In a regular tetrahedron, all dihedral angles at the edges and all trihedral angles at the vertices are equal
A tetrahedron has 4 faces, 4 vertices and 6 edges.
The basic formulas for a regular tetrahedron are given in the table.
Where:
S - Surface area of a regular tetrahedron
V - volume
h - height lowered to the base
r - radius of the circle inscribed in the tetrahedron
R - circumradius
a - edge length
Practical examples
Task.Find the surface area of a triangular pyramid with each edge equal to √3
Solution.
Since all the edges of a triangular pyramid are equal, it is regular. The surface area of a regular triangular pyramid is S = a 2 √3.
Then
S = 3√3
Answer: 3√3
Task.
All edges of a regular triangular pyramid are equal to 4 cm. Find the volume of the pyramid 
Solution.
Since in a regular triangular pyramid the height of the pyramid is projected to the center of the base, which is also the center of the circumscribed circle, then
AO = R = √3 / 3 a
AO = 4√3 / 3
So the height of the pyramid OM can be found from the right triangle AOM
AO 2 + OM 2 = AM 2
OM 2 = AM 2 - AO 2
OM 2 = 4 2 - (4√3 / 3) 2
OM 2 = 16 - 16/3
OM = √(32/3)
OM = 4√2 / √3
We find the volume of the pyramid using the formula V = 1/3 Sh
In this case, we find the area of the base using the formula S = √3/4 a 2
V = 1/3 (√3 / 4 * 16) (4√2 / √3)
V = 16√2/3
Answer: 16√2 / 3 cm
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?
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.
The tetrahedron is the simplest polygon figure. It consists of four faces, each of which is an equilateral triangle, with each side connected to the other by just one face. When studying the properties of this three-dimensional geometric figure, for clarity, it is best to make a tetrahedron model from paper.
How to glue a tetrahedron from paper?
To build a simple tetrahedron from paper we will need:
- the paper itself (thick, you can use cardboard);
- protractor;
- ruler;
- scissors;
- glue;
- paper tetrahedron, diagram.
Progress
- if the paper is very thick, then you should use a hard object, for example, the edge of a ruler, along the folds;
- in order to get a multi-colored tetrahedron, you can paint the edges or perform a scan on sheets of colored paper.
How to make a tetrahedron out of paper without gluing?
We present to your attention a master class that tells you how to assemble 6 paper tetrahedrons into a single module using the origami technique.
We will need:
- 5 pairs of square sheets of paper in various colors;
- scissors.
Progress
- We divide each sheet of paper into three equal parts, cut it and get strips, the aspect ratio of which is 1 to 3. As a result, we get 30 strips, from which we will fold the module.
- Place the strip face down in front of you, stretched horizontally. We bend it in half, unfold it and fold it towards the middle of the edge.
- On the far right edge, bend the corner so as to make an arrow, moving it 2-3 cm from the edge.
- We bend the left corner in the same way (photo of how to make a tetrahedron out of paper 3).
- We bend the upper right corner of the small triangle that resulted from the previous operation. This way, the sides of the folded edge will be at the same angle.
- Unfold the resulting fold.
- We unfold the left corner and, following the existing fold lines, turn the corner inward as shown in the photo.
- In the right corner, bend the top edge down so that it intersects with the fold made during operation No. 3.
- We turn the outer edge to the right again, using the fold made as a result of operation No. 3.
- We repeat the previous operations from the other end of the strip, but so that the small folds are at the parallel ends of the strip.
- We fold the resulting strip in half lengthwise and let it open spontaneously. The exact opening angle will become clear later, during final assembly of the model. The element is ready, now we do 29 more in the same way.
- We turn the link over so that its outer side is visible during assembly. We connect the two links by inserting the tongue into the pocket formed by the small inner corner.
- The connected links should form an angle of 60⁰, at which other links will be joined (photo of how to make a tetrahedron from paper 13).
- We add the third link to the second, and connect the second to the first. It turns out the end of the figure, at the top of which all three of its links are connected.
- Similarly, add three more links. The first tetrahedron is ready.
- The corners of the finished figure may not be exactly the same, so for a more accurate fit, individual corners of all subsequent tetrahedrons should be left open.
- The tetrahedrons should be connected to each other so that the corner of one passes through the hole in the other.
- Three tetrahedrons connected to each other.
- Four interconnected tetrahedrons.
- The module of five tetrahedrons is ready.
If you have mastered the tetrahedron, you can continue and make
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.