### A triangular pyramid with transformations

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# Geometric textbook "Polyhedra and sweep"    ''Polyhedrons and developments'' – this is a new set of teaching geometrical models, which is intended for demonstration of various polyhedrons and their developments. The set is arranged so that allows from each model of a polyhedron easily to obtain all types of its developments, and this in its turn gives the chance visually and logically to master inobtaining of the developments from polyhedrons and vice versa – from developments of polyhedrons.
''Polyhedrons and development''complement the geometric transformable models “Nane”. “Nane” models allow receiving numerous transformations of geometrical models, but with their help it is impossible to show the developments of stereometric bodies. The set of the models ''Polyhedrons and developments'' fills this gap and allows to demonstrate the geometric bodies. The models of a new set are arranged so that it is possible to place in them “Nane” models and combining both models, to show different opportunities of geometrical figures and bodies. Two main details of the components of a new set of models are the triangle and the square, which are connectedto each other with the help of a special lock. The set consists of ten triangles and ten squares,with the help ofwhich it is possible to obtain:

1. Triangular pyramid (tetrahedron) and cut-out shapes.
2. Quadrangular pyramid and cut-out shapes.
3. Pentagonal pyramid.
4. Triangular prism and cut-out shapes.
5. Abridged triangular pyramid and cut-out shapes.
6. Cube and cut-out shapes.
7. Truncated triangular prism and cut-out shapes.
8. Truncated rectangular prism and cut-out shapes.
9. Rectangular parallelepiped and cut-out shapes.
10. Correct octahedron (octahedron) and cut-out shapes.
11. Icosahedron (icosahedra) and cut-out shapes.
12. Variousstarpolyhedron.

Receiving different models of the triangular pyramid, for example, is as follows:

Method I.

We take four triangles, we connect so that to obtain a triangular pyramid.

Method II.

At first we display the development/cut-out shape of this model on a table, and then we receive the required model.

The originality of models is that from each one it is possible to receive all developments/cut-out shapes of this model. By means of these models it is possible to make and solve interesting tasks. Forexample:

Choose from the figures those, which are considered as the developments/cut-out shapes of a cube. To designate the tops of a triangular pyramid ABCD with the letters A, B, C and D, so that the triangular pyramid (octahedron) is turned out. The figure depicts all developments of the correct octahedron (octahedron) EABCDF. The tops of the octahedron are designated by letters E, A, B, C, D and F, so as to obtain an octahedron. By means of this set the educational process can be organized with high efficiency. For this purpose at first the pupils are given a task to draw the development of any polyhedron, then to carry out an inspection of correctness of the drawing (developments) with application of models. The training provided in such a way allows to acquire practical skills of receiving spatial bodies, and visually to display this process. It is also possible to carry out competitions on speed of assemblage of different polyhedrons, suggesting pupils to collect, for example, an icosahedron (icosahedra) and to receive its development.
Using these models in the course of geometry teaching, (especially stereometry), the teacher can reach more effective result, than working without them. The set can be applied as the constructor, it is also useful for pupils of all classes, and the use of models in pre-school groups can be organized in the form of geometrical games.

Thanks to the transformable qualities, the modelsgenerate an interest in geometry, promote logical and constructive thinking, and develop spatial imagination.

The possibilities to get new polygons are almost unlimited, wherein you can see for yourself.

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# Geometric educational instructional manual "Triangular Pyramid" and main transformations

Before starting working with the models it is necessary to be aware that for manipulations and transformations it is needed to work consistently only with each side separately. For the implementation of each of the following transformations it is needed to close consistently each of the sides up to the stop.

Getting the geometric figures out of triangular pyramid model

### To implement the conversion it is necessary to close consistently each side of a triangular pyramid ABCD up to the stop and get the correct triangular pyramid (tetrahedron).

1. Romb–is obtained from the original pyramid through BD side stretching/pulling 2.
Quadrat- is obtained from therhomb through AC side stretching/pulling and DB shortening 3. Trapezoiis obtained through stretching/pulling of the sides AB, as a result the diagonal AC and BD are stretched themselves d
4. Parallelogramisobtainedthroughstretching/pullingofthesidesDCandthediagonalAC. Here it is possible to demonstrate operation on vectors (addition and subtraction). 5.  Rectangle–is obtained stretching/pulling BD and shortening AC 6.  Triangle

а) Close each side of the model sequentially up to the stop - the starting point is obtained
б) All sides of the base of pyramid ABC should be stretched/pulled until the edges DA, DB, DC will fit the four centers lie in a planeof the triangle ABC

7.  Test of equality of triangles

AD must be disconnected at point D, and the AC and BD have to be stretched/pulled until the top of D does not coincide with the vertex A.
Watch the video "A triangular pyramid" with 0:56 seconds

8.  The characteristics of an isosceles triangle Starting position –is tetrahedron. In the beginning it is necessary to obtain a rhomb, and thenstretch/pull the rods CB, AB and DB.

9.  Pyramid

The pyramid can be obtained from the triangle (point 6) The top/vertex D starts to move away from the tops/vertexes A, B and C a triangular pyramid is being formed.

9.1 The correct triangular pyramid DА=DВ=DС и АВ=ВС=АС. 9.2 The projection of the triangle DAC in the plane of ABC is obtained by stretching DA and DC until DB does not become perpendicular to the plane of ABC.

9.3 The theorem on three perpendiculars/verticals. It is necessary to stretch/pull the BC and AC until the side of the CA will not be perpendicular to AB. Studying the theorem of three perpendiculars, the pupils are asked to convert the four triangles of the pyramid ABCD into the right triangles which is not that simple. When repeated attempts of pupils are unsuccessful, the teacher, being inferior in resourcefulness to pupils, but superior in knowledge, shows the miracle about three perpendiculars. 9.4

1.1   The planes (DCA) and (DBC) are perpendicular to the plane ABC. From this it follows that the line of intersection of the planes DC is perpendicular to the plane of ABC.

10A solution of the problem

Task. Given: a pyramid with ribs/links a, b, c, which are mutually perpendicular.Findthevolumeof the pyramid. 