Properties of the tetrahedron, types and formulas. Volume of a tetrahedron Regular tetrahedron drawing

Tetrahedron translated from Greek means “tetrahedron”. This geometric figure has four faces, four vertices and six edges. The faces are triangles. In fact, the tetrahedron is the first mention of polyhedra appeared long before the existence of Plato.

Today we’ll talk about the elements and properties of the tetrahedron, and also learn the formulas for finding the area, volume and other parameters of these elements.

Elements of a tetrahedron

A segment drawn from any vertex of a tetrahedron and dropped to the point of intersection of the medians of the opposite face is called a median.

The height of a polygon is a normal segment drawn from the opposite vertex.

A bimedian is a segment connecting the centers of intersecting edges.

Properties of the tetrahedron

1) Parallel planes that pass through two intersecting edges form a circumscribed parallelepiped.

2) A distinctive property of a tetrahedron is that the medians and bimedians of the figure meet at one point. It is important that the latter divides the medians in a ratio of 3:1, and bimedians - in half.

3) A plane divides a tetrahedron into two parts of equal volume if it passes through the middle of two intersecting edges.

Types of tetrahedron

The species diversity of the figure is quite wide. A tetrahedron can be:

• regular, that is, at the base an equilateral triangle;
• isohedral, in which all faces are the same in length;
• orthocentric, when the heights have a common intersection point;
• rectangular if the plane angles at the vertex are normal;
• proportionate, all bi heights are equal;
• frame if there is a sphere that touches the ribs;
• incentric, that is, the segments dropped from the vertex to the center of the inscribed circle of the opposite face have a common point of intersection; this point is called the center of gravity of the tetrahedron.

Let us dwell in detail on the regular tetrahedron, the properties of which are practically the same.

Based on the name, you can understand that it is called so because the faces are regular triangles. All the edges of this figure are congruent in length, and the faces are congruent in area. A regular tetrahedron is one of five similar polyhedra.

Tetrahedron formulas

The height of a tetrahedron is equal to the product of the root of 2/3 and the length of the edge.

The volume of a tetrahedron is found in the same way as the volume of a pyramid: the square root of 2 divided by 12 and multiplied by the length of the edge in the cube.

The remaining formulas for calculating the area and radii of circles are presented above.

Consider an arbitrary triangle ABC and a point D not lying in the plane of this triangle. Let's connect this point with the vertices of triangle ABC using segments. As a result, we get triangles ADC, CDB, ABD. The surface bounded by four triangles ABC, ADC, CDB and ABD is called a tetrahedron and is designated DABC.
The triangles that make up a tetrahedron are called its faces.
The sides of these triangles are called the edges of the tetrahedron. And their vertices are the vertices of a tetrahedron

The tetrahedron has 4 faces, 6 ribs And 4 peaks.
Two edges that do not have a common vertex are called opposite.
Often, for convenience, one of the faces of a tetrahedron is called basis, and the remaining three faces are side faces.

Thus, a tetrahedron is the simplest polyhedron whose faces are four triangles.

But it is also true that any arbitrary triangular pyramid is a tetrahedron. Then it is also true that a tetrahedron is called a pyramid with a triangle at its base.

Height of tetrahedron called a segment that connects a vertex with a point located on the opposite face and perpendicular to it.
Median of a tetrahedron called a segment that connects a vertex to the point of intersection of the medians of the opposite face.
Bimedian of a tetrahedron called a segment that connects the midpoints of the intersecting edges of a tetrahedron.

Since a tetrahedron is a pyramid with a triangular base, the volume of any tetrahedron can be calculated using the formula

• S– area of ​​any face,
• H– height lowered to this face

Regular tetrahedron - a special type of tetrahedron

A tetrahedron in which all faces are equilateral is called a triangle. correct.
Properties of a regular tetrahedron:

• All edges are equal.
• All plane angles of a regular tetrahedron are 60°
• Since each of its vertices is the vertex of three regular triangles, the sum of the plane angles at each vertex is 180°
• Any vertex of a regular tetrahedron is projected into the orthocenter of the opposite face (at the point of intersection of the altitudes of the triangle).

Let us be given a regular tetrahedron ABCD with edges equal to a. DH is its height.
Let us make additional constructions BM - the height of the triangle ABC and DM - the height of the triangle ACD.
The height of BM is equal to BM and is equal to
Consider the triangle BDM, where DH, which is the height of the tetrahedron, is also the height of this triangle.
The height of the triangle dropped to side MB can be found using the formula

, Where
BM=, DM=, BD=a,
p=1/2 (BM+BD+DM)=
Let's substitute these values ​​into the height formula. We get


Let's take out 1/2a. We get

Let's apply the difference of squares formula

After small transformations we get


The volume of any tetrahedron can be calculated using the formula
,
Where ,

Substituting these values, we get

Thus, the volume formula for a regular tetrahedron is

Where a–tetrahedron edge

Calculating the volume of a tetrahedron if the coordinates of its vertices are known

Let us be given the coordinates of the vertices of the tetrahedron

From the vertex we draw the vectors , , .
To find the coordinates of each of these vectors, subtract the corresponding beginning coordinate from the end coordinate. We get

TEXT TRANSCRIPT OF THE LESSON:

Good afternoon We continue to study the topic: “Parallelism of lines and planes.”

I think it is already clear that today we will talk about polyhedra - the surfaces of geometric bodies made up of polygons.

Namely about the tetrahedron.

We will study polyhedra according to plan:

1. definition of tetrahedron

2. elements of the tetrahedron

3. development of a tetrahedron

4. image on a plane

1. construct triangle ABC

2. point D not lying in the plane of this triangle

3. connect point D with segments to the vertices of triangle ABC. We get triangles DAB, DBC and DCA.

Definition: A surface made up of four triangles ABC, DAB, DBC and DCA is called a tetrahedron.

Designation: DABC.

Elements of a tetrahedron

The triangles that make up a tetrahedron are called faces, their sides are edges, and their vertices are called vertices of the tetrahedron.

How many faces, edges and vertices does a tetrahedron have?

A tetrahedron has four faces, six edges and four vertices

Two edges of a tetrahedron that do not have common vertices are called opposite.

In the figure, the edges AD and BC, BD and AC, CD and AB are opposite.

Sometimes one of the faces of a tetrahedron is isolated and called its base, and the other three are called side faces.

Development of a tetrahedron.

To make a tetrahedron from paper you will need the following development:

it needs to be transferred to thick paper, cut out, folded along the dotted lines and glued.

On a plane, a tetrahedron is depicted

In the form of a convex or non-convex quadrangle with diagonals. In this case, invisible edges are depicted with dashed lines.

In the first picture, AC is an invisible edge,

on the second - EK, LK and KF.

Let's solve several typical tetrahedron problems:

Find the development area of ​​a regular tetrahedron with an edge of 5 cm.

Solution. Let's draw the development of a tetrahedron

(a tetrahedron scan appears on the screen)

This tetrahedron consists of four equilateral triangles, therefore, the development area of ​​a regular tetrahedron is equal to the area of ​​the total surface of the tetrahedron or the area of ​​four regular triangles.

We find the area of ​​a regular triangle using the formula:

Then we get the area of ​​the tetrahedron equal to:

Let us substitute the length of the edge a = 5 cm into the formula,

it turns out

Answer: Development area of ​​a regular tetrahedron

Construct a section of the tetrahedron with a plane passing through points M, N and K.

a) Indeed, let us connect points M and N (belonging to the face ADC), points M and K (belonging to the face ADB), points N and K (faces DBC). The cross section of the tetrahedron is the triangle MKN.

b) Connect points M and K (belong to faces ADB), points K and N (belong to faces DCB), then continue lines MK and AB until they intersect and place point P. Line PN and point T lie in the same plane ABC and now we can construct the intersection of the straight line MK with each face. The result is a quadrilateral MKNT, which is the desired section.

All its faces are equal triangles. The development of an isohedral tetrahedron is a triangle divided by three midlines into four equal triangles. In an isohedral tetrahedron, the bases of the heights, the midpoints of the heights and the intersection points of the heights of the faces lie on the surface of one sphere (a sphere of 12 points) (An analogue of the Euler circle for a triangle).

Properties of an isohedral tetrahedron:

• All its faces are equal (congruent).
• Crossing edges are equal in pairs.
• Trihedral angles are equal.
• Opposite dihedral angles are equal.
• Two plane angles resting on the same edge are equal.
• The sum of the plane angles at each vertex is 180°.
• The development of a tetrahedron is a triangle or parallelogram.
• The described parallelepiped is rectangular.
• The tetrahedron has three axes of symmetry.
• The common perpendiculars of crossing edges are perpendicular in pairs.
• The midlines are perpendicular in pairs.
• The perimeters of the faces are equal.
• The areas of the faces are equal.
• The heights of the tetrahedron are equal.
• The segments connecting the vertices with the centers of gravity of opposite faces are equal.
• The radii of the circles circumscribed about the faces are equal.
• The center of gravity of the tetrahedron coincides with the center of the circumscribed sphere.
• The center of gravity coincides with the center of the inscribed sphere.
• The center of the circumscribed sphere coincides with the center of the inscribed sphere.
• The inscribed sphere touches the faces at the centers of the circles circumscribed about these faces.
• The sum of the outer unit normals (unit vectors perpendicular to the faces) is zero.
• The sum of all dihedral angles is zero.

Orthocentric tetrahedron

All heights dropped from vertices to opposite faces intersect at one point.

Properties of an orthocentric tetrahedron:

• The altitudes of the tetrahedron intersect at one point.
• The bases of the altitudes of the tetrahedron are the orthocenters of the faces.
• Every two opposite edges of a tetrahedron are perpendicular.
• The sums of the squares of opposite edges of a tetrahedron are equal.
• The segments connecting the midpoints of opposite edges of the tetrahedron are equal.
• The products of the cosines of opposite dihedral angles are equal.
• The sum of the squares of the areas of the faces is four times less than the sum of the squares of the products of opposite edges.
• U orthocentric tetrahedron The 9-point circles (Euler circles) of each face belong to one sphere (24-point sphere).
• U orthocentric tetrahedron the centers of gravity and the points of intersection of the heights of the faces, as well as the points dividing the segments of each height of the tetrahedron from the vertex to the point of intersection of the heights in a ratio of 2: 1, lie on one sphere (sphere of 12 points).

Rectangular tetrahedron

All edges adjacent to one of the vertices are perpendicular to each other. A rectangular tetrahedron is obtained by cutting off the tetrahedron with a plane from a cuboid.

Frame tetrahedron

This is a tetrahedron meeting any of the following conditions:

• there is a sphere touching all edges,
• the sums of the lengths of the crossing edges are equal,
• the sums of dihedral angles at opposite edges are equal,
• circles inscribed in faces touch in pairs,
• all quadrilaterals resulting from the development of a tetrahedron are described,
• perpendiculars raised to the faces from the centers of the circles inscribed in them intersect at one point.

Commensurate tetrahedron

Properties of a commensurate tetrahedron:

• The bi-heights are equal. The bialtitudes of a tetrahedron are the common perpendiculars to two of its intersecting edges (edges that do not have common vertices).
• Projection of a tetrahedron onto a plane perpendicular to any bimedians, there is a rhombus. Bimedians A tetrahedron is called the segments connecting the midpoints of its intersecting edges (which do not have common vertices).
• The faces of the described parallelepiped are equal in size.
• The following relations hold: $4a^2(a\_1)^2-\; (b^2+(b\_1)^2-c^2-(c\_1)^2)^2=4b^2(b\_1)^2-\; (c^2+(c\_1)\; ^2-a^2-(a\_1)^2)^2=4c^2(c\_1)^2-\; (a^2+(a\_1)^2-b^2-(b\_1)^2)^2$, Where $a$ And $a\_1$, $b$ And $b\_1$, $c$ And $c\_1$- lengths of opposite ribs.
• For each pair of opposite edges of a tetrahedron, the planes drawn through one of them and the middle of the second are perpendicular.
• A sphere can be inscribed into the described parallelepiped of a commensurate tetrahedron.

Incentric tetrahedron

In this type, the segments connecting the vertices of the tetrahedron with the centers of circles inscribed in opposite faces intersect at one point. Properties of an incentric tetrahedron:

• The segments connecting the centers of gravity of the faces of the tetrahedron with opposite vertices (medians of the tetrahedron) always intersect at one point. This point is the center of gravity of the tetrahedron.
• Comment. If in the last condition we replace the centers of gravity of the faces with the orthocenters of the faces, then it will turn into a new definition orthocentric tetrahedron. If we replace them with the centers of circles inscribed in the faces, sometimes called incenters, we get the definition of a new class of tetrahedra - incentric.
• The segments connecting the vertices of the tetrahedron with the centers of circles inscribed on opposite faces intersect at one point.
• The bisectors of the angles of two faces drawn to the common edge of these faces have a common base.
• The products of the lengths of opposite edges are equal.
• The triangle formed by the second intersection points of three edges emerging from one vertex with any sphere passing through the three ends of these edges is equilateral.

Regular tetrahedron

This is an isohedral tetrahedron, all of whose faces are regular triangles. It is one of Plato's five solids.

Properties of a regular tetrahedron:

• all edges of the tetrahedron are equal to each other,
• all faces of a tetrahedron are equal to each other,
• the perimeters and areas of all faces are equal.
• A regular tetrahedron is simultaneously orthocentric, frame, equilateral, incentric and proportional.
• A tetrahedron is regular if it belongs to any two of the following types of tetrahedra: orthocentric, frame, incentric, proportional, isohedral.
• A tetrahedron is regular if it is isohedral and belongs to one of the following types of tetrahedra: orthocentric, frame, incentric, proportional.
• An octahedron can be inscribed into a regular tetrahedron, moreover, four (out of eight) faces of the octahedron will be combined with four faces of the tetrahedron, all six vertices of the octahedron will be combined with the centers of six edges of the tetrahedron.
• A regular tetrahedron consists of one inscribed octahedron (in the center) and four tetrahedra (at the vertices), and the edges of these tetrahedra and the octahedron are half the size of the edges of the regular tetrahedron.
• A regular tetrahedron can be inscribed into a cube in two ways, with the four vertices of the tetrahedron aligned with the four vertices of the cube.
• A regular tetrahedron can be inscribed into an icosahedron, moreover, the four vertices of the tetrahedron will be combined with the four vertices of the icosahedron.
• The crossing edges of a regular tetrahedron are mutually perpendicular.

Volume of a tetrahedron

• The volume of a tetrahedron (taking into account the sign), the vertices of which are located at the points $\backslash mathbf(r)\_1\; (x\_1,y\_1,z\_1),$ $\backslash mathbf(r)\_2\; (x\_2,y\_2,z\_2),$ $\backslash mathbf(r)\_3\; (x\_3,y\_3,z\_3),$ $\backslash mathbf(r)\_4\; (x\_4,y\_4,z\_4),$ equals

$V\; =\; \backslash frac16$

\begin(vmatrix) 1 & x_1 & y_1 & z_1 \\ 1 & x_2 & y_2 & z_2 \\ 1 & x_3 & y_3 & z_3 \\ 1 & x_4 & y_4 & z_4 \end(vmatrix) = \frac16 \begin( vmatrix) x_2 - x_1 & y_2 - y_1& z_2 - z_1\\ x_3 - x_1 & y_3 - y_1& z_3 - z_1\\ x_4 - x_1 & y_4 - y_1& z_4 - z_1 \end(vmatrix), or

$V\; =\; \backslash frac(1)(3)\backslash \; S\; H,$

Where $S$ is the area of ​​any face, and $H$– the height lowered to this face.

• The volume of a tetrahedron in terms of edge lengths is expressed using the Cayley-Menger determinant:

$288\; \backslash cdot\; V^2\; =$

0 & 1 & 1 & 1 & 1 \\ 1 & 0 & d_(12)^2 & d_(13)^2 & d_(14)^2 \\ 1 & d_(12)^2 & 0 & d_( 23)^2 & d_(24)^2 \\ 1 & d_(13)^2 & d_(23)^2 & 0 & d_(34)^2 \\ 1 & d_(14)^2 & d_( 24)^2 & d_(34)^2 & 0

\end(vmatrix).

• This formula has a flat analogue for the area of ​​a triangle in the form of a variant of Heron's formula through a similar determinant.
• Volume of a tetrahedron through the lengths of two opposite edges a And b, like crossing lines that are spaced apart h from each other and form an angle with each other $\backslash phi$, is found by the formula:

$V\; =\; \backslash frac(1)(6)\; ab\; h\; \backslash sin\; \backslash phi\; .$

$V\; =\; \backslash frac(1)(3)\backslash \; abc\; \backslash sqrt\; (D)\; ,$

Where $$D=\backslash begin(vmatrix)$$

1 & \cos \gamma & \cos \beta \\ \cos \gamma & 1 & \cos \alpha \\ \cos \beta & \cos \alpha & 1 \end(vmatrix).

• The analogue for the plane of the last formula is the formula for the area of ​​a triangle in terms of the lengths of its two sides a And b, emerging from one vertex and forming an angle between themselves $\backslash gamma$:

$S\; =\; \backslash frac(1)(2)\backslash \; ab\; \backslash sqrt\; (D)\; ,$

Where $D=\backslash begin(vmatrix)$

1 & \cos \gamma \\ \cos \gamma & 1 \\ \end(vmatrix).

Tetrahedra in the microcosm

• A regular tetrahedron is formed by sp 3 -hybridization of atomic orbitals (their axes are directed to the vertices of the regular tetrahedron, and the nucleus of the central atom is located in the center of the described sphere of the regular tetrahedron), therefore many molecules in which such hybridization of the central atom takes place have the appearance of this polyhedron
• CH 4 methane molecule
• Sulfate ion SO 4 2-, phosphate ion PO 4 3-, perchlorate ion ClO 4 - and many other ions
• Diamond C is a tetrahedron with an edge equal to 2.5220 angstroms
• Fluorite CaF 2, tetrahedron with edge equal to 3, 8626 angstroms
• Sphalerite, ZnS, tetrahedron with edge equal to 3.823 angstroms
• Complex ions - , 2- , 2- , 2+
• Silicates, whose structures are based on the silicon-oxygen tetrahedron 4-

Tetrahedrons in nature

Some fruits, four of them on one hand, are located at the vertices of a tetrahedron that is close to regular. This design is due to the fact that the centers of four identical balls touching each other are located at the vertices of a regular tetrahedron. Therefore, ball-like fruits form a similar relative arrangement. For example, walnuts can be arranged in this way.

Tetrahedrons in technology

see also

• Simplex - n-dimensional tetrahedron

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Notes

Literature

• Matizen V. E., Dubrovsky. From the geometry of the tetrahedron “Kvant”, No. 9, 1988 P.66.
• Zaslavsky A. A. // Mathematical education, ser. 3 (2004), No. 8, pp. 78-92.

Correct Correct non-convex Convex Formulas, theorems, theories Other

Excerpt characterizing the Tetrahedron

On the fourth day, fires started on Zubovsky Val.
Pierre and thirteen others were taken to Krymsky Brod, to the carriage house of a merchant's house. Walking through the streets, Pierre was choking from the smoke, which seemed to be standing over the entire city. Fires were visible from different directions. Pierre did not yet understand the significance of the burning of Moscow and looked at these fires with horror.
Pierre stayed in the carriage house of a house near the Crimean Brod for four more days, and during these days he learned from the conversation of the French soldiers that everyone kept here expected the marshal's decision every day. Which marshal, Pierre could not find out from the soldiers. For the soldier, obviously, the marshal seemed to be the highest and somewhat mysterious link in power.
These first days, until September 8th, the day on which the prisoners were taken for secondary interrogation, were the most difficult for Pierre.

X
On September 8, a very important officer entered the barn to see the prisoners, judging by the respect with which the guards treated him. This officer, probably a staff officer, with a list in his hands, made a roll call of all the Russians, calling Pierre: celui qui n "avoue pas son nom [the one who does not say his name]. And, indifferently and lazily looking at all the prisoners, he ordered the guard it is proper for the officer to dress and tidy them up before leading them to the marshal. An hour later a company of soldiers arrived, and Pierre and thirteen others were led to the Maiden's Field. The day was clear, sunny after the rain, and the air was unusually clean. Smoke did not settle down as in that day when Pierre was taken out of the guardhouse of Zubovsky Val; smoke rose in columns in the clear air. The fires of the fires were nowhere to be seen, but columns of smoke rose from all sides, and all of Moscow, everything that Pierre could see, was one conflagration. On all sides one could see vacant lots with stoves and chimneys and occasionally the charred walls of stone houses. Pierre looked closely at the fires and did not recognize the familiar quarters of the city. In some places, surviving churches could be seen. The Kremlin, undestroyed, loomed white from afar with its towers and Ivan the Great. Nearby, the dome of the Novodevichy Convent glittered merrily, and the bell of the Gospel was especially loudly heard from there. This announcement reminded Pierre that it was Sunday and the feast of the Nativity of the Virgin Mary. But it seemed that there was no one to celebrate this holiday: everywhere there was devastation from the fire, and from the Russian people there were only occasionally ragged, frightened people who hid at the sight of the French.
Obviously, the Russian nest was ravaged and destroyed; but behind the destruction of this Russian order of life, Pierre unconsciously felt that over this ruined nest his own, completely different, but firm French order had been established. He felt this from the sight of those soldiers walking cheerfully and cheerfully, in regular rows, who escorted him with other criminals; he felt this from the sight of some important French official in a double carriage, driven by a soldier, driving towards him. He felt this from the cheerful sounds of regimental music coming from the left side of the field, and especially he felt and understood it from the list that the visiting French officer read this morning, calling out the prisoners. Pierre was taken by some soldiers, taken to one place or another with dozens of other people; it seemed that they could forget about him, mix him up with others. But no: his answers given during the interrogation came back to him in the form of his name: celui qui n "avoue pas son nom. And under this name, which Pierre was afraid of, he was now being led somewhere, with undoubted confidence written on them faces that all the other prisoners and he were the ones who were needed, and that they were being taken where they were needed. Pierre felt like an insignificant sliver caught in the wheels of an unknown to him, but correctly functioning machine.
Pierre and other criminals were led to the right side of the Maiden's Field, not far from the monastery, to a large white house with a huge garden. This was the house of Prince Shcherbatov, in which Pierre had often visited the owner before and in which now, as he learned from the conversation of the soldiers, the marshal, the Duke of Eckmuhl, was stationed.
They were led to the porch and one by one they were led into the house. Pierre was brought in sixth. Through a glass gallery, a vestibule, and an antechamber, familiar to Pierre, he was led into a long, low office, at the door of which stood an adjutant.
Davout sat at the end of the room above the table, glasses on his nose. Pierre came close to him. Davout, without raising his eyes, was apparently coping with some paper lying in front of him. Without raising his eyes, he quietly asked:
– Qui etes vous? [Who are you?]
Pierre was silent because he was unable to utter words. For Pierre, Davout was not just a French general; for Pierre Davout, he was a man known for his cruelty. Looking at the cold face of Davout, who, like a strict teacher, agreed to have patience for the time being and wait for an answer, Pierre felt that every second of delay could cost him his life; but he didn't know what to say. He did not dare say what he said during the first interrogation; revealing one's rank and position was both dangerous and shameful. Pierre was silent. But before Pierre could decide on anything, Davout raised his head, raised his glasses to his forehead, narrowed his eyes and looked intently at Pierre.
“I know this man,” he said in a measured, cold voice, obviously calculated to frighten Pierre. The cold that had previously run down Pierre's back gripped his head like a vice.
– Mon general, vous ne pouvez pas me connaitre, je ne vous ai jamais vu... [You couldn’t know me, general, I’ve never seen you.]
“C"est un espion russe, [This is a Russian spy,"] Davout interrupted him, addressing another general who was in the room and whom Pierre had not noticed. And Davout turned away. With an unexpected boom in his voice, Pierre suddenly spoke quickly.
“Non, Monseigneur,” he said, suddenly remembering that Davout was a Duke. - Non, Monseigneur, vous n"avez pas pu me connaitre. Je suis un officier militianaire et je n"ai pas quitte Moscow. [No, Your Highness... No, Your Highness, you could not know me. I am a police officer and I have not left Moscow.]
- Votre nom? [Your name?] - repeated Davout.
- Besouhof. [Bezukhov.]
– Qu"est ce qui me prouvera que vous ne mentez pas? [Who will prove to me that you are not lying?]
- Monseigneur! [Your Highness!] - Pierre cried out in a not offended, but pleading voice.
Davout raised his eyes and looked intently at Pierre. They looked at each other for several seconds, and this glance saved Pierre. In this view, apart from all the conditions of war and trial, a human relationship was established between these two people. Both of them in that one minute vaguely experienced countless things and realized that they were both children of humanity, that they were brothers.
At first glance for Davout, who only raised his head from his list, where human affairs and life were called numbers, Pierre was only a circumstance; and, not taking the bad deed into account on his conscience, Davout would have shot him; but now he already saw a person in him. He thought for a moment.
– Comment me prouverez vous la verite de ce que vous me dites? [How will you prove to me the truth of your words?] - Davout said coldly.
Pierre remembered Rambal and named his regiment, his last name, and the street on which the house was located.
“Vous n"etes pas ce que vous dites, [You are not what you say.],” Davout said again.
Pierre, in a trembling, intermittent voice, began to provide evidence of the truth of his testimony.
But at this time the adjutant entered and reported something to Davout.
Davout suddenly beamed at the news conveyed by the adjutant and began to button up. He apparently completely forgot about Pierre.
When the adjutant reminded him of the prisoner, he frowned, nodded towards Pierre and said to be led away. But Pierre didn’t know where they were supposed to take him: back to the booth or to the prepared place of execution, which his comrades showed him while walking along the Maiden’s Field.
He turned his head and saw that the adjutant was asking something again.
- Oui, sans doute! [Yes, of course!] - said Davout, but Pierre didn’t know what “yes” was.
Pierre did not remember how, how long he walked and where. He, in a state of complete senselessness and dullness, not seeing anything around him, moved his legs along with the others until everyone stopped, and he stopped. During all this time, one thought was in Pierre’s head. It was the thought of who, who, finally sentenced him to death. These were not the same people who interrogated him in the commission: not one of them wanted and, obviously, could not do this. It was not Davout who looked at him so humanly. Another minute and Davout would have realized that they were doing something wrong, but this moment was interrupted by the adjutant who entered. And this adjutant, obviously, did not want anything bad, but he might not have entered. Who was it that finally executed, killed, took his life - Pierre with all his memories, aspirations, hopes, thoughts? Who did this? And Pierre felt that it was no one.
It was an order, a pattern of circumstances.
Some kind of order was killing him - Pierre, depriving him of his life, of everything, destroying him.

From the house of Prince Shcherbatov, the prisoners were led straight down along the Devichye Pole, to the left of the Devichye Convent and led to a vegetable garden on which there was a pillar. Behind the pillar there was a large hole dug with freshly dug up earth, and a large crowd of people stood in a semicircle around the pit and the pillar. The crowd consisted of a small number of Russians and a large number of Napoleonic troops out of formation: Germans, Italians and French in different uniforms. To the right and left of the pillar stood fronts of French troops in blue uniforms with red epaulettes, boots and shakos.
The criminals were placed in a certain order, which was on the list (Pierre was sixth), and were led to a post. Several drums suddenly struck from both sides, and Pierre felt that with this sound it was as if part of his soul had been torn away. He lost the ability to think and think. He could only see and hear. And he had only one desire - the desire for something terrible to happen that had to be done as quickly as possible. Pierre looked back at his comrades and examined them.
The two men on the edge were shaven and guarded. One is tall and thin; the other is black, shaggy, muscular, with a flat nose. The third was a street servant, about forty-five years old, with graying hair and a plump, well-fed body. The fourth was a very handsome man, with a thick brown beard and black eyes. The fifth was a factory worker, yellow, thin, about eighteen, in a dressing gown.
Pierre heard that the French were discussing how to shoot - one at a time or two at a time? “Two at a time,” the senior officer answered coldly and calmly. There was movement in the ranks of the soldiers, and it was noticeable that everyone was in a hurry - and they were in a hurry not as they are in a hurry to do something understandable to everyone, but as they are in a hurry to finish a necessary, but unpleasant and incomprehensible task.
A French official in a scarf approached the right side of the line of criminals and read the verdict in Russian and French.
Then two pairs of Frenchmen approached the criminals and, at the officer’s direction, took two guards who were standing on the edge. The guards, approaching the post, stopped and, while the bags were brought, silently looked around them, as a wounded animal looks at a suitable hunter. One kept crossing himself, the other scratched his back and made a movement with his lips like a smile. The soldiers, hurrying with their hands, began to blindfold them, put on bags and tie them to a post.
Twelve riflemen with rifles stepped out from behind the ranks with measured, firm steps and stopped eight steps from the post. Pierre turned away so as not to see what would happen. Suddenly a crash and roar was heard, which seemed to Pierre louder than the most terrible thunderclaps, and he looked around. There was smoke, and the French with pale faces and trembling hands were doing something near the pit. They brought the other two. In the same way, with the same eyes, these two looked at everyone, in vain, with only their eyes, silently, asking for protection and, apparently, not understanding or believing what would happen. They could not believe, because they alone knew what their life was for them, and therefore they did not understand and did not believe that it could be taken away.
Pierre wanted not to look and turned away again; but again, as if a terrible explosion struck his ears, and along with these sounds he saw smoke, someone’s blood and the pale, frightened faces of the French, who were again doing something at the post, pushing each other with trembling hands. Pierre, breathing heavily, looked around him, as if asking: what is this? The same question was in all the glances that met Pierre’s gaze.

In this lesson we will look at the tetrahedron and its elements (tetrahedron edge, surface, faces, vertices). And we will solve several problems on constructing sections in a tetrahedron, using the general method for constructing sections.

Topic: Parallelism of lines and planes

Lesson: Tetrahedron. Problems on constructing sections in a tetrahedron

How to build a tetrahedron? Let's take an arbitrary triangle ABC. Any point D, not lying in the plane of this triangle. We get 4 triangles. The surface formed by these 4 triangles is called a tetrahedron (Fig. 1.). The internal points bounded by this surface are also part of the tetrahedron.

Rice. 1. Tetrahedron ABCD

Elements of a tetrahedron
A,B, C, D - vertices of a tetrahedron.
AB, A.C., AD, B.C., BD, CD - tetrahedron edges.
ABC, ABD, BDC, ADC - tetrahedron faces.

Comment: can be taken flat ABC behind tetrahedron base, and then point D is vertex of a tetrahedron. Each edge of the tetrahedron is the intersection of two planes. For example, rib AB- this is the intersection of planes ABD And ABC. Each vertex of a tetrahedron is the intersection of three planes. Vertex A lies in planes ABC, ABD, ADWITH. Dot A is the intersection of the three designated planes. This fact is written as follows: A= ABC ∩ ABD ∩ ACD.

Tetrahedron definition

So, tetrahedron is a surface formed by four triangles.

Tetrahedron edge- the line of intersection of two planes of the tetrahedron.

Make 4 equal triangles from 6 matches. It is impossible to solve the problem on a plane. And this is easy to do in space. Let's take a tetrahedron. 6 matches are its edges, four faces of the tetrahedron and will be four equal triangles. The problem is solved.

Given a tetrahedron ABCD. Dot M belongs to an edge of the tetrahedron AB, dot N belongs to an edge of the tetrahedron IND and period R belongs to the edge DWITH(Fig. 2.). Construct a section of a tetrahedron with a plane MNP.

Rice. 2. Drawing for problem 2 - Construct a section of a tetrahedron with a plane

Solution:
Consider the face of a tetrahedron DSun. On this face of the point N And P belong to the faces DSun, and therefore the tetrahedron. But according to the condition of the point N, P belong to the cutting plane. Means, NP- this is the line of intersection of two planes: the plane of the face DSun and cutting plane. Let's assume that straight lines NP And Sun not parallel. They lie in the same plane DSun. Let's find the point of intersection of the lines NP And Sun. Let's denote it E(Fig. 3.).

Rice. 3. Drawing for problem 2. Finding point E

Dot E belongs to the section plane MNP, since it lies on the line NP, and the straight line NP lies entirely in the section plane MNP.

Also point E lies in a plane ABC, because it lies on a straight line Sun out of plane ABC.

We get that EAT- line of intersection of planes ABC And MNP, since points E And M lie simultaneously in two planes - ABC And MNP. Let's connect the dots M And E, and continue straight EAT to the intersection with the line AC. Point of intersection of lines EAT And AC let's denote Q.

So in this case NPQМ- the required section.

Rice. 4. Drawing for problem 2. Solution of problem 2

Let us now consider the case when NP parallel B.C.. If straight NP parallel to some line, for example, a straight line Sun out of plane ABC, then straight NP parallel to the entire plane ABC.

The desired section plane passes through the straight line NP, parallel to the plane ABC, and intersects the plane in a straight line MQ. So the line of intersection MQ parallel to the line NP. We get NPQМ- the required section.

Dot M lies on the side ADIN tetrahedron ABCD. Construct a section of the tetrahedron with a plane that passes through the point M parallel to the base ABC.

Rice. 5. Drawing for problem 3 Construct a section of a tetrahedron with a plane

Solution:
Cutting plane φ parallel to the plane ABC according to the condition, this means that this plane φ parallel to lines AB, AC, Sun.
In plane ABD through the point M let's make a direct PQ parallel AB(Fig. 5). Straight PQ lies in a plane ABD. Similarly in the plane ACD through the point R let's make a direct PR parallel AC. Got a point R. Two intersecting lines PQ And PR plane PQR respectively parallel to two intersecting lines AB And AC plane ABC, which means planes ABC And PQR parallel. PQR- the required section. The problem is solved.

Given a tetrahedron ABCD. Dot M- internal point, point on the face of the tetrahedron ABD. N- internal point of the segment DWITH(Fig. 6.). Construct the intersection point of a line N.M. and planes ABC.

Rice. 6. Drawing for problem 4

Solution:
To solve this, we will construct an auxiliary plane DMN. Let it be straight DM intersects line AB at point TO(Fig. 7.). Then, SKD- this is a section of the plane DMN and tetrahedron. In plane DMN lies and straight N.M., and the resulting straight line SK. So if N.M. not parallel SK, then they will intersect at some point R. Dot R and there will be the desired intersection point of the line N.M. and planes ABC.

Rice. 7. Drawing for problem 4. Solution of problem 4

Given a tetrahedron ABCD. M- internal point of the face ABD. R- internal point of the face ABC. N- internal point of the edge DWITH(Fig. 8.). Construct a section of a tetrahedron with a plane passing through the points M, N And R.

Rice. 8. Drawing for problem 5 Construct a section of a tetrahedron with a plane

Solution:
Let us consider the first case, when the straight line MN not parallel to the plane ABC. In the previous problem we found the point of intersection of the line MN and planes ABC. This is the point TO, it is obtained using the auxiliary plane DMN, i.e. we do DM and we get a point F. We carry out CF and at the intersection MN we get a point TO.

Rice. 9. Drawing for problem 5. Finding point K

Let's make a direct KR. Straight KR lies both in the section plane and in the plane ABC. Getting the points P 1 And R 2. Connecting P 1 And M and as a continuation we get the point M 1. Connecting the dot R 2 And N. As a result, we obtain the desired section Р 1 Р 2 NM 1. The problem in the first case is solved.
Let us consider the second case, when the straight line MN parallel to the plane ABC. Plane MNP passes through a straight line MN parallel to the plane ABC and intersects the plane ABC along some straight line R 1 R 2, then straight R 1 R 2 parallel to the given line MN(Fig. 10.).

Rice. 10. Drawing for problem 5. The required section

Now let's draw a straight line R 1 M and we get a point M 1.Р 1 Р 2 NM 1- the required section.

So, we looked at the tetrahedron and solved some typical tetrahedron problems. In the next lesson we will look at a parallelepiped.

1. I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 p. : ill. Geometry. Grades 10-11: textbook for students of general education institutions (basic and specialized levels)

2. Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill. Geometry. Grades 10-11: Textbook for general education institutions

3. E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M.: Bustard, 008. - 233 p. :il. Geometry. Grade 10: Textbook for general education institutions with in-depth and specialized study of mathematics

Additional web resources

2. How to construct a cross section of a tetrahedron. Mathematics ().

3. Festival of pedagogical ideas ().

Do problems at home on the topic “Tetrahedron”, how to find the edge of a tetrahedron, faces of a tetrahedron, vertices and surface of a tetrahedron

1. Geometry. Grades 10-11: textbook for students of general education institutions (basic and specialized levels) I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 pp.: ill. Tasks 18, 19, 20 p. 50

2. Point E mid-rib MA tetrahedron MAVS. Construct a section of the tetrahedron with a plane passing through the points B, C And E.

3. In the tetrahedron MABC, point M belongs to the face AMV, point P belongs to the face BMC, point K belongs to the edge AC. Construct a section of the tetrahedron with a plane passing through the points M, R, K.

4. What shapes can be obtained as a result of the intersection of a tetrahedron with a plane?