How to find the volume of numbers. All formulas for volumes of geometric bodies

Measure all required distances in meters. The volume of many three-dimensional figures can be easily calculated using the appropriate formulas. However, all values ​​​​substituted into formulas must be measured in meters. Therefore, before plugging values ​​into the formula, make sure that they are all measured in meters, or that you have converted other units of measurement to meters.

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

Using an online calculator, you can correctly calculate the volume of a container such as a cylinder, barrel, tank, or the volume of liquid in any other horizontal cylindrical container.

Let's determine the amount of liquid in an incomplete cylindrical tank

All parameters are indicated in millimeters

L— Height of the barrel.

H— Liquid level.

D— Tank diameter.

Our online program will calculate the amount of liquid in the container, determine the surface area, free and total cubic capacity.

The determination of the main parameters of the cubic capacity of tanks (for example, a regular barrel or tank) should be made based on the geometric method for calculating the capacity of the cylinders. In contrast to methods for calibrating a container, where the volume is calculated in the form of real measurements of the amount of liquid using a measuring ruler (according to the readings of the meter rod).

V=S*L – formula for calculating the volume of a cylindrical tank, where:

L is body length.

S is the cross-sectional area of ​​the tank.

According to the results obtained, capacity calibration tables are created, which are also called calibration tables, which allow you to determine the weight of the liquid in the tank by specific gravity and volume. These parameters will depend on the filling level of the tank, which can be measured using a meter rod.

Our online calculator allows you to calculate the capacity of horizontal and vertical containers using a geometric formula. You can find out the useful capacity of the tank more accurately if you correctly determine all the main parameters that are listed above and are involved in the calculation.

How to correctly define master data

Determining the lengthL

Using a regular tape measure, you can measure the length L of a cylindrical tank with a non-flat bottom. To do this, you need to measure the distance between the intersecting lines of the bottom with the cylindrical body of the container. In the case of a horizontal tank with a flat bottom, then in order to determine the size L, it is enough to measure the length of the tank along the outside (from one edge of the tank to the other), and subtract the bottom thickness from the result obtained.

Determine the diameter D

The easiest way is to determine the diameter D of a cylindrical barrel. To do this, it is enough to use a tape measure to measure the distance between any two extreme points of the lid or edge.

If it is difficult to correctly calculate the diameter of the container, then in this case you can use the measurement of the circumference. To do this, use a regular tape measure to circle the entire tank around the circumference. To correctly calculate the circumference, two measurements are taken in each section of the tank. To do this, the surface being measured must be clean. Having found out the average circumference of our container - Lcr, we proceed to determining the diameter using the following formula:

This method is the simplest, since often measuring the diameter of a tank is accompanied by a number of difficulties associated with the accumulation of various types of equipment on the surface.

Important! It is best to measure the diameter in three different sections of the container, and then calculate the average value. Since often, these data can differ significantly.

Averaged values ​​after three measurements allow us to minimize the error in calculating the volume of a cylindrical tank. As a rule, used storage tanks undergo deformation during operation, may lose strength, and decrease in size, which leads to a decrease in the amount of liquid inside.

Determining the levelH

To determine the liquid level, in our case it is H, we need a meter rod. Using this measuring element, which is lowered to the bottom of the container, we can accurately determine the parameter H. But these calculations will be correct for tanks with a flat bottom.

As a result of calculating the online calculator, we get:

• Free volume in liters;
• Amount of liquid in liters;
• Volume of liquid in liters;
• Total tank area in m²;
• Bottom area in m²;
• Lateral surface area in m².

All values ​​are indicated in mm

H— Liquid level.

Y— The tank is tall.

L— Length of the container.

X— The reservoir is wide.

This program calculates the volume of liquid in rectangular containers of various sizes; it will also help calculate the surface area of ​​the reservoir, free and total volume.

Based on the results of the calculation, you will learn:

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

Technology for calculating the amount of liquid in tanks of various shapes

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

Calculating the volume of liquid in a small rectangular container can be done manually as follows. It is necessary to fill the entire tank with liquid to the brim. Then the volume of water in this case will become equal to the volume of the reservoir. Next, you should carefully drain all the water into separate containers. For example, in a special tank of the correct geometric shape or a measuring cylinder. Using 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 will quickly and accurately perform all calculations.

If the tank is large and it is impossible to measure the amount of liquid manually, then you can use the formula for the mass of a gas with a known molar mass. For example, the mass of nitrogen M = 0.028 kg/mol. These calculations are possible when the tank can be tightly closed (hermetically). Now, using a thermometer, we measure the temperature inside the tank, and the internal pressure with a pressure gauge. Temperature should 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 gas mass (m) by its temperature (T) and gas constant (R). Next, the result obtained should be divided into 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 are filled with clean water to a certain level. Many aquarium owners have repeatedly wondered how big their tank is and how the calculations can be done. The simplest and most reliable method is to use a tape measure and measure all the necessary parameters, which 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 an aquarium, which involves a longer process, using a 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=X*Y*H) or 100x50x60=3000000 cm³. Next, we need to convert the resulting result into liters. To do this, multiply the finished value by 0.001. From here it follows - 0.001x3000000 centimeters, and we get that the volume of our tank will be 300 liters. We have calculated the full capacity of the container, then we need to calculate the actual water level.

Each aquarium is filled significantly lower than its actual height, in order to avoid overflow of water, and closed with a lid taking into account the screed. For example, when our aquarium is 60 centimeters high, then the glued ties will be located 3-5 centimeters lower. With our size of 60 centimeters, slightly less than 10% of the volume of the container falls on 5-centimeter ties. From here we can calculate the actual volume of 300 liters - 10% = 270 liters.

Important! You should subtract a few percent, taking into account the volume of the glass; the dimensions of the aquarium or any other container are taken from the outside (without taking into account the thickness of the glass).

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

General review. Stereometry formulas!

Hello, dear friends! In this article I decided to make a general overview of the problems in stereometry that will be on Unified State Exam in Mathematics e. It must be said that the tasks from this group are quite varied, but not difficult. These are problems for finding geometric quantities: lengths, angles, areas, volumes.

Considered: cube, cuboid, prism, pyramid, compound polyhedron, cylinder, cone, ball. The sad fact is that some graduates do not even take on such problems during the exam itself, although more than 50% of them are solved simply, almost orally.

The rest require little effort, knowledge and special techniques. In future articles we will consider these tasks, don’t miss it, subscribe to blog updates.

To solve you need to know formulas for surface areas and volumes parallelepiped, pyramid, prism, cylinder, cone and sphere. There are no difficult problems, they are all solved in 2-3 steps, it is important to “see” what formula needs to be applied.

All the necessary formulas are presented below:

Ball or sphere. A spherical or spherical surface (sometimes simply a sphere) is the geometric locus of points in space equidistant from one point - the center of the ball.

Ball volume equal to the volume of a pyramid whose base has the same area as the surface of the ball, and the height is the radius of the ball

The volume of the sphere is one and a half times less than the volume of the cylinder circumscribed around it.

A circular cone can be obtained by rotating a right triangle around one of its legs, which is why a circular cone is also called a cone of revolution. See also Surface area of ​​a circular cone

Volume of a round cone equal to one third of the product of the base area S and the height H:

(H is the height of the cube edge)

A parallelepiped is a prism whose base is a parallelogram. Parallelepiped has six faces, and all of them are parallelograms. A parallelepiped whose four lateral faces are rectangles is called a straight parallelepiped. A right parallelepiped whose six faces are all rectangles is called rectangular.

Volume of a rectangular parallelepiped equal to the product of the area of ​​the base and the height:

(S is the area of ​​the base of the pyramid, h is the height of the pyramid)

A pyramid is a polyhedron, which has one face - the base of the pyramid - an arbitrary polygon, and the rest - side faces - triangles with a common vertex, called the top of the pyramid.

A section parallel to the base of the pyramid divides the pyramid into two parts. The part of the pyramid between its base and this section is a truncated pyramid.

Volume of a truncated pyramid equal to one third of the product of the height h (OS) by the sum of the areas of the upper base S1 (abcde), lower base of a truncated pyramid S2 (ABCDE) and the average proportional between them.

n - the number of sides of a regular polygon - the base of a regular pyramid
a - side of a regular polygon - base of a regular pyramid
h - height of a regular pyramid

A regular triangular pyramid is a polyhedron, which has one face - the base of the pyramid - a regular triangle, and the rest - the side faces - equal triangles with a common vertex. The height descends to the center of the base from the top.

Volume of a regular triangular pyramid equal to one third of the product of the area of ​​a regular triangle, which is the base S (ABC) to the height h (OS)

a - side of a regular triangle - base of a regular triangular pyramid
h - height of a regular triangular pyramid

Derivation of the formula for the volume of a tetrahedron

The volume of a tetrahedron is calculated using the classic formula for the volume of a pyramid. It is necessary to substitute the height of the tetrahedron and the area of ​​a regular (equilateral) triangle.

Volume of a tetrahedron- is equal to the fraction in the numerator of which the square root of two in the denominator is twelve, multiplied by the cube of the length of the edge of the tetrahedron

(h is the length of the side of the rhombus)

Circumference p is approximately three whole and one-seventh the length of the diameter of the circle. The exact ratio of a circle's circumference to its diameter is indicated by the Greek letter π

As a result, the perimeter of the circle or circumference is calculated using the formula

(r is the radius of the arc, n is the central angle of the arc in degrees.)

Scientists from different countries have worked for many years to create a unified system. For example, different countries had their own units for measuring distance: versts, feet, fathoms, miles. In the unified international system, distance is measured in meters. Mass is measured in kilograms instead of poods, pounds, and so on.

The cubic meter is a derivative, and this is also true for other units.

A cubic meter (m3) is a value equal to the volume of a cube with an edge length of 1 meter. Cubic meters are used to measure those physical bodies that are characterized by 3 measurement parameters:

• length;
• width;
• height.

To determine the volume of a body, you need to multiply all 3 parameters. To count smaller or larger objects, in addition to cubic meters (m 3), other units are used: cubic millimeters (mm 3), cubic centimeters (cm 3), cubic decimeters (dm 3), cubic kilometers (km 3), liters. Let's look at examples of calculating the volumes of bodies of different configurations.

Example 1. Find the volume of a box with a length of 2 m, a width of 4 m and a height of 3 m. The volume will be equal to: 2 m x 4 m x 3 m = 24 m 3

Example 2. Find the volume of a cylinder with a base diameter of 2 m and a height of 4 m. We calculate the area of ​​the circle, it is equal to πR 2. S = 3.14 x (1 m) 2 = 3.14 m 2. Find the volume: 3.14 m2 x 3m = 9.42 m3.

Example 3. Find the volume of a ball with a diameter of 3 m. To calculate the cubic meters in a ball, remember the formula.

V = 4/3πR 3. Substitute the given value and find the volume: 4/3 x 3.14 x (1.5 m) 3 = 14.13 m 3.

Corresponding cubic meter

To find the number of cubes in an irregularly shaped body, you need to divide it into components with the correct shape. Find their volumes and summarize the results obtained. Consider an object such as a tower with a cone-shaped roof.

We first find the cubic capacity of the working room, which has a cylindrical shape, then the cone-shaped roof using the above formulas. We add up the results obtained.

How to calculate the cubic capacity of materials?

To find out the volume of an edged board, you should take measurements of its three dimensions: length, width and thickness or height. We multiply the resulting values ​​and get the cubic capacity of one board. Then we multiply this volume by the number of boards in the pack.

There are 3 ways to calculate cubic capacity:

• batch;
• piece by piece;
• sampling.

Having chosen 1 calculation method, you must meet the following conditions:

• the front ends of the boards in the package must be aligned;
• the width of the package should not deviate from the specified length along the entire length;
• Laying boards overlapping is unacceptable;
• It is unacceptable to move the boards inside or outside the package by an amount greater than 100 mm.

From the side of the aligned ends, the height of the package h 1 is measured. Find the actual height h. It will be equal to h 1 - ab, where a is the number of spacers between the boards, b is the thickness of one spacer.

The width of the package is measured along the center line dividing the height in half. The permissible measurement error is ±10 mm.

Method 2 speaks for itself. Each board is measured, all volumes are calculated and then added up.

Method 3 is used for large quantities of wood. Its cubic capacity is calculated using average indicators taken for the entire batch.

The accuracy of calculating the cubic capacity of unedged lumber depends on the type of tree, its type and degree of processing. It often happens that among these boards there are also edged ones.

To facilitate the task of calculating volumes, specially designed tables - the so-called cubeturns - will help.

Methods for converting cubic meters to other cubic units

When calculating volumes, it is necessary to adhere to the same units of measurement. If the data is presented in other units, and the final result must be obtained in cubes, then it will be enough to do the conversion correctly.

If V is measured in mm 3, cm 3, dm 3, l, then we convert to m 3 accordingly:

• 1 m 3 = 1 mm 3 x x 0.000000001 = 1 mm 3 x 10 -9;
• 1 m 3 = 1 cm 3 x 0.000001 = 1 cm 3 x 10 -6;
• 1 m 3 = 1 dm 3 x 0.001 = 1 dm 3 x 10 -3. The same translation is used for liters, since 1 liter contains 1 dm 3.

To find cubes of a substance, knowing its mass, you need to find its density using a table or determine it manually. Dividing the given mass M (kg) by the density index P (kg/m3), we obtain V material (m3).

Knowledge to determine volumes is necessary for both specialists and ordinary people in everyday life.