "the program is suspended". The problem is quite possible with a low memory capacity.
The unit of measurement of information is the byte and its derivatives (kilobyte, megabyte, terabyte).
1 kB = 1024 B
1 MB = 1024 KB
1 GB = 1024 MB
The oddity in this allocation system follows from the binary coding system of information.
The size of the memory of the storage device is its main characteristic. Comparing the amount of information transferred and the storage capacity of the drive, you can determine the possibility of its further operation.
The concept of "volume" is so large that one can fully understand its versatility by solving applied problems, interesting and fascinating.
Similar articles
Trending Now