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TEXT EXPLANATION OF THE LESSON:
Recall the definition of a prism.
A PRISM is a polyhedron whose two faces (bases) are equal polygons located in parallel planes, and the other faces (sides) are parallelograms.
A prism is called straight when the lateral edges of the prism are perpendicular to the bases.
A right prism is called regular if its bases are regular polygons.
The side faces of the prism are parallelograms.
Let's prove the theorem.
The volume of a straight prism is equal to the base area multiplied by the height.
First we prove the theorem for a triangular prism, and then for an arbitrary one.
Given: straight prism
Prove: V = Sos. h.
Proof:
1. BCDB1C1D1 is a straight prism. AC BD (we choose the height that divides ΔBCD into two triangles) we draw the plane (CAA1) (BCD), we get two prisms, the bases of which are right-angled triangles. Then V1 is the volume of the prism BCAB1C1A1 and is equal to SBCA.h
V2 is the volume of the ACDA1C1D1 prism and is equal to SACD.h
Then the volume of the prism ВСDB1C1D1 will be equal to the sum of the volumes of the prism BCAB1C1A1 and ACDA1C1D1;
And since the sum of the areas of triangles BCA and ACD is equal to the area of triangle BCD, then the volume of the prism will be equal to the product of the height and the base area of BCD. Q.E.D.
2. Consider an arbitrary n-gonal prism with base area S, it can be divided into straight triangular prisms with height h.
Therefore, V1, V2, V3,…,Vn-2 are the volumes of triangular prisms,
S1, S2, S3,…, Sn-2 - base areas of triangular prisms.
This means that the volume of an n-gonal prism will be equal to the sum of the volumes of all triangular prisms.
It follows that the volume will be equal to the product of the height of the prism and the sum of the areas of the bases of triangular prisms.
This convex pentagonal prism can be divided into three straight triangular prisms. Find the volume of each prism and add these volumes. We take out the common factor h, we get that the volume of the pentagonal prism will be equal to the product of the height and the sum of the areas of the bases of the triangular prisms. The sum of the areas of the bases of triangular prisms is equal to the area of the base of this prism, which means that the volume of this prism is equal to the product of the height and the base.
The theorem has been proven.
Problem solving
Find the volume of a regular n-gonal prism with each edge equal to a if a) n=3; b) n=4; c) n=6.d) n=8
Regular n-gonal prism,
a is the edge of the prism.
Since each edge is equal to a by condition, then the height of the prism h in a straight prism, which is the edge of the prism, is also equal to a
The volume of the prism is found by the formula:
The base of a regular n-gonal prism, with n = 3, is a regular triangle, the area of \u200b\u200bwhich is found by the formula.
Then the volume is
b) n = 4, that is, the base is a quadrilateral, and since the prism is regular, then it is a square, and by condition all the edges of the prism are equal, which means that a regular quadrangular prism is a cube, therefore V =
c) n=6. We are looking for the volume of a regular hexagonal prism by the formula:
(this is the formula, since the base is a regular hexagon, its area can only be expressed through side a).
d) n=8. We are looking for the volume of a regular octagonal prism by the formula:
We are looking for the area of \u200b\u200bthe base by the formula:
(this is the formula, since the base is a regular octagon, its area can only be expressed through side a).
Answer: a) V = ; b) V = ;
c) V = 1.5. ; d) V = (2+2) . .
In a regular triangular prism, a section is drawn through the side of the lower base and the vertex of the upper base opposite to it, making an angle of 60 with the plane of the base. Find the volume of the prism if the side is a.
Regular triangular prism with
side a.
Cross section ABC1
Let us construct SC AB, segment C1K in the plane of section AC1B. According to the three perpendiculars theorem -
S1K AB; C1KS=60°.
From ΔC1KS: (the ratio of the sides of the triangle - CC1 to SC is equal to the tangent of 60 degrees and is equal to the square root of three)
Consider the triangle ΔSKB, it is rectangular since the SK is the height drawn to the point K, then by definition of the sine of the acute angle of a right triangle, we have = sin ∠СBК, the angle of the SVK is 60 degrees, since the triangle at the base is regular, which means that all its angles are equal.
CK = BC sin60 °, since BC = a, and the sine of 60 degrees is equal, then,
Then we substitute the value of SC into the formula CC1, we get
And the area of an equilateral triangle is calculated by the formula.
Drawn from the spatial angles of the bases perpendicular to its opposite sides. From the points of their intersection, a vertical line is drawn, which will be the axis prisms. When building trihedral prism you need to choose the right point of view. The subject should be depicted in such a way that it looks three-dimensional, with two visible planes and a front edge slightly offset to the side. trihedral prism with such a turn it will be the most expressive, voluminous and expedient, provided that the object is located in the optimal perspective.
Great difficulties are experienced in determining the values of the segments of the faces in the foreshortening based on prisms. To avoid errors, it is recommended to use an additional circle ( plan, top view), on which, in accordance with the apparent position of the object, the spatial angles of the base are precisely determined prisms. Thus, for the correct prismatic, it is necessary to construct a cylindrical scheme, followed by the construction of facets in it.
Building trihedral prism should start with a horizontal ( it must be carried out strictly horizontally). This makes it possible to correctly determine the position of the surface of the bases of the prism with respect to the axis of the body. After that, a vertical axial line should be drawn. Marking the radius of the base, draw a circle ( ellipse) in perspective (Fig. 39). To correctly determine the spatial points of the corners of the base on the ellipse, it is necessary to draw a circle above it, in accordance with the radius of the ellipse, along one axis. When drawing it, check how correctly it is drawn, since on a distorted circle it will be impossible to accurately determine the spatial points and sizes of the segments of the faces. The correctness of the surface of the base of the prism and the entire object as a whole will largely depend on how correctly they are defined on the circle.
Having accurately determined the apparent position of the points of the spatial angles of the base of the prism on the circle, transfer them to the ellipse. To determine its upper, base, it is necessary to repeat the ellipse, after which, connecting the spatial points of the bases with vertical edges, a trihedral prism is constructed. On a prism, the circumference (ellipse) of the lower base should be somewhat wider than the upper one.
When constructing an object on a plane, one should strictly observe and. For greater expressiveness of its volume-spatial characteristics, the near edges should be highlighted with more contrast, weakening and softening them as they move away. During a long, hours-long lesson, you can gradually get rid of all auxiliary. during the construction process, it should be performed by lightly pressing on, so that as you refine it, you can correct and delete unnecessary.
The sequence of drawing a hexagonal prism
A hexagonal prism is characterized by twelve points of spatial angles of the base and six ribs. Its axis is determined drawn from opposite spatial angles of the base, where the point of their intersection will be the center through which the axis of the prism passes. To correctly determine its spatial angles, as well as when constructing a trihedral prism, it is necessary to start work with constructing an ellipse and a circle under it. In accordance with the apparent position of the object at a given point of view, it is necessary to correctly determine the points of the spatial angles of a regular hexagon on a circle. It is necessary to pay attention to the rotation of the prism, you should not draw a hexagonal prism with a symmetrical arrangement of its planes. Therefore, when choosing a drawing place, you need to sit down so that the object looks the most expressive, voluminous, as, for example, shown in Fig. 40.
the construction of a hexagonal prism is carried out in the same way as with triangular prism. The difficulty lies in the correct determination from a visible position cut edges, their relations. In this case, you should also use the auxiliary circle in the plan at the lower base of the prism, as shown in Fig.40. Having built the circle of the base of the prism, you need to determine six spatial angles along the circle. In this case, it is important to correctly set aside equal segments, taking into account the rotation of the prism, i.e. from a visible position. Connecting the dots with light , it is necessary to trace the parallelism of opposite sides. Having received the points of the spatial angles of the base, in the same way as in the first case, they should be transferred to the lower base of the ellipse. It should be noted that when transferring the spatial angles to the base of the ellipse, take into account reduction of its far half, although these changes are insignificant. The main thing is to prevent the reverse .
By connecting all points on the grounds, proceed to check the work performed. Errors found are corrected without delay. In order to achieve the greatest expressiveness spatial need near vertical and horizontal strengthen the ribs, and weaken the distant ones. If it is necessary to continue work on helpers should be eliminated building with an eraser.
A trihedral pyramid (Fig. 41) is characterized by three points of spatial angles of the base, a point of top and six ribs.
For the right pyramids should begin with the construction of its base, which is similar to the construction of a prismatic . By connecting the points of the spatial angles of the basethe height of the natural model. Then you should connect the top with the spatial corners of the base.
Subsequence drawing p iramides
- First stage.The size of the pyramid and its spatial position, the main proportions of the pyramid, the degree of turn of its faces are determined.
tetrahedral pyramid ( fig.42), in contrast to the trihedral one, is characterized by four points of the spatial angles of the base, a point of the top and eight edges. The constructive axis of the pyramid, similarly to the trihedral one, is determined by the connection of their opposite spatial angles. From the point of intersection, a vertical (axial) line is drawn, on which the point of the top of the pyramid should be indicated. When constructing a pyramid in a horizontal position, attention should be paid to the position of the axis of the pyramid in relation to the center of its base (Fig. 43). In this case, the plane of the base of the pyramid with respect to its constructive axis must be strictly at a right angle, that is, perpendicular, regardless of the position of the object at a given point of view. The structure of the body structure also remains unchanged.
Prismatic polyhedron is a generalization of the prism in spaces of dimension 4 and higher. n-dimensional prismatic polyhedron is constructed from two ( n− 1 )-dimensional polytopes carried over to the next dimension.
Prismatic elements n-dimensional polyhedron are doubled from the elements ( n− 1 )-dimensional polyhedron, then new elements of the next level are created.
Let's take n-dimensional polyhedron with elements f i (\displaystyle f_(i)) (i-dimensional edge, i = 0, ..., n). Prismatic ( n + 1 (\displaystyle n+1))-dimensional polyhedron will have 2 f i + f − 1 (\displaystyle 2f_(i)+f_(-1)) dimension elements i(at f − 1 = 0 (\displaystyle f_(-1)=0), f n = 1 (\displaystyle f_(n)=1)).
By dimensions:
- Take a polygon n peaks and n parties. Get a prism with 2 n pinnacles, 3 n ribs and 2+n (\displaystyle 2+n) faces.
- We take a polyhedron with v peaks, e ribs and f faces. We get a (4-dimensional) prism with 2 v vertices, edges, faces and 2+f (\displaystyle 2+f) cells.
- We take a 4-dimensional polyhedron with v peaks, e ribs f faces and c cells. We get a (5-dimensional) prism with 2 v peaks, 2e+v (\displaystyle 2e+v) ribs 2 f + e (\displaystyle 2f+e)(2-dimensional) faces, 2 c + f (\displaystyle 2c+f) cells and 2+c (\displaystyle 2+c) hypercells.
Uniform prismatic polyhedra
Right n-polytope represented by the Schläfli symbol ( p, q, ..., t), can form a uniform prismatic polyhedron of dimension ( n+ 1 ) represented by the direct product of two Schläfli symbols: ( p, q, ..., t}×{}.
By dimensions:
- A prism from a 0-dimensional polyhedron is a line segment, represented by the empty Schläfli symbol ().
- A prism from a 1-dimensional polyhedron is a rectangle obtained from two line segments. This prism is represented as a product of the Schläfli symbols ()×(). If the prism is a square, the notation can be abbreviated: ()×() = (4).
- a polygonal prism is a 3-dimensional prism obtained from two polygons (one obtained by parallel translation of the other) that are connected by rectangles. From a regular polygon ( p) you can get a homogeneous n-coal prism represented by the product ( p)×(). If a p= 4 , the prism becomes a cube: (4)×() = (4, 3).
- A 4-dimensional prism obtained from two polyhedra (one obtained by parallel translation of the other), with connecting 3-dimensional prismatic cells. From a regular polyhedron ( p, q) one can obtain a homogeneous 4-dimensional prism represented by the product ( p, q)×(). If the polyhedron is a cube and the sides of the prism are also cubes, the prism becomes a tesseract: (4, 3)×() = (4, 3, 3).
Higher-dimensional prismatic polyhedra also exist as direct products of any two polyhedra. The dimension of a prismatic polyhedron is equal to the product of the dimensions of the elements of the product. The first example of such a product exists in 4-dimensional space and is called duoprisms, which are obtained by multiplying two polygons. Regular duoprisms are represented by the symbol ( p}×{ q}.
| Polygon | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mosaic |
More word meanings and translation OCTAGONAL PRISM from English into Russian in English-Russian dictionaries.
What is and translation of OCTAGONAL PRISM from Russian into English in Russian-English dictionaries.
More meanings of this word and English-Russian, Russian-English translations for OCTAGONAL PRISM in dictionaries.
- PRISM - f. prism
Russian-English Dictionary of the Mathematical Sciences - PRISM
Russian-American English Dictionary - PRISM - prism through a prism (rd.) - in the light (of)
English-Russian-English Dictionary of General Vocabulary - Collection of the best dictionaries - PRISM
Russian-English Dictionary of General Subjects - PRISM - prism
Russian Learner's Dictionary - PRISM - well. prism through a prism (rd.) - in the light (of)
Russian-English dictionary - PRISM - well. prism ♢ through a prism (rd.) — in the light (of)
Russian-English Smirnitsky abbreviations dictionary - PRISM - V block, prism, V
Russian-English Dictionary of Mechanical Engineering and Automation of Production - PRISM - husband. prism .. - through a prism
Russian-English Concise Dictionary of General Vocabulary - PRISM - prism, (in the calculation of foundations) wedge
Russian-English Dictionary of Construction and New Construction Technologies - PRISM
British Russian-English Dictionary - PRISM - prism; through ~y (rd.) in the light (of); ~atic prismatic
Russian-English Dictionary - QD - PRISM - husband. prism through the prism of prisms | a - f. prism through ~y (rd.) in the light (of) ~atic prismatic
- PRISM - prism prism
Russian-English Dictionary Socrates - OCTAGONAL INSERT - octagonal insert
Modern Russian-English dictionary of mechanical engineering and production automation - OCTAGONAL STAR - lat. stella octangula
Big Russian-English Dictionary - STELLA OCTANGULA
- SLIDING TRIANGLE - sliding prism; collapse prism
Big English-Russian Dictionary - RUBBLE TOE - stone resistant prism; stone drainage prism
Big English-Russian Dictionary - ROCKFILL TOE - rockfill thrust prism; rockfill drainage prism
Big English-Russian Dictionary - PRISM - noun prism prism prism
Big English-Russian Dictionary - OCTAGONAL PRISM - mat. octagonal prism
Big English-Russian Dictionary - OBLIQUE PRISM - oblique prism, oblique prism
Big English-Russian Dictionary - KNIFE EDGE - 1. reference prism 2. prismatic (knife) support 3. cutting edge of a knife or cutter knife edge prism (scales) > to be ...
Big English-Russian Dictionary - KNIFE-EDGE - noun. 1) knife edge; smth. sharp cutting 2) reference prism (weights, etc.) 3) ridge (mountains, dunes, glaciers, etc.)
Big English-Russian Dictionary - GIB - I noun; reduce from Gilbert cat Syn: tomcat II n.; those. wedge, counter wedge; gib; bar gib arm ≈ …
Big English-Russian Dictionary - EDGE - 1. noun. 1) a) edge, edge; edge, border cutting edge ≈ sharp edge jagged, ragged edge ≈ jagged edge at, ...
Big English-Russian Dictionary - DOWNSTREAM TOE OF DAM - 1. thrust prism of the downstream slope of the dam; drainage prism of the downstream slope of the dam; bottom tooth of the dam 2. sole [lower edge] of the downstream slope ...
Big English-Russian Dictionary - ANALYSER - noun. 1) analyzer (electronic device) 2) tester 3) physical. diffusing prism ∙ Syn: analyzer analyzer tester (physical) diffusing prism …
Big English-Russian Dictionary - STELLA OCTANGULA - lat. octagonal star; stellated octahedron
- PRISM - 1) prism 2) reflective prism 3) prismatic 4) prism. prism of the second order - a prism of the second kind prism over polyhedron - a prism over a polyhedron right truncated ...
English-Russian Scientific and Technical Dictionary - OCTAGONAL PRISM - math. octagonal prism
English-Russian Scientific and Technical Dictionary - OCTAGONAL INSERT - octagonal insert
Modern English-Russian dictionary of mechanical engineering and production automation - OPTICAL - devices in which the radiation of any region of the spectrum (ultraviolet, visible, infrared) is converted (transmitted, reflected, refracted, polarized). Paying tribute to the historical tradition, optical…
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