Instructions
Method 2. Let's assume that cuboid is a cube. A cube is a rectangular parallelepiped, each face is represented by a square. Therefore, all its sides are equal. Then to calculate the length of its diagonal it will be expressed as follows:
Sources:
- rectangle diagonal formula
Parallelepiped - special case a prism in which all six faces are parallelograms or rectangles. Parallelepiped with rectangular edges also called rectangular. A parallelepiped has four intersecting diagonals. If three edges a, b, c are given, you can find all the diagonals of a rectangular parallelepiped by performing additional constructions.
Instructions
Find the diagonal of the parallelepiped m. To do this, find the unknown hypotenuse in a, n, m: m² = n² + a². Substitute known values, then calculate the square root. The result obtained will be the first diagonal of the parallelepiped m.
In the same way, draw sequentially all the other three diagonals of the parallelepiped. Also, for each of them, perform additional construction of diagonals of adjacent faces. Considering the right triangles formed and applying the Pythagorean theorem, find the values of the remaining diagonals.
Video on the topic
Sources:
- finding a parallelepiped
The hypotenuse is the side opposite right angle. Legs are the sides of a triangle adjacent to a right angle. Applied to triangles ABC and ACD: AB and BC, AD and DC–, AC is the common hypotenuse for both triangles (the desired diagonal). Therefore, AC = square AB + square BC or AC b = square AD + square DC. Substitute the side lengths rectangle into the above formula and calculate the length of the hypotenuse (diagonal rectangle).
For example, the sides rectangle ABCD are equal to the following values: AB = 5 cm and BC = 7 cm. The square of the diagonal AC of a given rectangle according to the Pythagorean theorem: AC squared = square AB + square BC = 52+72 = 25 + 49 = 74 sq.cm. Use a calculator to calculate the value square root 74. You should get 8.6 cm (rounded value). Please note that according to one of the properties rectangle, its diagonals are equal. So the length of the second diagonal BD rectangle ABCD is equal to the length of diagonal AC. For the above example, this value
It's called a parallelepiped quadrangular prism, whose bases are parallelograms. The height of a parallelepiped is the distance between the planes of its bases. In the figure, the height is shown by the segment 
. There are two types of parallelepipeds: straight and inclined. As a rule, a math tutor first gives the appropriate definitions for a prism and then transfers them to a parallelepiped. We will do the same.
Let me remind you that a prism is called straight if its side edges are perpendicular to the bases; if there is no perpendicularity, the prism is called inclined. This terminology is also inherited by the parallelepiped. A right parallelepiped is nothing more than a type of right prism, side rib which coincides with the height. Definitions of such concepts as face, edge and vertex, which are common to the entire family of polyhedra, are preserved. The concept of opposite faces appears. A parallelepiped has 3 pairs of opposite faces, 8 vertices and 12 edges.
The diagonal of a parallelepiped (the diagonal of a prism) is a segment connecting two vertices of a polyhedron and not lying on any of its faces.
Diagonal section - a section of a parallelepiped passing through its diagonal and the diagonal of its base.
Properties inclined parallelepiped
:
1) All its faces are parallelograms, and the opposite faces are equal parallelograms.
2)
The diagonals of a parallelepiped intersect at one point and bisect at this point.
3)
Each parallelepiped consists of six triangular pyramids of equal volume. To show them to the student, the math tutor must cut off half of it from the parallelepeded diagonal section and divide it separately into 3 pyramids. Their foundations must lie in different faces the original parallelepiped. A math tutor will find application of this property in analytical geometry. It is used to display the volume of the pyramid through mixed work vectors.
Formulas for the volume of a parallelepiped:
1) , where is the area of the base, h is the height.
2) Volume of a parallelepiped equal to the product area cross section on the side edge.
Math tutor: As you know, the formula is common to all prisms and if the tutor has already proven it, there is no point in repeating the same thing for a parallelepiped. However, when working with an average-level student (the formula is not useful to a weak student), it is advisable for the teacher to act exactly the opposite. Leave the prism alone and carry out a careful proof for the parallelepiped.
3) , where is the volume of one of the six triangular pyramid of which the parallelepiped consists.
4) If , then
The area of the lateral surface of a parallelepiped is the sum of the areas of all its faces:
The total surface of a parallelepiped is the sum of the areas of all its faces, that is, the area + two areas of the base: .
About the work of a tutor with an inclined parallelepiped:
A math tutor does not often work on problems involving an inclined parallelepiped. The likelihood of them appearing on the Unified State Exam is quite low, and the didactics are indecently poor. A more or less decent problem on the volume of an inclined parallelepiped calls serious problems, associated with determining the location of point H - the base of its height. In this case, the math tutor can be advised to cut the parallelepiped to one of its six pyramids (about which we're talking about in property No. 3), try to find its volume and multiply it by 6.
If the side edge of a parallelepiped has equal angles with the sides of the base, then H lies on the bisector of angle A of the base ABCD. And if, for example, ABCD is a rhombus, then
Math tutor tasks:
1) The faces of a parallelepiped are equal to each other with a side of 2 cm and an acute angle. Find the volume of the parallelepiped.
2) In an inclined parallelepiped, the side edge is 5 cm. The section perpendicular to it is a quadrilateral with mutually perpendicular diagonals having lengths of 6 cm and 8 cm. Calculate the volume of the parallelepiped.
3) In an inclined parallelepiped it is known that , and in ABCD the base is a rhombus with a side of 2 cm and an angle . Determine the volume of the parallelepiped.
Mathematics tutor, Alexander Kolpakov
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, to reach a common opinion about the essence of paradoxes scientific community so far it has not been possible... we 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, mathematical apparatus The use of 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 measurements of time and do not go to reciprocal quantities. 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. For the next time interval, equal to 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, the logical paradox is 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 in different moments time, but distance cannot be determined 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.” Reasonable beings will never understand such absurd logic. This is the level 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 with 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 stack and hand it to the mathematician" mathematical set salaries." We explain to the mathematics 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 the help of 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 measurements. 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 mathematical operation does not depend on the size of the number, the unit of measurement used and who performs the 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.
In geometry, the following types of parallelepipeds are distinguished: rectangular parallelepiped (the faces of the parallelepiped are rectangles); right parallelepiped (its side faces act as rectangles); inclined parallelepiped (its side faces act as perpendiculars); a cube is a parallelepiped with absolutely identical dimensions, and the faces of the cube are squares. Parallelepipeds can be either inclined or straight.
The main elements of a parallelepiped are that the two faces of the represented geometric figure, which do not have a common edge are opposite, and those that do are adjacent. The vertices of the parallelepiped, which do not belong to the same face, act opposite to each other. A parallelepiped has a dimension - these are three edges that have a common vertex.
The segment that connects opposite vertices, is called a diagonal. The four diagonals of a parallelepiped, intersecting at one point, are simultaneously divided in half.
In order to determine the diagonal of a parallelepiped, you need to determine the sides and edges, which are known from the conditions of the problem. With three known ribs A , IN , WITH draw a diagonal in the parallelepiped. According to the property of a parallelepiped, which says that all its angles are right, the diagonal is determined. Construct a diagonal from one of the faces of the parallelepiped. The diagonals must be drawn in such a way that the diagonal of the face, the desired diagonal of the parallelepiped and famous rib, created a triangle. After a triangle is formed, find the length of this diagonal. The diagonal in the other resulting triangle acts as the hypotenuse, so it can be found using the Pythagorean theorem, which must be taken under the square root. This way we know the value of the second diagonal. In order to find the first diagonal of a parallelepiped in the formed right triangle, it is also necessary to find the unknown hypotenuse (following the Pythagorean theorem). Using the same example, sequentially find the remaining three diagonals existing in the parallelepiped, performing additional constructions of diagonals that form right triangles and solve using the Pythagorean theorem.
A rectangular parallelepiped (PP) is nothing more than a prism, the base of which is a rectangle. For a PP, all diagonals are equal, which means that any of its diagonals is calculated using the formula:
a, c - sides of the base of the PP;
c is its height.
Another definition can be given by considering the Cartesian rectangular system coordinates:
The PP diagonal is the radius vector of any point in space, given by coordinates x, y and z in Cartesian system coordinates This radius vector to the point is drawn from the origin. And the coordinates of the point will be the projections of the radius vector (diagonals of the PP) onto coordinate axes. The projections coincide with the vertices of this parallelepiped.
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 side faces of the previous view 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 the necessary values of the edges by 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 side(base sides) and hypotenuse (diagonals). It is equal to the sine of 30º. That is unknown party the base will be defined 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, which 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 of an 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 the 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. The largest angle of a parallelogram is 120º, since it forms 180º with the 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 the known diagonal of the body, and the second leg will be “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. It's not there again known quantities. 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 a right triangle. The hypotenuse in it will be a known edge, 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).
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).