How to find the volume of a regular hexagonal prism. Prism base area: from triangular to polygonal

In a triangle B E B1 :

• B B1 = h
• B E = 2 ⋅ a- because E O = O B = a
• ∠ E B B1 = 90 ∘ - according to the properties of the correct straightness

Thus, it turns out that the triangle B E B1 rectangular. According to the properties of a right triangle

E B1 = B B2 1 +B E2 − − − − − − − − − − √ = h2 + 4 ⋅ a2 − − − − − − − − √

If h = a, so then

E B1 = 5 √ ⋅a

After similar reasoning we obtain that F C1 =A D1 = B E1 = C F1 = D A1 = h2 + 4 ⋅ a2 − − − − − − − − √ .

We find O F1

In a triangle F O F1 :

• F F1 = h
• F O = a
• ∠ O F F1 = 90 ∘ - according to the properties of a regular prism

Thus, it turns out that the triangle F O F1 rectangular. According to the properties of a right triangle

O F1 = F F2 1 +O F2 − − − − − − − − − − √ = h2 + a2 − − − − − − √

If h = a, so then

Determining the volumes of geometric bodies is one of the important problems of spatial geometry. This article discusses the question of what a prism with a hexagonal base is, and also provides a formula for the volume of a regular hexagonal prism.

Definition of a prism

From the point of view of geometry, a prism is a figure in space that is formed by two identical polygons located in parallel planes. And also several parallelograms that connect these polygons into a single figure.

In three-dimensional space, a prism of arbitrary shape can be obtained by taking any polygon and segment. Moreover, the latter will not belong to the plane of the polygon. Then, by placing this segment from each vertex of the polygon, you can obtain a parallel transfer of the latter to another plane. The figure formed in this way will be a prism.

To have a clear idea of ​​the class of figures under consideration, we present a drawing of a quadrangular prism.

Many people know this figure as a parallelepiped. It can be seen that two identical prism polygons are squares. They are called the bases of the figure. Its other four sides are rectangles, that is, it special case parallelograms.

Hexagonal prism: definition and types

Before giving the formula for how the volume of a hexagonal regular prism is determined, it is necessary to clearly understand what kind of figure we are talking about. has a hexagon at the base. That is, a flat polygon with six sides and the same number of angles. The sides of the figure, as for any prism, are generally parallelograms. Let us immediately note that the hexagonal base can be represented by both regular and irregular hexagons.

The distance between the bases of the figure is its height. In what follows we will denote it by the letter h. Geometrically, the height h is a segment perpendicular to both bases. If this is perpendicular:

• omitted from the geometric center of one of the bases;
• intersects the second base also at the geometric center.

The figure in this case is called a straight line. In any other case, the prism will be oblique or inclined. The difference between these types of hexagonal prism can be seen at a glance.

A right hexagonal prism is a figure that has regular hexagons at its base. Moreover, it is direct. Let's take a closer look at its properties.

Elements of a regular hexagonal prism

To understand how to calculate the volume of a regular hexagonal prism (the formula is given below in the article), you also need to understand what elements the figure consists of, as well as what properties it has. To make it easier to analyze the figure, we show it in the figure.

Its main elements are faces, edges and vertices. The quantities of these elements obey Euler's theorem. If we denote P - the number of edges, B - the number of vertices and G - faces, then we can write the equality:

Let's check it out. The number of faces of the figure in question is 8. Two of them are regular hexagons. The six faces are rectangles, as can be seen from the figure. The number of vertices is 12. Indeed, 6 vertices belong to one base, and 6 to another. According to the formula, the number of edges should be 18, which is fair. 12 edges lie at the bases and 6 form sides of rectangles parallel to each other.

Moving on to obtaining the formula for the volume of a regular hexagonal prism, you should focus on one important property of this figure: the rectangles that form lateral surface, are equal to each other and perpendicular to both bases. This leads to two important consequences:

1. The height of the figure is equal to the length of its side edge.
2. Any lateral section made using a cutting plane that is parallel to the bases is a regular hexagon equal to these bases.

Hexagon area

You can intuitively guess that this area of ​​the base of the figure will appear in the formula for the volume of a regular hexagonal prism. Therefore, in this paragraph of the article we will find this area. A regular hexagon divided into 6 equal triangles whose vertices intersect at its geometric center is shown below:

Each of these triangles is equilateral. It's not very difficult to prove this. Since the entire circle has 360 o, the angles of the triangles near the geometric center of the hexagon are equal to 360 o /6 = 60 o. The distances from the geometric center to the vertices of the hexagon are the same.

The latter means that all 6 triangles will be isosceles. Since one of the corners isosceles triangles is equal to 60 o, which means that the other two angles are also equal to 60 o. ((180 o -60 o)/2) - equilateral triangles.

Let us denote the length of the side of the hexagon by the letter a. Then the area of ​​one triangle will be equal to:

S 1 = 1/2*√3/2*a*a = √3/4*a 2 .

The formula is derived from the standard expression for the area of ​​a triangle. Then the area S 6 for the hexagon will be:

S 6 = 6*S 1 = 6*√3/4*a 2 = 3*√3/2*a 2 .

Formula for determining the volume of a regular hexagonal prism

To write down the formula for the volume of the figure in question, you should take into account the above information. For an arbitrary prism, the volume of space limited by its faces is calculated as follows:

That is, V is equal to the product of the base area S o and the height h. Since we know that the height h is equal to the length of the side edge b for a hexagonal regular prism, and the area of ​​its base corresponds to S 6, then the formula for the volume of a regular hexagonal prism will take the form:

V 6 = 3*√3/2*a 2 *b.

An example of solving a geometric problem

Dana hexagonal correct prism. It is known that it is inscribed in a cylinder with a radius of 10 cm. The height of the prism is twice the side of its base. You need to find the volume of the figure.

To find the required value, you need to know the length of the side and side edge. When examining a regular hexagon, it was shown that its geometric center is located in the middle of the circle described around it. The radius of the latter is equal to the distance from the center to any of the vertices. That is, he equal to length sides of a hexagon. These arguments lead to the following results:

a = r = 10 cm;

b = h = 2*a = 20 cm.

Substituting these data into the formula for the volume of a regular hexagonal prism, we get the answer: V 6 ≈5196 cm 3 or about 5.2 liters.

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 at the present time, come to general opinion the scientific community has not yet succeeded in understanding the essence of paradoxes... they 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. From a physical point of view, this 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 at a 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? Remain in constant units of time and do not switch to reciprocal units. 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, 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 at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data 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 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 “mind 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. Let us apply mathematical set theory to 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 dirt, crystal structure and atomic arrangement of 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 the digits of a 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, it means it 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 a fool who doesn’t know physics. She just has a strong stereotype of perceiving 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.

A prism is one of the three-dimensional figures whose properties are studied at school in the course of spatial geometry. In this article we will consider a specific prism - a hexagonal one. What kind of figure is this, how to find the volume of a regular hexagonal prism and its surface area? The answers to these questions are contained in the article.

Prism figure

Suppose that we have an arbitrary polygon with the number of sides n, which is located in some plane. For each vertex of this polygon we will construct a vector that will not lie in the plane of the polygon. Using this operation, we will obtain n identical vectors, the vertices of which form a polygon exactly equal to the original one. A figure bounded by two identical polygons and parallel lines connecting their vertices is called a prism.

The faces of the prism are two bases, represented by polygons with n sides, and n side parallelogram surfaces. The number of edges P of a figure is related to the number of its vertices B and faces G by Euler’s formula:

For a polygon with n sides, we get n + 2 faces and 2 * n vertices. Then the number of edges will be equal to:

P = B + G - 2 = 2 * n + n + 2 - 2 = 3 * n

The simplest prism is triangular, that is, its base is a triangle.

The classification of prisms is quite diverse. So, they can be regular and irregular, rectangular and oblique, convex and concave.

Hexagonal prism

This article is devoted to the issue of the volume of a regular hexagonal prism. First, let's take a closer look at this figure.

As the name suggests, the base of a hexagonal prism is a polygon with six sides and six angles. In the general case, a great variety of such polygons can be made, but for practice and for solving geometric problems, one single case is important - a regular hexagon. All its sides are equal to each other, and each of the 6 angles is 120 o. This polygon can be easily constructed by dividing the circle into 6 equal parts with three diameters (they should intersect at angles of 60 o).

A regular hexagonal prism presupposes not only the presence of a regular polygon at its base, but also the fact that all sides the shapes must be rectangles. This is only possible if the side faces are perpendicular to the hexagonal bases.

A regular hexagonal prism is a fairly perfect figure that is found in everyday life and nature. One has only to think about the shape of a honeycomb or a hex wrench. Hexagonal prisms are also common in the field of nanotechnology. For example, crystal lattices HPU and C32, which are realized under certain conditions in titanium and zirconium, as well as the graphite lattice, have the shape of hexagonal prisms.

Surface area of ​​a hexagonal prism

Let us now move directly to the issue of calculating the area and volume of the prism. First, let's calculate the surface area of ​​this figure.

The surface area of ​​any prism is calculated using the following equation:

That is, the required area S is equal to the sum of the areas of the two bases S o and the area of ​​the lateral surface S b . To determine the value of S o, you can proceed in two ways:

• Calculate it yourself. To do this, the hexagon is divided into 6 equilateral triangles. Knowing that the area of ​​one triangle is equal to half the product of the height and the base (the length of the side of the hexagon), you can find the area of ​​the polygon in question.
• Take advantage well-known formula. It is shown below:

S n = n / 4 * a 2 * ctg(pi / n)

Here a is the side length of a regular polygon with n vertices.

Obviously, both methods lead to the same result. For a regular hexagon, the area is:

S o = S 6 = 3 * √3 * a 2 / 2

It’s easy to find the lateral surface area; to do this, multiply the base of each rectangle a by the height of the prism h, multiply the resulting value by the number of such rectangles, that is, by 6. As a result:

Using the formula for the total surface area, for a regular hexagonal prism we obtain:

S = 3 * √3 * a 2 + 6 * a * h = 3 * a * (√3 * a + 2 * h)

How to find the volume of a prism?

Volume is physical quantity, which reflects the area of ​​space occupied by the object. For a prism, this value can be calculated using the following formula:

This expression answers the question of how to find the volume of a prism of arbitrary shape, that is, it is necessary to multiply the base area S o by the height of the figure h (the distance between the bases).

Note that the above expression is valid for any prism, including concave and oblique figures formed by irregular polygons at the base.

Formula for the volume of a hexagonal regular prism

On this moment we have considered all the necessary theoretical calculations to obtain an expression for the volume of the prism in question. To do this, it is enough to multiply the area of ​​the base by the length of the side edge, which is the height of the figure. As a result, the hexagonal prism will take the form:

V = 3 * √3 * a 2 * h / 2

Thus, calculating the volume of the prism in question requires knowledge of only two quantities: the length of the side of its base and the height. These two quantities uniquely determine the volume of the figure.

Comparison of volumes and cylinder

It was said above that the base of a hexagonal prism can be easily constructed using a circle. It is also known that if you increase the number of sides of a regular polygon, its shape will approach a circle. In this regard, it is of interest to calculate how much the volume of a regular hexagonal prism differs from this value for a cylinder.

To answer this question, you need to calculate the side length of a hexagon inscribed in a circle. It can be easily shown that it is equal to the radius. Let us denote the radius of the circle by the letter R. Let us assume that the height of the cylinder and the prism is equal to a certain value h. Then the volume of the prism is equal to next value:

V p = 3 * √3 * R 2 * h / 2

The volume of a cylinder is determined by the same formula as the volume for an arbitrary prism. Considering that the area of ​​the circle is equal to pi * R 2, for the volume of the cylinder we have:

Let's find the ratio of the volumes of these figures:

V p / V с = 3 * √3 * R 2 * h / 2 / (pi * R 2 * h) = 3 * √3 / (2 * pi)

Pi is 3.1416. Substituting it, we get:

Thus, the volume of a regular hexagonal prism is about 83% of the volume of the cylinder into which it is inscribed.