Rectangle- one of the simplest flat figures, and a rectangular parallelepiped is the same simple figure, but in space (Fig. 1). They are very similar.
As similar as a circle and a ball.
Rice. 1. Rectangle and parallelepiped
A conversation about areas begins with the area of a rectangle, and about volumes - with volume rectangular parallelepiped.
If we know how to find the area of a rectangle, then this allows us to find the area of any figure.
We can divide this figure into 3 rectangles and find the area of each, and therefore the entire figure. (Fig. 2.)
Rice. 2. Figure
Rice. 3. A figure whose area is equal to seven rectangles
Even if the figure is not divided exactly into rectangles, this can be done with any accuracy and the area can be calculated approximately.
The area of this figure (Fig. 3) is approximately equal to the sum of the areas of seven rectangles. The inaccuracy is due to the upper small figures. If you increase the number of rectangles, the inaccuracy will decrease.
That is rectangle is a tool for calculating the areas of any shapes.
The same situation when we're talking about about volumes.
Any figure can be laid out with rectangular parallelepipeds or bricks. The smaller these bricks are, the more accurately we can calculate the volume (Fig. 4, Fig. 5).
Rice. 4. Calculating area using cuboids
A rectangular parallelepiped is a tool for calculating the volumes of any shapes.
Rice. 5. Calculating area using small parallelepipeds
Let's remember a little.
A square with a side of 1 unit (Fig. 6) has an area of 1 square unit. The original linear unit can be any: centimeter, meter, kilometer, mile.
For example, 1 cm2 is the area of a square with a side of 1 cm.
Rice. 6. Square and rectangle
Area of a rectangle- this is the number of such squares that will fit into it. (Fig. 6.)
Let's put it down unit squares the length of a rectangle in one row. It turned out to be 5 pieces.
The height fits 3 squares. This means that there are three rows in total, each with five squares.
The total area is .
It is clear that there is no need to place single squares inside the rectangle every time.
It is enough to multiply the length of one side by the length of the other.
Or in general view:
The situation is very similar with the volume of a rectangular parallelepiped.
The volume of a cube with a side of 1 unit is 1 cubic unit. Again, the original linear quantities can be anything: millimeters, centimeters, inches.
For example, 1 cm 3 is the volume of a cube with a side of 1 cm, and 1 km 3 is the volume of a cube with a side of 1 km.
Let's find the volume of a rectangular parallelepiped with sides 7 cm, 5 cm, 4 cm. (Fig. 7.)
Rice. 7. Rectangular parallelepiped
The volume of our rectangular parallelepiped is the number of unit cubes that fit into it.
Place a row of single cubes with a side of 1 cm along the long side on the bottom. Fits 7 pieces. Already from experience working with a rectangle, we know that only 5 such rows will fit on the bottom, 7 pieces in each. That is, in total:
Let's call this layer. How many of these layers can we stack on top of each other?
It depends on the height. It is equal to 4 cm. This means that 4 layers of 35 pieces are laid in each. Total:
Where did we get the number 35 from? This is 75. That is, we got the number of cubes by multiplying the lengths of all three sides.
But this is the volume of our rectangular parallelepiped.
Answer: 140
Now we can write the formula in general form. (Fig. 8.)
Rice. 8. Volume of a parallelepiped
Volume of a rectangular parallelepiped with sides , , equal to the product all three sides.
If the lengths of the sides are given in centimeters, then the volume will be in cubic centimeters(cm 3).
If in meters, then the volume is in cubic meters (m3).
Similarly, volume can be measured in cubic millimeters, kilometers, etc.
A glass cube with a side of 1 m is completely filled with water. What is the mass of water? (Fig. 9.)
Rice. 9. Cube
The cube is a unit. Side - 1 m. Volume - 1 m 3.
If we know how much 1 cubic meter of water weighs (abbreviated to cubic meter), then the problem is solved.
But if we don’t know this, then it’s not difficult to calculate.
Side length.
Let's calculate the volume in dm 3.
But 1 dm3 has a separate name, 1 liter. That is, we have 1000 liters of water.
We all know that the mass of one liter of water is 1 kg. That is, we have 1000 kg of water, or 1 ton.
It is clear that such a cube filled with water cannot be moved by any ordinary person.
Answer: 1 t.
Rice. 10. Refrigerator
The refrigerator is 2 meters high, 60 cm wide and 50 cm deep. Find its volume.
Before we use the volume formula - the product of the lengths of all sides - it is necessary to convert the lengths into the same units of measurement.
We can convert everything into centimeters.
Accordingly, we will get the volume in cubic centimeters.
I think you will agree that volume in cubic meters is more understandable.
A person has a hard time distinguishing a number with five zeros from a number with six zeros, but one is 10 times larger than the other.
Often we need to convert one unit of volume to another. For example, cubic meters to cubic decimeters. It's hard to remember all these ratios. But this is not necessary. It is enough to understand the general principle.
For example, how many cubic centimeters are in a cubic meter?
Let's see how many cubes with a side of 1 centimeter will fit into a cube with a side of 1 m. (Fig. 11.)
Rice. 11. Cube
100 pieces are placed in one row (after all, there are 100 cm in one meter).
100 rows or cubes are laid in one layer.
A total of 100 layers can be placed.
Thus,
That is, if linear quantities are related by the relation “there are 100 cm in one meter,” then to obtain the relation for cubic quantities, you need to raise 100 to the 3rd power (). And you don’t need to draw cubes every time.
Students often ask indignantly: “How will this be useful to me in life?” On any topic of each subject. The topic about the volume of a parallelepiped is no exception. And this is where you can just say: “It will come in handy.”
How, for example, can you find out whether a package will fit in a postal box? Of course, you can choose the right one by trial and error. What if this is not possible? Then calculations will come to the rescue. Knowing the capacity of the box, you can calculate the volume of the parcel (at least approximately) and answer the question posed.
Parallelepiped and its types
If we literally translate its name from ancient Greek, it turns out that this is a figure consisting of parallel planes. There are the following equivalent definitions of a parallelepiped:
- a prism with a base in the form of a parallelogram;
- a polyhedron, each face of which is a parallelogram.
Its types are distinguished depending on what figure lies at its base and how the lateral ribs are directed. IN general case talk about inclined parallelepiped, whose base and all faces are parallelograms. If the previous type side faces become rectangles, then it will need to be called direct. And rectangular and the base also has 90º angles.
Moreover, in geometry they try to depict the latter in such a way that it is noticeable that all the edges are parallel. Here, by the way, is the main difference between mathematicians and artists. It is important for the latter to convey the body in compliance with the law of perspective. And in this case, the parallelism of the ribs is completely invisible.
About the introduced notations
In the formulas below, the notations indicated in the table are valid.
Formulas for an inclined parallelepiped
First and second for areas:
The third is to calculate the volume of a parallelepiped:
Since the base is a parallelogram, to calculate its area you will need to use the appropriate expressions.
Formulas for a rectangular parallelepiped
Similar to the first point - two formulas for areas:
And one more for volume:
First task
Condition. Given a rectangular parallelepiped, the volume of which needs to be found. The diagonal is known - 18 cm - and the fact that it forms angles of 30 and 45 degrees with the plane of the side face and the side edge, respectively.
Solution. To answer the problem question, you will need to know all the sides in three right triangles. They will give required values edges along which you need to calculate the volume.
First you need to figure out where the 30º angle is. To do this, you need to draw a diagonal of the side face from the same vertex from where the main diagonal of the parallelogram was drawn. The angle between them will be what is needed.
The first triangle that will give one of the values of the sides of the base will be the following. It contains the required side and two drawn diagonals. It's rectangular. Now we need to use the relation opposite leg(base sides) and hypotenuse (diagonals). It is equal to the sine of 30º. That is, the unknown side of the base will be determined as the diagonal multiplied by the sine of 30º or ½. Let it be designated by the letter “a”.
The second will be a triangle containing a known diagonal and an edge with which it forms 45º. It is also rectangular, and you can again use the ratio of the leg to the hypotenuse. In other words, side edge to diagonal. It is equal to the cosine of 45º. That is, “c” is calculated as the product of the diagonal and the cosine of 45º.
c = 18 * 1/√2 = 9 √2 (cm).
In the same triangle you need to find another leg. This is necessary in order to then calculate the third unknown - “in”. Let it be designated by the letter “x”. It can be easily calculated using the Pythagorean theorem:
x = √(18 2 - (9√2) 2) = 9√2 (cm).
Now we need to consider another right triangle. It already contains known parties“c”, “x” and the one that needs to be counted, “b”:
in = √((9√2) 2 - 9 2 = 9 (cm).
All three quantities are known. You can use the formula for volume and calculate it:
V = 9 * 9 * 9√2 = 729√2 (cm 3).
Answer: the volume of the parallelepiped is 729√2 cm 3.
Second task
Condition. You need to find the volume of a parallelepiped. In it, the sides of the parallelogram that lies at the base are known to be 3 and 6 cm, as well as its acute angle - 45º. The side rib has an inclination to the base of 30º and is equal to 4 cm.
Solution. To answer the question of the problem, you need to take the formula that was written for the volume inclined parallelepiped. But both quantities are unknown in it.
The area of the base, that is, of a parallelogram, will be determined by a formula in which you need to multiply the known sides and the sine of the acute angle between them.
S o = 3 * 6 sin 45º = 18 * (√2)/2 = 9 √2 (cm 2).
The second unknown quantity is height. It can be drawn from any of the four vertices above the base. It can be found from a right triangle in which the height is the leg and the side edge is the hypotenuse. In this case, an angle of 30º lies opposite unknown height. This means that we can use the ratio of the leg to the hypotenuse.
n = 4 * sin 30º = 4 * 1/2 = 2.
Now all the values are known and the volume can be calculated:
V = 9 √2 * 2 = 18 √2 (cm 3).
Answer: the volume is 18 √2 cm 3.
Third task
Condition. Find the volume of a parallelepiped if it is known that it is straight. The sides of its base form a parallelogram and are equal to 2 and 3 cm. Sharp corner there is 60º between them. The minor diagonal of a parallelepiped is larger diagonal grounds.
Solution. In order to find out the volume of a parallelepiped, we use the formula with the base area and height. Both quantities are unknown, but they are easy to calculate. The first one is height.
Since the smaller diagonal of the parallelepiped is the same size as larger base, then they can be designated by one letter d. Larger angle a parallelogram is 120º, since it forms 180º with an acute one. Let the second diagonal of the base be designated by the letter “x”. Now for the two diagonals of the base we can write the cosine theorems:
d 2 = a 2 + b 2 - 2av cos 120º,
x 2 = a 2 + b 2 - 2ab cos 60º.
It makes no sense to find values without squares, since later they will be raised to the second power again. After substituting the data, we get:
d 2 = 2 2 + 3 2 - 2 * 2 * 3 cos 120º = 4 + 9 + 12 * ½ = 19,
x 2 = a 2 + b 2 - 2ab cos 60º = 4 + 9 - 12 * ½ = 7.
Now the height, which is also the side edge of the parallelepiped, will turn out to be a leg in the triangle. The hypotenuse will be known diagonal body, and the second leg - “x”. We can write the Pythagorean Theorem:
n 2 = d 2 - x 2 = 19 - 7 = 12.
Hence: n = √12 = 2√3 (cm).
Now the second unknown quantity is the area of the base. It can be calculated using the formula mentioned in the second problem.
S o = 2 * 3 sin 60º = 6 * √3/2 = 3√3 (cm 2).
Combining everything into the volume formula, we get:
V = 3√3 * 2√3 = 18 (cm 3).
Answer: V = 18 cm 3.
Fourth task
Condition. It is required to find out the volume of a parallelepiped that meets the following conditions: the base is a square with a side of 5 cm; the side faces are rhombuses; one of the vertices located above the base is equidistant from all the vertices lying at the base.
Solution. First you need to deal with the condition. There are no questions with the first point about the square. The second, about rhombuses, makes it clear that the parallelepiped is inclined. Moreover, all its edges are equal to 5 cm, since the sides of the rhombus are the same. And from the third it becomes clear that the three diagonals drawn from it are equal. These are two that lie on the side faces, and the last one is inside the parallelepiped. And these diagonals are equal to the edge, that is, they also have a length of 5 cm.
To determine the volume, you will need a formula written for an inclined parallelepiped. There are again no known quantities in it. However, the area of the base is easy to calculate because it is a square.
S o = 5 2 = 25 (cm 2).
The situation with height is a little more complicated. It will be like this in three figures: a parallelepiped, quadrangular pyramid And isosceles triangle. This last circumstance should be taken advantage of.
Since it is the height, it is a leg in right triangle. The hypotenuse in it will be famous rib, and the second leg equal to half diagonals of the square (height is also the median). And the diagonal of the base is easy to find:
d = √(2 * 5 2) = 5√2 (cm).
The height will need to be calculated as the difference between the second power of the edge and the square of half the diagonal and then remember to take the square root:
n = √ (5 2 - (5/2 * √2) 2) = √(25 - 25/2) = √(25/2) = 2.5 √2 (cm).
V = 25 * 2.5 √2 = 62.5 √2 (cm 3).
Answer: 62.5 √2 (cm 3).
Math lesson in 5th grade. (Vilenkin)
Subject: Volumes. Volume of a rectangular parallelepiped.
Target: 1. Consolidate knowledge on this topic when solving problems. Prepare for test work. Give the ratio of volume units.
2. Repeat the properties of multiplication, simplification of expressions, parts of a parallelepiped.
3. Educate environmental aspect, attention.
Equipment: on the board: topic, task for oral counting; Handout: models of parallelepiped, cube, matchbox; for children: cheat sheets, rulers, two-color signal circles,
During the classes.
Organizing time.
Good afternoon, happy hour, we have mathematics. On the desk: rulers, cheat sheets, notebooks, textbooks.
Oral counting (warm-up) No. 806 – in rows “in a chain”,
- apply distributive property multiplication:
(x + 8) 20 on the board
247 123 – 147 123
- simplify:
20a – 19a 4x + x – 2x
13v - 27 + 13v - 10v
Communicate the topic and purpose.
— What geometric figures did you get acquainted with? Today we will repeat how to find the volume of a rectangular parallelepiped and the units of volume. Getting ready for the test.
IV. Repetition of what has been learned. cube models,
— Show top, back, bottom and front edges. parallelepiped
— Show two faces that have a common edge,
— Show vertical edges.
(show 2 or 3 students at the same time)
Game "Yes - no"
— Any cube is a rectangular parallelepiped (+) signal
— A rectangular parallelepiped has 10 vertices (-, 8) circles
– 6 edges (+) – 12 edges (+)
— Each face of the cube is a square (+)
— If the length of a rectangular parallelepiped is not equal to its height, then it cannot be a cube (+)
— The volume of a rectangular parallelepiped is equal to the product of its three dimensions (+)
Find the formula.
- calculate the volume matchbox, cube, parallelepiped. visibility
— additional material“How much air does a person need to breathe?”
With each inhalation, a person introduces 9 liters of air into his lungs in 1 minute. This amounts to 9 * 60 per hour, i.e. 540 liters. Let’s round up to 500 liters or half a cubic meter and find out that a person inhales 12 m³ of air per day. This volume is 14 kg.
In one day, a person passes through his body more air than food: no one eats even 3 kg per day, but we inhale 14 kg. If we consider that the inhaled air consists of 4/5 nitrogen, which is useless for breathing, then it seems that our body consumes only 3 kg, i.e. approximately the same amount as food (solid and liquid).
Do I need any other proof of the need to renew the air in the living room?
- No. 804, 801 - on the board,
— How to calculate the volume of a parallelepiped or cube?
— In what units is volume measured?
VI. Ratio of volume units.“cheat sheets” Write in “cheat sheets”. flyleaf
— Game “The Weakest Link” — No. 802,
— Task on cards.
— Express in cubic cm:
6 dm³, 287 dm³
5 dm³ 23 cm³ 16000 mm³
5 dm³ 635 cm³ 2 dm³ 80 cm³
— Express in cubic dm:
6m³ 580cm³ 7m³ 15dm³
VII. Repetition of what has been learned. № 808
VIII. Result:— What do you remember from the lesson?
— Who worked for 5? by 4?
IX. Homework : § 21, No. 822 (a, b), No. 823.
Mathematics
5th grade
21. Volumes.
If you fill the mold with wet sand, and then turn it over and remove it, you will get figures that have the same volume (Fig. 83). If the mold is filled with water, the volume of water will be equal to volume each sand figure.
Rice. 83
To compare the volumes of two vessels, you can fill one of them with water and pour it into the second vessel. If the second vessel is filled and there is no water left in the first vessel, then the volumes of the vessels are equal. If water remains in the first vessel, then its volume is greater than the volume of the second vessel. And if it is not possible to fill the second vessel with water, then the volume of the first vessel is less than the volume of the second.
The following units are used to measure volumes: cubic millimeter (mm3), cubic centimeter (cm3), cubic decimeter (dm3), cubic meter (m3), cubic kilometer (km3).
For example: a cubic centimeter is the volume of a cube with an edge of 1 cm (Fig. 84).
Rice. 84
A cubic decimeter is also called a liter.
The figure in Figure 85 consists of 4 cubes with an edge of 1 cm. This means that its volume is 4 cm3.
Rice. 85
Let us derive a rule for calculating the volume of a rectangular parallelepiped.
Formulas for the volumes of parallelepipeds and cubes
Let a rectangular parallelepiped have a length of 4 cm, a width of 3 cm and a height of 2 cm (Fig. 86, a). Let's divide it into two layers 1 cm thick (Fig. 86, b). Each of these layers consists of 3 columns 4 cm long (Fig. 86, c), and each column consists of 4 cubes with an edge of 1 cm (Fig. 86, d). This means that the volume of each column is 4 cm3, each layer is 4 3 (cm3), and the entire rectangular parallelepiped is (4 3) 2, that is, 24 cm3.
Rice. 86
To find the volume of a rectangular parallelepiped, you need to multiply its length by its width and height.
The formula for the volume of a rectangular parallelepiped is
where V is volume; a, b, c - measurements.
If the edge of a cube is 4 cm, then the volume of the cube is 4 4 4 = 43 (cm3), that is, 64 cm3.
If the edge of a cube is equal to a, then the volume V of the cube is equal to a a a = a3.
This means that the formula for the volume of a cube has the form
That is why the entry a3 is called the cube of a.
The volume of a cube with an edge of 1 m is equal to 1 m3. And since 1 m = 10 dm, then 1 m3 = 103 dm3, that is, 1 m3 = 1000 dm3 = 1000 l.
In the same way we find that
1 l = 1 dm3 = 1000 cm3; 1 cm3 = 1000 mm3;
1 km3 = 1,000,000,000 m3 (see figure).
Self-test questions
- The figure consists of 19 cubes with a side of 1 cm each; what is the volume of this figure?
- What is a cubic centimeter; cubic meter?
- What is another name for cubic decimeter?
- How many cubic centimeters is 1 liter?
- How many liters is a cubic meter equal?
- How many cubic meters in cubic kilometers?
- Write the formula for the volume of a rectangular parallelepiped.
- What does the letter V mean in this formula; letters a, b, c?
- Write the formula for the volume of a cube.
Do the exercises
819. Figures are made from cubes with an edge of 1 cm (Fig. 87). Find the volumes and surface areas of these figures.
Rice. 87
820. Find the volume of a rectangular parallelepiped if:
- a) a = 6 cm, b = 10 cm, c = 5 cm;
- b) a = 30 dm, b = 20 dm, c = 30 dm;
- c) a = 8 dm, b = 6 m, c = 12 m;
- d) a = 2 dm 1 cm, b = 1 dm 7 cm, c = 8 cm;
- e) a = 3 m, b = 2 dm, c = 15 cm.
821. Square bottom edge of a rectangular parallelepiped is 24 cm2. Determine the height of this parallelepiped if its volume is 96 cm3.
822. The volume of the room is 60 m3. The height of the room is 3 m, the width is 4 m. Find the length of the room and the area of the floor, ceiling, and walls.
823. Find the volume of a cube whose edge is 8 dm; 3 dm 6 cm.
824. Find the volume of a cube if its surface area is 96 cm2.
825. Express:
- a) in cubic centimeters: 5 dm3 635 cm3; 2 dm3 80 cm3;
- b) in cubic decimeters: 6 m3 580 dm3; 7 m3 15 dm3;
- c) in cubic meters and decimeters: 3270 dm3; 12,540,000 cm3.
826. The height of the room is 3 m, width 5 m and length 6 m. How many cubic meters of air are in the room?
827. The length of the aquarium is 80 cm, the width is 45 cm, and the height is 55 cm. How many liters of water must be poured into this aquarium so that the water level is 10 cm below the top edge of the aquarium?
828. The rectangular parallelepiped (Fig. 88) is divided into two parts. Find the volume and surface area of the entire parallelepiped and both of its parts. Is the volume of a parallelepiped equal to the sum of the volumes of its parts? Can this be said about their surface areas? Explain why.
Rice. 88
829. Calculate orally:
830. Restore the chain of calculations:
831. Find the meaning of the expression:
- a) 23 + Z2;
- b) 33 + 52;
- c) 43 + 6;
- d) 103 - 10.
832. How many tens are there in the quotient:
- a) 1652: 7;
- b) 774: 6;
- c) 1632: 12;
- d) 2105: 5?
833. Do you agree with the statement:
- a) any cube is also a rectangular parallelepiped;
- b) if the length of a rectangular parallelepiped is not equal to its height, then it cannot be a cube;
- c) each face of a cube is a square?
834. Four identical barrels hold 26 buckets of water. How many buckets of water can 10 of these barrels hold?
835. In how many ways from 7 beads different colors can you make a necklace (with a clasp)?
836. Name in a rectangular parallelepiped (Fig. 89):
- a) two faces having a common edge;
- b) top, back, front and bottom edges;
- c) vertical ribs.
Rice. 89
837. Solve the problem:
- Find the area of each plot if the area of the first plot is 5 times more area the second, and the area of the second is 252 hectares less area first.
- Find the area of each plot if the area of the second plot is 324 hectares greater than the area of the first plot, and the area of the first plot is 7 times less than the area of the second.
838. Follow these steps:
- 668 (3076 + 5081);
- 783 (66 161 — 65 752);
- 2 111 022: (5960 — 5646);
- 2 045 639: (6700 — 6279).
839. In Rus', in the old days, a bucket (about 12 liters), a shtof (a tenth of a bucket) was used as units of volume measurement; in the USA, England and other countries a barrel (about 159 liters), a gallon (about 4 liters), a bushel (about 36 l), pint (from 470 to 568 cubic centimeters). Compare these units. Which ones are larger than 1 m3?
840. Find the volumes of the figures shown in Figure 90. The volume of each cube is 1 cm3.
Rice. 90
841. Find the volume of a rectangular parallelepiped (Fig. 91).
Rice. 91
842. Find the volume of a rectangular parallelepiped if its dimensions are 48 dm, 16 dm and 12 dm.
843. The barn, shaped like a rectangular parallelepiped, is filled with hay. The length of the barn is 10 m, width 6 m, height 4 m. Find the mass of hay in the barn if the mass of 10 m3 of hay is 6 quintals.
844. Express in cubic decimeters:
- 2 m3 350 dm3;
- 3 m3 7 dm3;
- 4 m3 30 dm3;
- 18,000 cm3;
- 210,000 cm3.
845. The volume of a rectangular parallelepiped is 1248 cm3. Its length is 13 cm and its width is 8 cm. Find the height of this parallelepiped.
846. Using the formula V = abc calculate:
- a) V, if a - 3 dm, b = 4 dm, c = 5 dm;
- b) a, if V = 2184 cm3, b = 12 cm, c = 13 cm;
- c) b, if V = 9200 cm3, a = 23 cm, c = 25 cm;
- d) ab, if V = 1088 dm3, c = 17 cm.
What is the meaning of ab?
847. Father older than my son for 21 years. Write down a formula expressing - the age of the father - through b - the age of the son. Find using this formula:
- a) a, if b = 10;
- b) a, if b = 18;
- c) b, if a = 48.
848. Find the meaning of the expression:
- a) 700,700 - 6054 (47,923 - 47,884) - 65,548;
- b) 66,509 + 141,400: (39,839 - 39,739) + 1985;
- c) (851 + 2331) : 74 - 34;
- d) (14,084: 28 - 23) 27 - 12,060;
- e) (102 + 112 + 122) : 73 + 895;
- f) 2555: (132 + 142) + 35.
849. Calculate from the table (Fig. 92):
- a) how many times does the number 9 appear;
- b) how many times do the numbers 6 and 7 appear in the table (not counting them separately);
- c) how many times do the numbers 5, 6 and 8 appear (not counting them individually).
Rice. 92
Stories about the history of the emergence and development of mathematics
200 years ago in different countries, including in Russia, were used various systems units for measuring length, mass and other quantities. The relationships between the measures were complex, there were different definitions for units of measurement.
For example, to this day in Great Britain there are two different “tons” (2000 and 2940 pounds), more than 50 different “bushels”, etc. This hampered the development of science and trade between countries, so there is a need to introduce a unified system of measures , convenient for all countries, with simple relationships between units.
Such a system - it was called the metric system of measures - was developed in France. Basic unit of length, 1 meter (from Greek word“metron” - measure), defined as a forty-millionth fraction of the Earth’s circumference, the basic unit of mass, 1 kilogram - as the mass of 1 dm3 clean water. The remaining units were determined through these two, the ratios between units of the same value were equal to 10, 100, 1000, etc.
The metric system of measures has been adopted by most countries of the world; in Russia its introduction began in 1899. Great contribution to the introduction and dissemination metric system measures in our country belong to Dmitry Ivanovich Mendeleev, the great Russian chemist.
However, according to tradition, even today the old units are sometimes used. sailors measure distances in miles (1852 m) and cables (a tenth of a mile, that is, about 185 m), speed - in knots (1 mile per hour). The mass of diamonds is measured in carats (200 mg, that is, a fifth of a gram is the mass of a wheat grain). The volume of oil is measured in barrels (159 l), etc.
This can be done different ways, it all depends on what quantities and objects we have.
So, the first method, which is suitable exclusively for a rectangular parallelepiped.
To determine the volume of a parallelepiped you will need its height, width and length.
Since rectangles form a parallelepiped, let's mark their length and width with the letters a and b, respectively. Then the area of the rectangle will be calculated as a*b.
The height of a parallelepiped is the height of the side edge, and since the height is a constant value, to find the volume you need to multiply the base area of the parallelepiped by the height. This is expressed by the following formula: V = a*b*c = S*c, where c is the height.
Let's look at an example. Let's say we have a parallelepiped with a base length and width of 5 and 8 cm, and its height is 11 cm. It is necessary to calculate the volume.
Find the area of the base: 5*8=40 sq. cm. Now we multiply the resulting value by the height 40*11=440 cubic meters. cm is the volume of the figure.
Second way.
Since the base of the parallelepiped is geometric figure parallelogram, you need to determine its area. To find the area of a parallelogram depending on the known data, you can use the following formulas:
- S = a*h, where a is the side of the parallelogram, h is the height drawn to a.
- S = a*b*sinα, where a and b are the sides of the figure, α is the angle between these sides.
After that. How did you figure it out? How to find the area of a parallelogram, you can begin to find the volume of our parallelepiped. To do this we use the formula:
V = S*h, where S is the base area obtained earlier, h is the height of our parallelepiped.
Let's look at an example.
We are given a parallelepiped with a height of 50 cm, the base (parallelogram) of which has a side equal to 23 cm and the height drawn to this side is 8 cm. We substitute the above formula:
S = 23*8 = 184 sq. cm.
Now we substitute the formula to find the volume of a parallelepiped:
V = 184*50 = 9,200 cubic meters
Mathematics lesson ‘Volume of a rectangular parallelepiped’ (5th grade)
Answer: the volume of this parallelepiped is 9200 cubic centimeters.
Third way.
This option is only suitable for rectangular type parallelepiped, sides whose bases will be equal. To do this, you just need to cube these sides.
V = a3, i.e. cubed
Given a parallelepiped with a base side of 12. This means that the volume of this figure is calculated by the following formula V = 123 = 1728 cc cm.
Either method is very simple. The main thing is to arm yourself with a calculator and perform all the calculations correctly. Good luck!
volume of a rectangular parallelepiped
S1*2 + S2*2 + S3*2 = S
Parallelepiped base
The calculator will calculate and write out the solution in detail and with comments. All you have to do is copy the line solution of the parallelepiped into your notebook. A detailed text solution with explanations will allow you to understand the methodology for solving such problems and, if necessary, answer questions by giving a detailed and competent answer.
Calculation of volume and area of a parallelogram is an elementary basis for many technical and everyday calculations!
Volumes. Volume of a rectangular parallelepiped
For example, to calculate repairs in a room, calculate data for heating or air conditioning.
rectangular parallelogram
The formula used in our calculator will find volume of a rectangular parallelepiped. And if your parallelepiped has oblique edges, instead of the length of the corresponding oblique edge, you must enter the value of the height of this part of the figure.
Formula for the volume of a rectangular parallelepiped
To find it, you need to know the dimensions of the ribs: height, width and length. According to the formula, the dimensions of the parallelepiped faces must be multiplied in any order.
The volume can be expressed in liters or cubic cm, cubic millimeters.
Formula for the surface area of a parallelepiped
S1*2 + S2*2 + S3*2 = S
Using the formula for the area of a parallelepiped, you need to find the areas of all sides of the parallelepiped and then add them up. Opposite sides, faces, and edges of a parallelepiped are equal to each other, so when calculating areas you can use multiplication by two.
Parallelepiped base
In some cases, the base area of the parallelepiped is known, then in order to find the volume it is enough to multiply the base area by the height. ! IMPORTANT! - this is true only for a rectangular parallelepiped.
How to find the volume of a parallelepiped?
The easiest way to find the volume is by entering three known values into columns online calculator volume! Then - press the button - you will get the result)!
The calculator will calculate volume of parallelepiped abcda1b1c1d1 and will describe the decision in detail and with comments.
Volume of a rectangular parallelepiped
All you have to do is copy the line solution of the parallelepiped into your notebook. A detailed text solution with explanations will allow you to understand the methodology for solving such problems and, if necessary, answer questions by giving a detailed and competent answer.
Calculation of volume and area of a parallelogram is an elementary basis for many technical and everyday calculations! For example, to calculate repairs in a room, calculate data for heating or air conditioning.
A parallelogram is a three-dimensional geometric figure that has six sides, each side being a parallelogram. The sides of a parallelogram are usually called faces. If all the faces of a parallelepiped have the shape of a rectangle, then this is already rectangular parallelogram! This figure is designated by the letters abcda1b1c1d1.
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ...discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes...were involved in the study of the issue mathematical analysis, set theory, new physical and philosophical approaches; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. WITH physical point From a perspective, it looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs with constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Stay in constant units time measurements and do not go to reciprocals. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia logical paradox it can be overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Such absurd logic sentient beings never understand. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “screw me, I’m in the house”, or rather “mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them to reality. This umbilical cord is money. Apply mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” We explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: on different coins there is different quantities mud, crystal structure and the arrangement of atoms in each coin is unique...
And now I have the most interest Ask: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of numbers given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. WITH a large number 12345 I don’t want to fool my head, let’s look at the number 26 from the article about . Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don't think this girl is stupid, no knowledgeable in physics. She just has an arch stereotype of perception graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
>> Lesson 31. Formula for the volume of a rectangular parallelepiped
A rectangular parallelepiped is a spatial figure limited rectangles.
Many objects from the environment have a parallelepiped shape: a box, cubes, TV, wardrobe, etc..
