Concrete mortars are used for the construction of the foundation, the erection of walls and the pouring of the floor. Before the start of activities, it is important to choose the design of the foundation, correctly calculate the total level of costs and determine the required amount of building materials. Knowing how to calculate the volume of concrete, you can determine the estimated cost of construction activities, accurately plan the duration of the concrete work and avoid unexpected costs. Let us dwell in detail on the methodology for performing calculations for various, as well as walls and floors.
Strip foundation scheme
What methods can calculate the volume of concrete
The execution of construction work is preceded by the development of the project. At this stage, the type of foundation base is determined, and the volume of concrete solution required for the construction of the base is calculated. At the design stage, the need for a solution for pouring monolithic walls and a concrete floor is calculated. Determination of the cubic capacity of the concrete mix necessary for the performance of work is carried out according to the volume of the concreted structures of the building.
Various methods are used to perform calculations.:
- manual. It is based on the calculation of the volumes of the foundation, main walls and floor. The calculation is made on a conventional calculator according to school formulas for calculating volume and does not take into account the concrete shrinkage coefficient. The value obtained differs slightly from the results of calculations using software tools;
- program. The initial data entered into the program about the type of foundation, its dimensions, design features and brand of concrete are promptly processed. As a result, a fairly accurate result is produced, which you can focus on when purchasing building materials for building a foundation base, building walls or pouring a floor.
Features when calculating the volume of concrete To obtain an accurate result, it is not enough to take into account only the internal size of the formwork. The second method is more accurate, since the online calculator takes into account all the data: the type of foundation, the section of the foundation base, the presence of a reinforcing cage and the grade of mortar.
Getting ready to determine the volume of concrete - how to calculate without errors
When preparing to perform calculations, it should be remembered that the need for a concrete mixture is determined in cubic meters, and not in kilograms, tons or liters. As a result of manual or software calculations, the volume of the binder solution will be determined, and not its mass. One of the main mistakes that novice developers make is to perform calculations before the type of foundation is determined.
The decision on the design of the foundation is made after the following works have been completed:
- the production of geodetic measures to determine the properties of the soil, the level of freezing and the location of aquifers;
- calculation of the load capacity of the base. It is determined on the basis of weight, design features of the structure and natural factors.
How to calculate the amount (volume) of concrete mix - type of foundation being built;
- foundation dimensions, its configuration;
- brand of mixture used for concreting;
- soil freezing depth.
The accuracy with which the volume of concrete is calculated depends on the data used for the calculation.
They are different for each type of foundation.:
- when calculating the tape base, its dimensions and shape are taken into account;
- for a columnar base, it is important to know the number of concrete columns and their dimensions;
- you can calculate the cube of concrete for a solid slab by its thickness and dimensions.
The accuracy of the result obtained depends on the completeness of the data used for the calculation.
How to calculate concrete in cubes for a foundation
For all types of foundations, the need for concrete is determined by a formula that takes into account the total volume of foundation structures being erected. In this case, the part of the foundation poured into the ground is also taken into account without fail. To perform calculations, one should be guided by the dimensions indicated in the project documentation.
- columnar;
- grillage.
Determining the need for concrete mortar for each type of foundation has its own characteristics.
How to calculate a cube of concrete for a tape base
strip foundation calculator The base of the tape type is quite popular. It is used for the construction of private houses, outbuildings and country houses. The structure is a solid concrete strip reinforced with steel bars. Monolithic tape follows the contour of the building, including internal partitions.
Table of composition and proportions of concrete mix The calculation of the volume of concrete for a monolithic strip foundation is carried out using a simple formula V = AxBxP. Let's decrypt it:
- V - the need for a concrete solution, expressed in cubic meters;
- A is the thickness of the foundation tape;
- B - height of the tape base, including the underground part;
- P is the perimeter of the formed tape contour.
Multiplying these parameters together, we calculate the total cubic capacity of the concrete solution.
Consider the calculation algorithm for a strip base with dimensions of 6x8 m, a thickness of 0.5 m and a height of 1.2 m. Perform the calculation according to the following algorithm:
- Calculate the perimeter by doubling the length of the sides 2x(6+8)=28m.
- Calculate the cross-sectional area by multiplying the thickness and height of the tape 0.5x1.2 \u003d 0.6 m 2.
- Determine the volume by multiplying the perimeter by the cross-sectional area 28x0.6 \u003d 16.8 m 3.
The result obtained has a small error due to the fact that reinforced concrete reinforcement and mixture shrinkage during vibration compaction are not taken into account.
Strip foundation scheme How to calculate a cube of concrete for a pile type foundation
The base in the form of concrete columns is one of the simplest. It consists of reinforced concrete supports evenly spaced along the contour of the building, including at the corners of the building, as well as at the intersection of internal partitions with walls. Part of the support elements is located in the ground and transfers the load from the mass of the structure to the soil. The calculation algorithm provides for determining the total need for concrete by multiplying the volume of individual columns by their number.
For calculations, use the formula - V \u003d Sxn, which is deciphered as follows:
- V is the amount of solution for pouring columns;
- S is the cross-sectional area of the support element;
- n is the total number of pile columns.
Using the example of the requirements of a project involving the installation of 40 piles with a diameter of 0.3 m and a total length of 1.8 m, we calculate the required amount of concrete:
- Calculate the area of the pile by multiplying the coefficient 3.14 by the square of the radius - 3.14x0.15x0.15 \u003d 0.07065 m 2.
- Calculate the volume of one support by multiplying its area by the length - 0.07065x1.8 \u003d 0.127 m 3.
- Determine the required amount of the mixture by multiplying the volume of one pile by the total number of supports 0.127x40 \u003d 5.08 m 3.
How to calculate the cube of concrete With a rectangular section of the supporting columns, to calculate the cross section, it is necessary to multiply the width and thickness of the element.
How to calculate concrete for a column base with a reinforced concrete grillage
To improve the strength characteristics of the columnar base, the protruding parts of the supports are combined with a reinforced concrete structure, which is called a grillage. It is made in the form of a solid reinforced concrete tape or slab, in which the column heads are concreted.
How to calculate the volume of concrete for the construction of a strip foundation and piles - Determine the cross-sectional area of the grillage by multiplying its thickness by the height;
- Calculate the volume of the grillage by multiplying the cross-sectional area by the length of the structure.
The obtained value corresponds to the need for a concrete mixture for concreting the grillage base.
We calculate the volume of concrete for the foundation in the form of a solid slab
The slab type base is used on difficult soils with a high moisture concentration. Buildings without a basement are erected on it. This design allows you to evenly distribute the load from the mass of the structure on the soil and provide increased rigidity and stability of the object under construction. The use of reinforcement allows you to increase the strength of the slab foundation. The structure is a reinforced concrete slab in the shape of a rectangular parallelepiped.
How to calculate the volume of concrete Explanation of designations:
- V is the volume of concrete composition for pouring the slab;
- S is the area of the slab base in cross section;
- L is the length of the foundation structure.
For a foundation 12 m long, 10 m wide and 0.5 m thick, consider the calculation algorithm:
- Determine the area by multiplying the width of the slab by its thickness 10x0.5 \u003d 5 m 2.
- Calculate the volume of the base by multiplying the length of the structure by the area 12x5 \u003d 60 m 3.
The resulting value corresponds to the need for concrete mix. If the slab foundation has a complex configuration, then it should be divided on the plan into simpler figures, and then the area and volume should be calculated for each.
How to correctly calculate the cube of concrete for the construction of walls
Calculator for calculating the amount of concrete for a strip foundation on the site For the construction of massive buildings, strong boxes are constructed from concrete reinforced with steel reinforcement. To determine the need for building materials, builders face the task of calculating the volume of concrete for such structures. To perform calculations, use the following formula - V \u003d (S-S1) x H.
Let us decipher the notation included in the formula:
- V - the amount of concrete mix for building walls;
- S is the total area of the wall surface;
- S1 - total area of window and door openings;
- H is the height of the concreted wall box.
When performing calculations, the total area of the openings is determined by summing up the individual openings. The calculation algorithm is reminiscent of determining the need for concrete for a slab base and can easily be done independently using a calculator.
How to calculate a cube of concrete for pouring a floor
How to calculate the volume of concrete for the floor To increase the load capacity of the floor and ensure its flatness, a concrete screed is performed. After the concrete has hardened, such a surface serves as the basis for laying floor coverings or ceramic tiles. To prevent cracking, the thickness of the formed concrete screed is 5–10 cm. This is due to the fact that the thinner material cracks during operation. It is important to correctly calculate the cube of concrete so that the formed screed is strong and has the thickness provided for by the project.
The formula for determining the amount of solution V \u003d Sxh is easily deciphered:
- V is the amount of poured material;
- S is the total area of the concreted screed;
- h is the thickness of the concrete base.
Let's figure out how to perform calculations for a room with dimensions of 6x8 m and a concrete base thickness of 0.06 m:
- Determine the area of the floor surface by multiplying the length and width of the room - 6x8 \u003d 48 m 2.
- Calculate the volume of poured concrete composition to form a screed by multiplying the area by the layer thickness - 48x0.06 = 2.88 m 3.
Guided by the above algorithm, you can easily determine the amount of concrete composition for concreting the floor. There are situations when the rough surface has a slope. In this case, the formed screed has a different thickness over the area of \u200b\u200bthe room. In this situation, you can use the average layer thickness, which reduces the accuracy of the calculations.
Conclusion - why you need to know how to calculate a cube of concrete
Being engaged in construction and planning to independently produce a concrete solution or purchase it at the enterprises of reinforced concrete products in the required quantity, it is important to know how to calculate the volume of concrete. This will allow you to predict the amount of upcoming expenses, purchase building materials in a timely manner, and complete the work on schedule. You can make calculations both manually on a calculator and using software tools. The main thing is to master the calculation methodology and use reliable data to determine the amount of concrete.
For simple bodies, the volume is a positive value, the numerical value of which has the following properties:
1. Equal bodies have equal volumes.
2. If the body is divided into parts that are simple bodies, then the volume of this body is equal to the sum of the volumes of its parts.
3. The volume of a cube whose edge is equal to a unit of length is equal to one.
If the cube referred to in the definition has an edge of 1 cm, then the volume is measured in cubic centimeters; if the edge of the cube is , then the volume is measured in cubic
meters; if the edge of the cube is 1 km, then the volume is measured in cubic kilometers, and so on.
Figure 181 shows a simple body - a quadrangular pyramid SABCD. The volume of this pyramid, based on property 2, is equal to the sum of the volumes of the pyramids SABC and SADC.
59. The volume of a parallelepiped, prism and pyramid.
The volume of a rectangular parallelepiped is found by the formula
where are the edges of a rectangular parallelepiped. Based on this formula, you can get a formula for the volume of a cube. The volume of a cube is found by the formula
where a is the edge of the cube.
It is sometimes said that the volume of a rectangular parallelepiped is equal to the product of its linear dimensions or the product of the area of its base and its height. The last statement is also true for any parallelepiped.
Figure 182 shows an inclined parallelepiped. Its volume is , where is the area of the base, and the height of the inclined parallelepiped.
You can derive a rule for finding the volume of any prism (including oblique).
The volume of a prism is equal to the product of the area of its base and its height;
In the case of a straight prism (Fig. 183), its height coincides with the side edge and the volume of the straight prism is equal to the product of the base area and the side edge.
The volume of any pyramid is found by the formula
where S is the area of the base, H is the height of the pyramid.
Figure 184 shows a regular tetrahedron SABC with edge a. Its volume is
Example. In an inclined parallelepiped, the base and side face are rectangles, the areas of which are respectively equal and the angle between their planes is 80°. One side face of the parallelepiped has area Find the volume of the parallelepiped.
Solution. Let the sides of the parallelepiped be rectangles. Then the edge AD is perpendicular to the face. Further calculations can be performed without finding the lengths of these segments. We have Multiplying these equalities term by term, we get whence
60. Volume of a cylinder and a cone.
The volume of any body is determined as follows. A given body has volume V if there exist simple bodies containing it and simple bodies contained in it with volumes arbitrarily little different from V.
Applying this definition to finding the volumes of a cylinder and a cone, one can prove theorems.
The volume of a cylinder is equal to the product of the area of the base and the height, i.e.
If the radius of the base of the cylinder is R, and the height is H, then the formula for its volume is:
The volume of a cone is equal to one third of the product of the area of the base and the height "i.e..
If the radius of the base of the cone is H, and the height is II, then its volume is found by the formula
The volume of a truncated cone can be found by the formula
where the radii of the bases, H is the height of the truncated cone. The volume of the truncated cone shown in Figure 185 is found by the formula
61. The general formula for the volumes of bodies of revolution.
The volume of the sphere and its parts. To derive the formula for the volume of a body of revolution, Cartesian coordinates in space are introduced, taking the axis of the body as the axis. The plane intersects the surface of the body along a line for which the x-axis is the axis of symmetry. Let the equation of that part of the line that is located above the x-axis (Fig. 186).
1. Calculation of the volume of the cube
a- side of cube
The formula for the volume of a cube, ( V ):
2. Find by the formula, the volume of a rectangular parallelepiped
a, b, c- sides of the parallelepiped
Still sometimes the side of the parallelepiped is called an edge.
The formula for the volume of a parallelepiped, ( V):
3. Formula for calculating the volume of a ball, sphere
R — ball radius
Using the formula, if the radius is given, you can find the volume of the ball, ( V):
4. How to calculate the volume of a cylinder?
h- cylinder height
r- base radius
Using the formula, find the volume of the cylinder, if known - its base radius and height, ( V):
5. How to find the volume of a cone?
R- base radius
H- cone height
The formula for the volume of a cone, if the radius and height are known ( V):
7. The formula for the volume of a truncated cone
r- top base radius
R- bottom base radius
h- cone height
The formula for the volume of a truncated cone, if known - the radius of the lower base, the radius of the upper base and the height of the cone ( V):
8. Volume of a regular tetrahedron
A regular tetrahedron is a pyramid in which all faces are equilateral triangles.
a- edge of a tetrahedron
The formula for calculating the volume of a regular tetrahedron ( V):
9. Volume of a regular quadrangular pyramid
A pyramid whose base is square and whose faces are equal, isosceles triangles, is called a regular quadrangular pyramid.
a- base side
h- the height of the pyramid
The formula for calculating the volume of a regular quadrangular pyramid, ( V):
10. Volume of a regular triangular pyramid
A pyramid whose base is an equilateral triangle and whose faces are equal, isosceles triangles, is called a regular triangular pyramid.
a- base side
h- the height of the pyramid
The formula for the volume of a regular triangular pyramid, if given - the height and side of the base ( V):
11. Find the volume of a regular pyramid
A pyramid at the base, which is a regular polygon and faces equal triangles, is called regular.
h- the height of the pyramid
a side of the base of the pyramid
n- the number of sides of the polygon at the base
The formula for the volume of a regular pyramid, knowing the height, side of the base and the number of these sides ( V):
All formulas for volumes of geometric bodies
Geometry, Algebra, Physics
Volume formulas
The volume of a geometric figure- a quantitative characteristic of the space occupied by a body or substance. In the simplest cases, the volume is measured by the number of unit cubes that fit in the body, that is, cubes with an edge equal to a unit of length. The volume of a body or the capacity of a vessel is determined by its shape and linear dimensions.
Cube volume formula
1) The volume of a cube is equal to the cube of its edge.
V- cube volume
H is the height of the edge of the cube
Pyramid volume formula
1) The volume of the pyramid is equal to one third of the product of the base area S (ABCD) and the height h (OS).
V- the volume of the pyramid
S- area of the base of the pyramid
h- the height of the pyramid
Cone volume formulas
1) The volume of a cone is equal to one third of the product of the area of the base and the height.
2) The volume of a cone is equal to one third of the product of pi (3.1415) times the square of the radius of the base times the height.
V is the volume of the cone
S is the area of the base of the cone
h- cone height
π - pi (3.1415)
r- cone radius
Cylinder volume formulas
1) The volume of a cylinder is equal to the product of the area of the base and the height.
2) The volume of the cylinder is equal to the product of the number pi (3.1415) times the square of the radius of the base and the height.
V- cylinder volume
S is the base area of the cylinder
h- cylinder height
π - pi (3.1415)
r is the radius of the cylinder
Sphere volume formula
1) The volume of a sphere is calculated using the formula below.
V- the volume of the ball
π - pi (3.1415)
R- ball radius
Volume formula for a tetrahedron
1) The volume of a tetrahedron is equal to a fraction in the numerator of which is the square root of two times the cube of the length of the edge of the tetrahedron, and in the denominator twelve.
Volume formulas
Volume formulas and online volume calculators
volume formula.
Volume Formula necessary to calculate the parameters and characteristics of a geometric figure.
figure volume is a quantitative characteristic of the space occupied by a body or substance. In the simplest cases, the volume is measured by the number of unit cubes that fit in the body, that is, cubes with an edge equal to a unit of length. The volume of a body or the capacity of a vessel is determined by its shape and linear dimensions.
Parallelepiped.
The volume of a cuboid is equal to the product of the area of the base and the height.
Cylinder.
The volume of a cylinder is equal to the product of the area of the base and the height.
The volume of a cylinder is equal to the product of pi (3.1415) times the square of the radius of the base times the height.
Pyramid.
The volume of the pyramid is equal to one third of the base area S (ABCDE) multiplied by the height h (OS).
Correct pyramid- this is a pyramid, at the base of which lies a regular polygon, and the height passes through the center of the inscribed circle to the base.
Regular triangular pyramid is a pyramid whose base is an equilateral triangle and whose faces are equal isosceles triangles.
Regular quadrangular pyramid It is a pyramid whose base is a square and whose faces are equal isosceles triangles.
Tetrahedron is a pyramid in which all faces are equilateral triangles.
Truncated pyramid.
The volume of the truncated pyramid is equal to one third of the product of the height h (OS) and the sum of the areas of the upper base S 1 (abcde), the lower base of the truncated pyramid S 2 (ABCDE) and the average proportional between them.
Calculating the volume of a cube is easy - you need to multiply the length, width and height. Since the length of the cube is equal to the width and equal to the height, the volume of the cube is s 3 .
Cone- this is a body in Euclidean space, obtained by the union of all rays emanating from one point (the vertex of the cone) and passing through a flat surface.
Frustum obtained by drawing a section parallel to the base of a cone.
V \u003d 1/3 πh (R 2 + Rr + r 2)
The volume of a sphere is one and a half times less than the volume of a cylinder circumscribed around it.
Prism.
The volume of a prism is equal to the product of the area of the base of the prism times the height.
Ball sector.
The volume of the spherical sector is equal to the volume of the pyramid, the base of which has the same area as the part of the spherical surface cut out by the sector, and the height is equal to the radius of the ball.
ball layer- this is the part of the ball enclosed between two secant parallel planes.
ball segment- this is the part of the ball cut off from it by some plane, called a spherical or spherical segment
Volume Formula
The formula for the volume of a cube, ball, pyramid, parallelogram, cylinder, tetrahedron, cone, prism and volumes of other geometric shapes.
In the course of solid geometry, one of the main questions is how to calculate the volume of a particular geometric body. It all starts with a simple box and ends with a ball.
In life, too, often have to deal with similar problems. For example, to calculate the volume of water that fits in a bucket or barrel.
Properties valid for the volume of each body
- This value is always a positive number.
- If the body can be divided into parts so that there are no intersections, then the total volume is equal to the sum of the volumes of the parts.
- Equal bodies have the same volumes.
- If the smaller body is completely placed in the larger one, then the volume of the first is less than the second.
General designations for all bodies
Each of them has edges and bases, heights are built in them. Therefore, such elements are identically designated for them. That is how they are written in the formulas. How to calculate the volume of each of the bodies - we will learn further and apply new skills in practice.
Some formulas have other values. Their designation will be discussed when the need arises.
Prism, box (straight and oblique) and cube
These bodies are combined because they are very similar in appearance, and the formulas for how to calculate the volume are identical:
V = S * h.
Only S will differ. In the case of a parallelepiped, it is calculated as for a rectangle or square. In a prism, the base can be a triangle, a parallelogram, an arbitrary quadrilateral, or another polygon.
For a cube, the formula is greatly simplified because all its dimensions are equal:
V = a 3 .
Pyramid, tetrahedron, truncated pyramid
For the first of these bodies, there is such a formula to calculate the volume:
V \u003d 1/3 * S * n.
The tetrahedron is a special case of the triangular pyramid. All edges are equal in it. Therefore, again, a simplified formula is obtained:
V = (а 3 * √2) / 12, or V = 1/ 3 S h
The pyramid becomes truncated when its upper part is cut off. Therefore, its volume is equal to the difference between the two pyramids: the one that would be intact, and the remote top. If it is possible to find out both bases of such a pyramid (S 1 - more and S 2 - less), then it is convenient to use this formula to calculate the volume:
Cylinder, cone and truncated cone
V \u003d π * r 2 * h.
The situation with the cone is somewhat more complicated. There is a formula for it:
V = 1/3 π * r 2 * h. It is very similar to the one indicated for the cylinder, only the value is reduced by a factor of three.
Just as with a truncated pyramid, the situation is not easy with a cone that has two bases. The formula for calculating the volume of a truncated cone looks like this:
V \u003d 1/3 π * h * (r 1 2 + r 1 r 2 + r 2 2). Here r 1 is the radius of the lower base, r 2 is the upper (smaller).
Ball, ball segments and sector
These are the most difficult formulas to remember. For the volume of the ball, it looks like this:
V = 4/3 π *r 3 .
In tasks, there is often a question of how to calculate the volume of a spherical segment - a part of a sphere that is, as it were, cut parallel to the diameter. In this case, the following formula will come to the rescue:
V \u003d π h 2 * (r - h / 3). In it, h is taken as the height of the segment, that is, the part that goes along the radius of the ball.
The sector is divided into two parts: a cone and a spherical segment. Therefore, its volume is defined as the sum of these bodies. The formula after transformation looks like this:
V = 2/3 pr 2 * h. Here h is also the height of the segment.
Task examples
About the volumes of a cylinder, a ball and a cone
Condition: the diameter of the cylinder (1 body) is equal to its height, the diameter of the ball (2 body) and the height of the cone (3 body), check the proportionality of the volumes V 1: V 2: V 3 = 3:2:1
Solution. First you need to write down three formulas for volumes. Then take into account that the radius is half the diameter. That is, the height will be equal to two radii: h = 2r. After making a simple substitution, it turns out that the formulas for volumes will look like this:
V 1 \u003d 2 π r 3, V 3 \u003d 2/3 π r 3. The formula for the volume of a sphere does not change because it does not include the height.
Now it remains to write down the volume ratios and make the reduction 2π and r 3 . It turns out that V 1: V 2: V 3 \u003d 1: 2/3: 1/3. These numbers can easily be written as 3:2:1.
About the volume of the ball
Condition: there are two watermelons with radii of 15 and 20 cm, which is more profitable to eat: the first four or the second eight?
Solution. To answer this question, you need to find the ratio of the volumes of parts that will come from each watermelon. Given that they are balls, two formulas for volumes need to be written down. Then take into account that from the first one everyone will get only a fourth part, and from the second - an eighth.
It remains to write the ratio of the volumes of the parts. It will look like this:
(V 1: 4) / (V 2: 8) = (1/3 π r 1 3) / (1/6 π r 2 3). After the transformation, only the fraction remains: (2 r 1 3) / r 2 3 . After substituting the values and calculating, the fraction 6750/8000 is obtained. From it it is clear that the part from the first watermelon will be less than from the second.
Answer. It is more profitable to eat an eighth of a watermelon with a radius of 20 cm.
About the volumes of the pyramid and the cube
Condition: there is a clay pyramid with a rectangular base 8x9 cm and a height of 9 cm, a cube was made from the same piece of clay, what is its edge?
Solution. If we designate the sides of the rectangle with the letters b and c, then the area of \u200b\u200bthe base of the pyramid is calculated as their product. Then the formula for its volume is:
The formula for the volume of a cube is written in the article above. These two values are equal: V 1 = V 2 . It remains to equate the right parts of the formulas and make the necessary calculations. It turns out that the edge of the cube will be equal to 6 cm.
About the volume of a parallelepiped
Condition: it is required to make a box with a capacity of 0.96 m 3, its width and length are known - 1.2 and 0.8 meters, what should be its height?
Solution. Since the base of the parallelepiped is a rectangle, its area is defined as the product of the length (a) and the width (b). Therefore, the formula for volume looks like this:
From it it is easy to determine the height by dividing the volume by the area. It turns out that the height should be equal to 1 m.
Answer. The height of the box is one meter.
How to calculate the volume of various geometric bodies?
In the course of solid geometry, one of the main tasks is how to calculate the volume of a particular geometric body. It all starts with a simple box and ends with a ball.
Instruction
If a student is trying to calculate the volume of a rectangle, then specify: we are talking about a specific figure - or its volumetric analogue, rectangular. Find out also: what exactly is required to be found according to the conditions of the problem - volume, or length. In addition, find out: what part of the figure in question is meant - the whole figure, face, edge, vertex, side or.
To calculate the volume of a rectangular, multiply its length, width and height () together. That is, use the formula:
where: a, b and c are the length, width and height of the parallelepiped (respectively), and V is its volume.
First bring all the lengths of the sides to one unit of measurement, then the volume of the parallelepiped will turn out in the corresponding "cubic" units.
What will be the capacity of a water tank having dimensions:
length - 2 meters;
width - 1 meter 50 centimeters;
height - 200 centimeters.
1. We give the lengths of the sides to meters: 2; 1.5; 2.
2. We multiply the received numbers: 2 * 1.5 * 2 = 6 (cubic).
If the problem is still about a rectangle, then you probably need to calculate its area. To do this, simply multiply the length of the rectangle by its width. That is, apply the formula:
where:
a and b are the lengths of the sides of the rectangle,
S is the area of the rectangle.
Use the same formula if the problem has a cuboid face - according to the definition, it also has the shape of a rectangle.
The volume of the cube is 27 m³. What is the area of the rectangle formed by the side of the cube?
A parallelepiped is called inclined, the side faces of which are not perpendicular to the faces of the base. In this case, the volume is equal to the product of the base area and the height - V=Sh. Inclined height parallelepiped- a perpendicular segment, lowered from any upper vertex to the corresponding side of the base of the face (that is, the height of any side face).
A cube is a right parallelepiped in which all edges are equal, and all six faces are. The volume is equal to the product of the base area and the height - V=Sh. The base is a square whose base area is equal to the product of its two sides, that is, the size of the side in. The height of the cube is the same value, so in this case the volume will be the value of the edge of the cube raised to the third - V=a³.
note
The bases of a parallelepiped are always parallel to each other, this follows from the definition of a prism.
Useful advice
The dimensions of a box are the lengths of its edges.
The volume is always equal to the product of the area of the base and the height of the parallelepiped.
The volume of an inclined parallelepiped can be calculated as the product of the size of the side edge and the area of the section perpendicular to it.
To calculate the volume of any body, you need to know its linear dimensions. This applies to figures such as prism, pyramid, sphere, cylinder and cone. Each of these figures has its own definition of volume.
You will need
- - ruler;
- - knowledge of the properties of three-dimensional figures;
- - formulas for the area of a polygon.
Instruction
For example, in order to find the volume, the base of which is a right-angled triangle with legs 4 and 3 cm, and a height of 7 cm, perform the following calculations:
calculate the area of the rectangle that is the base of the prism. To do this, multiply the lengths of the legs, and divide the result by 2. Sbase = 3∙4/2=6 cm²;
multiply the base area by the height, this will be the volume of the prism V=6∙7=42 cm³.
To calculate the volume of a pyramid, find the products of the area of its base and the height, and multiply the result by 1/3 V=1/3∙Sbase∙H. The height of a pyramid is a segment lowered from its top to the base plane. The most common are the so-called regular pyramids, the top is projected into the center of the base, which is the right one.
For example, in order to find the volume of a pyramid based on a regular hexagon with a side of 2 cm and a height of 5 cm, do the following:
using the formula S \u003d (n / 4) a² ctg (180º / n), where n are the sides of a regular polygon, and is the length of one of the sides, find the area of \u200b\u200bthe base. S=(6/4) 2² ctg(180º/6)≈10.4 cm²;
calculate the volume of the pyramid using the formula V=1/3∙Sbase∙H=1/3∙10.4∙5≈17.33 cm³.
Find the volume in the same way as prisms, through the product of the area of one of the bases and its height V = Son ∙ H. When calculating, keep in mind that the base of the cylinder is a circle, the area of \u200b\u200bwhich is equal to Son \u003d 2 ∙ π ∙ R², where π ≈ 3.14, and R is the radius of the circle, which is the base of the cylinder.
Find the volume of the cone by analogy with the pyramid using the formula V=1/3∙Sbase∙H. The base of the cone is a circle, the area of which is found as described for the cylinder.
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A ball is called the simplest three-dimensional figure of a geometrically regular shape, all points of space inside the boundaries of which are removed from its center by a distance not exceeding the radius. The surface formed by the set of points as far as possible from the center is called a sphere. To quantify the measure of the space contained within the sphere, a parameter is intended, which is called the volume of the ball.
Instruction
If it is required to measure the volume of the ball not theoretically, but only with improvised means, then this can be done, for example, by determining the volume of water displaced by it. This method is applicable in the case when it is possible to place the ball in any container commensurate with it - a beaker, a glass, a jar, a bucket, a barrel, a pool, etc. In this case, before placing the ball, mark the water level, do this again after it is completely immersed, and then find the difference between the marks. Usually, a factory-made measuring container has divisions showing the volume in liters and units derived from it -, etc. If the value obtained is necessary in and multiple units of volume, then proceed from the fact that one liter corresponds to one cubic decimeter or one thousandth of a cubic meter.
If it is known from which the ball is made, and the density of this material can be found, for example, from a reference book, then the volume can be determined by weighing this object. Simply divide the weighing result by the manufacturing reference density: V=m/p.
If the radius of the ball is known from the conditions of the problem or it can be measured, then the corresponding mathematical formula can be used to calculate the volume. Multiply quadruple Pi by the third power of the radius, and divide the result by three: V=4*π*r³/3. For example, with a radius of 40 cm, the volume of the ball will be 4 * 3.14 * 40³ / 3 = 267946.67 cm³ ≈ 0.268 m³.
Diameter is often easier to measure than radius. In this case, there is no need to halve it for use with the formula from the previous step - the formula itself is better. In accordance with the converted formula, multiply the number Pi by the diameter to the third power, and divide the result by six: V=π*d³/6. For example, 50cm should have a volume of 3.14*50³/6 = 65416.67cm³ ≈ 0.654m³.
Due to some circumstances, it may be necessary to make a rectangular sheet from a sheet square, for example, during the manufacture of many origami paper crafts. But not always at hand there is a pencil and a ruler. However, there are ways in which you can get square with nothing but ingenuity.
You will need
- - rectangle;
- - ruler;
- - pencil;
- - scissors.
Instruction
A rectangle is a geometric figure in which all four corners are right and pairs of sides are parallel to each other. opposite sides rectangle in length between themselves, and between pairs - different. The square differs from the previous figure only in that all four sides are the same.
To square from rectangle You can also use a pencil. For example, sides rectangle are 30 cm (length) and 20 cm (width). Then square will have sides with a smaller value, i.e. 20 cm. Measure on the top long side rectangle 20 cm. Do the same, but only with the bottom side. Connect the dots with a ruler. If necessary, cut off the excess, resulting in square with sides 20 cm.
Do square from rectangle possible even if there are no drawing accessories. Lay in front of you and bend one of its right angles (it can be any angle) strictly in half. If you put the resulting figure on the long side, then there will be a rectangular trapezoid, visually consisting of a triangle and another rectangle. Bend the resulting rectangle into a triangle (it will be double due to the folded), smooth with your fingers and cut or carefully tear it off. Unfold the paper, which will be square. From the little left rectangle you can get it again square, only smaller. It is permissible to use the same methods.
The rectangle may have slightly different
General review. Formulas of stereometry!
Hello dear friends! In this article, I decided to make a general overview of the problems in stereometry, which will be USE in mathematics e. It must be said that the tasks from this group are quite diverse, but not difficult. These are tasks for finding geometric quantities: lengths, angles, areas, volumes.
Considered: a cube, a rectangular parallelepiped, a prism, a pyramid, a compound polyhedron, a cylinder, a cone, a ball. It is sad that some graduates do not even take on such tasks at the exam itself, although more than 50% of them are solved elementarily, almost verbally.
The rest require little effort, knowledge and special techniques. In future articles, we will consider these tasks, do not miss it, subscribe to the blog update.
To solve, you need to know surface area and volume formulas parallelepiped, pyramid, prism, cylinder, cone and sphere. There are no complex tasks, they are all solved in 2-3 steps, it is important to "see" what formula needs to be applied.
All necessary formulas are presented below:
Ball or sphere. A spherical or spherical surface (sometimes simply a sphere) is the locus of points in space that are equidistant from one point - the center of the ball.
Ball volume equal to the volume of the pyramid, the base of which has the same area as the surface of the ball, and the height is the radius of the ball
The volume of a sphere is one and a half times less than the volume of a cylinder circumscribed around it.
A round cone can be obtained by rotating a right triangle around one of its legs, so a round cone is also called a cone of revolution. See also Surface area of a circular cone
Volume of a round cone is equal to one third of the product of the base area S and the height H:
(H - cube edge height)
A parallelepiped is a prism whose base is a parallelogram. The parallelepiped has six faces, and all of them are parallelograms. A parallelepiped whose four lateral faces are rectangles is called a right parallelepiped. A right box in which all six faces are rectangles is called a rectangular box.
Volume of a cuboid is 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 with 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 is equal to one third of the product of the height h (OS) by the sum of the areas of the upper base S1 (abcde), the lower base of the truncated pyramid S2 (ABCD) and the average proportional between them.
| 1. | V= |
n - the number of sides of a regular polygon - the bases of a regular pyramid
a - side of regular polygon - bases of regular pyramid
h - the height of the regular pyramid
A regular triangular pyramid is a polyhedron with one face - the base of the pyramid - a regular triangle, and the rest - 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 is equal to one third of the product of the area of an equilateral triangle, which is the base S (ABC) to the height h (OS)
a - side of a regular triangle - bases of a regular triangular pyramid
h - the 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 classical 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 into it.
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 about three whole and one seventh the length of the diameter of a circle. The exact ratio of the circumference of a circle to its diameter is denoted by the Greek letter π
As a result, the perimeter of a circle or the circumference of a circle is calculated by the formula
| π rn |
(r is the radius of the arc, n is the central angle of the arc in degrees.)
