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What shape is the parallelepiped. Rectangular parallelepiped. Definition: concept of volume

The prism is called parallelepiped if its bases are parallelograms. Cm. Fig.1.

Properties of the box:

    The opposite faces of the parallelepiped are parallel (i.e. lie in parallel planes) and equal.

    The diagonals of the parallelepiped intersect at one point and bisect that point.

Adjacent faces of a box are two faces that have a common edge.

Opposite faces of a parallelepiped– faces that do not have common edges.

Opposite vertices of the box are two vertices that do not belong to the same face.

Diagonal of the box A line segment that connects opposite vertices.

If the lateral edges are perpendicular to the planes of the bases, then the parallelepiped is called direct.

A right parallelepiped whose bases are rectangles is called rectangular. A prism all of whose faces are squares is called cube.

Parallelepiped A prism whose bases are parallelograms.

Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the plane of the base.

cuboid is a right parallelepiped whose bases are rectangles.

Cube is a rectangular parallelepiped with equal edges.

Parallelepiped a prism is called, the base of which is a parallelogram; thus, the parallelepiped has six faces and all of them are parallelograms.

Opposite faces are pairwise equal and parallel. The parallelepiped has four diagonals; they all intersect at one point and divide in half at it. Any face can be taken as a base; the volume is equal to the product of the base area and the height: V = Sh.

A parallelepiped whose four lateral faces are rectangles is called a right parallelepiped.

A right parallelepiped, in which all six faces are rectangles, is called rectangular. Cm. Fig.2.

The volume (V) of a right parallelepiped is equal to the product of the base area (S) and the height (h): V = Sh .

For a rectangular parallelepiped, in addition, the formula V=abc, where a,b,c are edges.

The diagonal (d) of a cuboid is related to its edges by the relationship d 2 \u003d a 2 + b 2 + c 2 .

cuboid- a parallelepiped whose lateral edges are perpendicular to the bases, and the bases are rectangles.

Properties of a cuboid:

    In a cuboid, all six faces are rectangles.

    All dihedral angles of a cuboid are right angles.

    The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions (lengths of three edges that have a common vertex).

    The diagonals of a rectangular parallelepiped are equal.

A rectangular parallelepiped, all of whose faces are squares, is called a cube. All edges of a cube are equal; the volume (V) of a cube is expressed by the formula V=a 3, where a is the edge of the cube.

In geometry, the key concepts are plane, point, line and angle. Using these terms, any geometric figure can be described. Polyhedra are usually described in terms of simpler shapes that lie in the same plane, such as a circle, triangle, square, rectangle, etc. In this article, we will consider what a parallelepiped is, describe the types of parallelepipeds, its properties, what elements it consists of, and also give the basic formulas for calculating the area and volume for each type of parallelepiped.

Definition

A parallelepiped in three-dimensional space is a prism, all sides of which are parallelograms. Accordingly, it can have only three pairs of parallel parallelograms or six faces.

To visualize the box, imagine a regular standard brick. A brick is a good example of a cuboid that even a child can imagine. Other examples are multi-story prefabricated houses, cabinets, appropriately shaped food storage containers, etc.

Varieties of the figure

There are only two types of parallelepipeds:

  1. Rectangular, all side faces of which are at an angle of 90 o to the base and are rectangles.
  2. Inclined, the side faces of which are located at a certain angle to the base.

What elements can this figure be divided into?

  • As in any other geometric figure, in a parallelepiped, any 2 faces with a common edge are called adjacent, and those that do not have it are called parallel (based on the property of a parallelogram that has pairwise parallel opposite sides).
  • The vertices of a parallelepiped that do not lie on the same face are called opposite vertices.
  • The segment connecting such vertices is a diagonal.
  • The lengths of the three edges of a cuboid that join at one vertex are its dimensions (namely, its length, width, and height).

Shape Properties

  1. It is always built symmetrically with respect to the middle of the diagonal.
  2. The intersection point of all diagonals divides each diagonal into two equal segments.
  3. Opposite faces are equal in length and lie on parallel lines.
  4. If you add the squares of all dimensions of the box, the resulting value will be equal to the square of the length of the diagonal.

Calculation formulas

The formulas for each particular case of a parallelepiped will be different.

For an arbitrary parallelepiped, the assertion is true that its volume is equal to the absolute value of the triple scalar product of the vectors of three sides emanating from one vertex. However, there is no formula for calculating the volume of an arbitrary parallelepiped.

For a rectangular parallelepiped, the following formulas apply:

  • V=a*b*c;
  • Sb=2*c*(a+b);
  • Sp=2*(a*b+b*c+a*c).
  • V is the volume of the figure;
  • Sb - side surface area;
  • Sp - total surface area;
  • a - length;
  • b - width;
  • c - height.

Another special case of a parallelepiped in which all sides are squares is a cube. If any of the sides of the square is denoted by the letter a, then the following formulas can be used for the surface area and volume of this figure:

  • S=6*a*2;
  • V=3*a.
  • S is the area of ​​the figure,
  • V is the volume of the figure,
  • a - the length of the face of the figure.

The last kind of parallelepiped we are considering is a straight parallelepiped. What is the difference between a cuboid and a cuboid, you ask. The fact is that the base of a rectangular parallelepiped can be any parallelogram, and the base of a straight line can only be a rectangle. If we designate the perimeter of the base, equal to the sum of the lengths of all sides, as Po, and designate the height as h, we have the right to use the following formulas to calculate the volume and areas of the full and lateral surfaces.

Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, this can be represented as a rectangle in which one side denotes lettuce, the other side denotes water. The sum of these two sides will denote borscht. The diagonal and area of ​​such a "borscht" rectangle are purely mathematical concepts and are never used in borscht recipes.


How do lettuce and water turn into borscht in terms of mathematics? How can the sum of two segments turn into trigonometry? To understand this, we need linear angle functions.


You won't find anything about linear angle functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work whether we know they exist or not.

Linear angular functions are the laws of addition. See how algebra turns into geometry and geometry turns into trigonometry.

Is it possible to do without linear angular functions? You can, because mathematicians still manage without them. The trick of mathematicians lies in the fact that they always tell us only about those problems that they themselves can solve, and never tell us about those problems that they cannot solve. See. If we know the result of the addition and one term, we use subtraction to find the other term. Everything. We do not know other problems and we are not able to solve them. What to do if we know only the result of the addition and do not know both terms? In this case, the result of addition must be decomposed into two terms using linear angular functions. Further, we ourselves choose what one term can be, and the linear angular functions show what the second term should be in order for the result of the addition to be exactly what we need. There can be an infinite number of such pairs of terms. In everyday life, we do very well without decomposing the sum; subtraction is enough for us. But in scientific studies of the laws of nature, the expansion of the sum into terms can be very useful.

Another law of addition that mathematicians don't like to talk about (another trick of theirs) requires the terms to have the same unit of measure. For lettuce, water, and borscht, these may be units of weight, volume, cost, or unit of measure.

The figure shows two levels of difference for math. The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the area of ​​units of measurement, which are shown in square brackets and are indicated by the letter U. This is what physicists do. We can understand the third level - the differences in the scope of the described objects. Different objects can have the same number of the same units of measure. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same notation for the units of measurement of different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or in connection with our actions. letter W I will mark the water with the letter S I will mark the salad with the letter B- borsch. Here's what the linear angle functions for borscht would look like.

If we take some part of the water and some part of the salad, together they will turn into one serving of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals will turn out. What then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we do not understand what, it is not clear why, and we understand very poorly how this relates to reality, because of the three levels of difference, mathematicians operate on only one. It will be more correct to learn how to move from one unit of measurement to another.

And bunnies, and ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar problem for adults. What do you get when you add bunnies and money? There are two possible solutions here.

First option. We determine the market value of the bunnies and add it to the available cash. We got the total value of our wealth in terms of money.

Second option. You can add the number of bunnies to the number of banknotes we have. We will get the amount of movable property in pieces.

As you can see, the same addition law allows you to get different results. It all depends on what exactly we want to know.

But back to our borscht. Now we can see what will happen for different values ​​of the angle of the linear angle functions.

The angle is zero. We have salad but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. Zero borsch can also be at zero salad (right angle).


For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This is because addition itself is impossible if there is only one term and the second term is missing. You can relate to this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so discard your logic and stupidly cram the definitions invented by mathematicians: "division by zero is impossible", "any number multiplied by zero equals zero" , "behind the point zero" and other nonsense. It is enough to remember once that zero is not a number, and you will never have a question whether zero is a natural number or not, because such a question generally loses all meaning: how can one consider a number that which is not a number. It's like asking what color to attribute an invisible color to. Adding zero to a number is like painting with paint that doesn't exist. They waved a dry brush and tell everyone that "we have painted." But I digress a little.

The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but little water. As a result, we get a thick borscht.

The angle is forty-five degrees. We have equal amounts of water and lettuce. This is the perfect borscht (may the cooks forgive me, it's just math).

The angle is greater than forty-five degrees but less than ninety degrees. We have a lot of water and little lettuce. Get liquid borscht.

Right angle. We have water. Only memories remain of the lettuce, as we continue to measure the angle from the line that once marked the lettuce. We can't cook borscht. The amount of borscht is zero. In that case, hold on and drink water while it's available)))

Here. Something like this. I can tell other stories here that will be more than appropriate here.

The two friends had their shares in the common business. After the murder of one of them, everything went to the other.

The emergence of mathematics on our planet.

All these stories are told in the language of mathematics using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, let's return to the trigonometry of borscht and consider projections.

Saturday, October 26, 2019

Wednesday, August 7, 2019

Concluding the conversation about , we need to consider an infinite set. Gave in that the concept of "infinity" acts on mathematicians, like a boa constrictor on a rabbit. The quivering horror of infinity deprives mathematicians of common sense. Here is an example:

The original source is located. Alpha denotes a real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take an infinite set of natural numbers as an example, then the considered examples can be represented as follows:

To visually prove their case, mathematicians have come up with many different methods. Personally, I look at all these methods as the dances of shamans with tambourines. In essence, they all come down to the fact that either some of the rooms are not occupied and new guests are settled in them, or that some of the visitors are thrown out into the corridor to make room for the guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Moving an infinite number of visitors takes an infinite amount of time. After we have vacated the first guest room, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will already be from the category of "the law is not written for fools." It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an "infinite hotel"? An infinity inn is an inn that always has any number of vacancies, no matter how many rooms are occupied. If all the rooms in the endless hallway "for visitors" are occupied, there is another endless hallway with rooms for "guests". There will be an infinite number of such corridors. At the same time, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, on the other hand, are not able to move away from banal everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to "shove the unpushed".

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers exist - one or many? There is no correct answer to this question, since we ourselves invented numbers, there are no numbers in Nature. Yes, Nature knows how to count perfectly, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers exist. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on a shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take a unit from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:

I have written down the operations in algebraic notation and in set theory notation, listing the elements of the set in detail. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same one is added.

Option two. We have many different infinite sets of natural numbers on the shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:

The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If one infinite set is added to another infinite set, the result is a new infinite set consisting of the elements of the first two sets.

The set of natural numbers is used for counting in the same way as a ruler for measurements. Now imagine that you have added one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - this is your own business. But if you ever run into mathematical problems, consider whether you are on the path of false reasoning, trodden by generations of mathematicians. After all, mathematics classes, first of all, form a stable stereotype of thinking in us, and only then they add mental abilities to us (or vice versa, they deprive us of free thinking).

pozg.ru

Sunday, August 4, 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical basis of the mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it weak for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, personally I got the following:

The rich theoretical basis of modern mathematics does not have a holistic character and is reduced to a set of disparate sections, devoid of a common system and evidence base.

I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole cycle of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.

May we have many BUT consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter a, the subscript with a number will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sexual characteristic" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set BUT on gender b. Notice that our "people" set has now become the "people with gender" set. After that, we can divide the sexual characteristics into male bm and women's bw gender characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, it does not matter which one is male or female. If it is present in a person, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.

After multiplication, reductions and rearrangements, we got two subsets: the male subset bm and a subset of women bw. Approximately the same way mathematicians reason when they apply set theory in practice. But they do not let us in on the details, but give us the finished result - "a lot of people consists of a subset of men and a subset of women." Naturally, you may have a question, how correctly applied mathematics in the above transformations? I dare to assure you that in fact the transformations are done correctly, it is enough to know the mathematical justification of arithmetic, Boolean algebra and other sections of mathematics. What it is? Some other time I will tell you about it.

As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.

As you can see, units of measurement and common math make set theory a thing of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. The mathematicians did what the shamans once did. Only shamans know how to "correctly" apply their "knowledge". This "knowledge" they teach us.

Finally, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its 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 infinitely quickly overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls 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 one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of a flying arrow:

A flying arrow is motionless, since at each 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, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
I will show the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, and there are without a bow. After that, we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

Now let's do a little trick. Let's take "solid in a pimple with a bow" and unite these "whole" by color, selecting red elements. We got a lot of "red". Now a tricky question: are the received sets "with a bow" and "red" the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid pimply with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a bump), decorations (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics. Here's what it looks like.

The letter "a" with different indices denotes different units of measurement. In parentheses, units of measurement are highlighted, according to which the "whole" is allocated at the preliminary stage. The unit of measurement, according to which the set is formed, is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dances of shamans with tambourines. Shamans can "intuitively" come to the same result, arguing it with "obviousness", because units of measurement are not included in their "scientific" arsenal.

With the help of units of measurement, it is very easy to break one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

In this lesson, everyone will be able to study the topic "Rectangular box". At the beginning of the lesson, we will repeat what an arbitrary and straight parallelepipeds are, recall the properties of their opposite faces and diagonals of the parallelepiped. Then we will consider what a cuboid is and discuss its main properties.

Topic: Perpendicularity of lines and planes

Lesson: Cuboid

A surface composed of two equal parallelograms ABCD and A 1 B 1 C 1 D 1 and four parallelograms ABB 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1 is called parallelepiped(Fig. 1).

Rice. 1 Parallelepiped

That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (bases), they lie in parallel planes so that the side edges AA 1, BB 1, DD 1, CC 1 are parallel. Thus, a surface composed of parallelograms is called parallelepiped.

Thus, the surface of a parallelepiped is the sum of all the parallelograms that make up the parallelepiped.

1. Opposite faces of a parallelepiped are parallel and equal.

(the figures are equal, that is, they can be combined by overlay)

For example:

ABCD \u003d A 1 B 1 C 1 D 1 (equal parallelograms by definition),

AA 1 B 1 B \u003d DD 1 C 1 C (since AA 1 B 1 B and DD 1 C 1 C are opposite faces of the parallelepiped),

AA 1 D 1 D \u003d BB 1 C 1 C (since AA 1 D 1 D and BB 1 C 1 C are opposite faces of the parallelepiped).

2. The diagonals of the parallelepiped intersect at one point and bisect that point.

The diagonals of the parallelepiped AC 1, B 1 D, A 1 C, D 1 B intersect at one point O, and each diagonal is divided in half by this point (Fig. 2).

Rice. 2 The diagonals of the parallelepiped intersect and bisect the intersection point.

3. There are three quadruples of equal and parallel edges of the parallelepiped: 1 - AB, A 1 B 1, D 1 C 1, DC, 2 - AD, A 1 D 1, B 1 C 1, BC, 3 - AA 1, BB 1, SS 1, DD 1.

Definition. A parallelepiped is called straight if its lateral edges are perpendicular to the bases.

Let the side edge AA 1 be perpendicular to the base (Fig. 3). This means that the line AA 1 is perpendicular to the lines AD and AB, which lie in the plane of the base. And, therefore, rectangles lie in the side faces. And the bases are arbitrary parallelograms. Denote, ∠BAD = φ, the angle φ can be any.

Rice. 3 Right box

So, a right box is a box in which the side edges are perpendicular to the bases of the box.

Definition. The parallelepiped is called rectangular, if its lateral edges are perpendicular to the base. The bases are rectangles.

The parallelepiped АВСДА 1 В 1 С 1 D 1 is rectangular (Fig. 4) if:

1. AA 1 ⊥ ABCD (lateral edge is perpendicular to the plane of the base, that is, a straight parallelepiped).

2. ∠BAD = 90°, i.e., the base is a rectangle.

Rice. 4 Cuboid

A rectangular box has all the properties of an arbitrary box. But there are additional properties that are derived from the definition of a cuboid.

So, cuboid is a parallelepiped whose lateral edges are perpendicular to the base. The base of a cuboid is a rectangle.

1. In a cuboid, all six faces are rectangles.

ABCD and A 1 B 1 C 1 D 1 are rectangles by definition.

2. Lateral ribs are perpendicular to the base. This means that all the side faces of a cuboid are rectangles.

3. All dihedral angles of a cuboid are right angles.

Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, i.e., the dihedral angle between the planes ABB 1 and ABC.

AB is an edge, point A 1 lies in one plane - in the plane ABB 1, and point D in the other - in the plane A 1 B 1 C 1 D 1. Then the considered dihedral angle can also be denoted as follows: ∠А 1 АВD.

Take point A on edge AB. AA 1 is perpendicular to the edge AB in the plane ABB-1, AD is perpendicular to the edge AB in the plane ABC. Hence, ∠A 1 AD is the linear angle of the given dihedral angle. ∠A 1 AD \u003d 90 °, which means that the dihedral angle at the edge AB is 90 °.

∠(ABB 1, ABC) = ∠(AB) = ∠A 1 ABD= ∠A 1 AD = 90°.

It is proved similarly that any dihedral angles of a rectangular parallelepiped are right.

The square of the diagonal of a cuboid is equal to the sum of the squares of its three dimensions.

Note. The lengths of the three edges emanating from the same vertex of the cuboid are the measurements of the cuboid. They are sometimes called length, width, height.

Given: ABCDA 1 B 1 C 1 D 1 - a rectangular parallelepiped (Fig. 5).

Prove: .

Rice. 5 Cuboid

Proof:

The line CC 1 is perpendicular to the plane ABC, and hence to the line AC. So triangle CC 1 A is a right triangle. According to the Pythagorean theorem:

Consider a right triangle ABC. According to the Pythagorean theorem:

But BC and AD are opposite sides of the rectangle. So BC = AD. Then:

As , a , then. Since CC 1 = AA 1, then what was required to be proved.

The diagonals of a rectangular parallelepiped are equal.

Let us designate the dimensions of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =

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parallelepiped, parallelepiped photo
Parallelepiped(ancient Greek παραλληλ-επίπεδον from other Greek παρ-άλληλος - “parallel” and other Greek ἐπί-πεδον - “plane”) - a prism, the base of which is a parallelogram, or (equivalently) a polyhedron, which has six faces and each of them - parallelogram.

  • 1 Types of box
  • 2 Basic elements
  • 3 Properties
  • 4 Basic formulas
    • 4.1 Right box
    • 4.2 Cuboid
    • 4.3 Cube
    • 4.4 Arbitrary box
  • 5 mathematical analysis
  • 6 Notes
  • 7 Links

Types of box

cuboid

There are several types of parallelepipeds:

  • A cuboid is a cuboid whose faces are all rectangles.
  • An oblique box is a box whose side faces are not perpendicular to the bases.

Main elements

Two faces of a parallelepiped that do not have a common edge are called opposite, and those that have a common edge are called adjacent. Two vertices of a parallelepiped that do not belong to the same face are called opposite. The segment connecting opposite vertices is called the diagonal of the parallelepiped. The lengths of three edges of a cuboid that have a common vertex are called its dimensions.

Properties

  • The parallelepiped is symmetrical about the midpoint of its diagonal.
  • Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided by it in half; in particular, all the diagonals of the parallelepiped intersect at one point and bisect it.
  • Opposite faces of a parallelepiped are parallel and equal.
  • The square of the length of the diagonal of a cuboid is equal to the sum of the squares of its three dimensions.

Basic Formulas

Right parallelepiped

The area of ​​the lateral surface Sb \u003d Po * h, where Ro is the perimeter of the base, h is the height

The total surface area Sp \u003d Sb + 2So, where So is the area of ​​\u200b\u200bthe base

Volume V=So*h

cuboid

Main article: cuboid

The area of ​​the side surface Sb=2c(a+b), where a, b are the sides of the base, c is the side edge of the rectangular parallelepiped

Total surface area Sp=2(ab+bc+ac)

Volume V=abc, where a, b, c - measurements of a rectangular parallelepiped.

Cube

Surface area:
Volume: , where is the edge of the cube.

Arbitrary box

The volume and ratios in a skew box are often defined using vector algebra. The volume of a parallelepiped is equal to the absolute value of the mixed product of three vectors defined by the three sides of the parallelepiped emanating from one vertex. The ratio between the lengths of the sides of the parallelepiped and the angles between them gives the statement that the Gram determinant of these three vectors is equal to the square of their mixed product:215.

In mathematical analysis

In mathematical analysis, an n-dimensional rectangular parallelepiped is understood as a set of points of the form

Notes

  1. Dvoretsky's Ancient Greek-Russian Dictionary "παραλληλ-επίπεδον"
  2. Gusyatnikov P.B., Reznichenko S.V. Vector algebra in examples and problems. - M.: Higher school, 1985. - 232 p.

Links

Wiktionary has an article "parallelepiped"
  • cuboid
  • Parallelepiped, educational film

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