EntranceRussian-English translation octagonal prism. Prism (geometry) Prism and all its formulas
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 transferred 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 face, 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 with n peaks and n parties. We get a prism with 2 n peaks, 3 n ribs and 2 + n (\displaystyle 2+n) edges.
- We take a polyhedron with v peaks, e ribs and f edges. 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 edges and c cells. We get a (5-dimensional) prism with 2 v peaks, 2 e + 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.
Homogeneous prismatic polyhedra
Correct n-polyhedron represented by the Schläfli symbol ( p, q, ..., t), can form a homogeneous 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 segments. This prism is represented as the product of Schläfli symbols ()×(). If the prism is a square, the notation can be shortened: ()×() = (4).
- A polygonal prism is a 3-dimensional prism obtained from two polygons (one obtained by translating the other in parallel) that are connected by rectangles. From a regular polygon ( p) you can get a homogeneous n-coal prism represented by the product ( p)×(). If 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) we 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 turns into a tesseract: (4, 3)×() = (4, 3, 3).
Prismatic polyhedra of higher dimensions 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 the product of two polygons. Regular duoprisms are represented by the symbol ( p}×{ q}.
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More meanings of the word and translation of OCTAGONAL PRISM from English into Russian in English-Russian dictionaries.
What is and the 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 - Prism
Russian-American English Dictionary - PRISM - prism through a prism (rd.) - in the light (of)
English-Russian-English dictionary general vocabulary- Collection of the best dictionaries - PRISM
Russian-English dictionary of general topics - PRISM - prism
Russian Learner's Dictionary - PRISMA - w. prism through the prism (rd.) - in the light (of)
Russian-English dictionary - PRISMA - w. prism ♢ through the prism (rd.) – in the light (of)
Russian-English Smirnitsky abbreviations dictionary - PRISM - V block, prism, V
Russian-English dictionary of mechanical engineering and production automation - PRISMA - husband. prism .. - through a prism
Russian-English short dictionary in general vocabulary - PRISM - prism, (in foundation calculations) wedge
Russian-English dictionary on construction and new construction technologies - PRISM - Prism
British Russian-English Dictionary - PRISM - prism; through ~y (rd.) in the light (of); ~atic prismatic
Russian-English Dictionary - QD - PRISMA - husband. prism through the prism of prism|a - g. prism through ~y (rd.) in the light (of) ~atic prismatic
- PRISM - prism prism
Russian-English Dictionary Socrates - OCTAGONAL CUTTING PLATE - octagonal insert
Modern Russian-English dictionary of mechanical engineering and production automation - OCTAGONAL STAR - lat. stella octangula
Large Russian-English Dictionary - STELLA OCTANGULA
- SLIDING TRIANGLE - sliding prism; collapse prism
Big English-Russian dictionary - RUBBLE TOE - stone persistent prism; stone drainage prism
Large English-Russian Dictionary - ROCKFILL TOE - rockfill thrust prism; rockfill drainage prism
Large English-Russian Dictionary - PRISM - noun prism prism prism prism
Large English-Russian Dictionary - OCTAGONAL PRISM - mat. octagonal prism
Large English-Russian Dictionary - OBLIQUE PRISM - oblique prism, oblique prism
Large English-Russian Dictionary - KNIFE EDGE - 1. support prism 2. prismatic (knife) support 3. cutting edge of a knife or cutter tip of a knife prism (scales) > to be ...
Large English-Russian Dictionary - KNIFE-EDGE - noun. 1) knife edge; smb. sharp cutting 2) supporting prism (scales, etc.) 3) ridge (mountains, dunes, glacier, etc.)
Large English-Russian Dictionary - GIB - I noun; decrease from Gilbert cat Syn: tomcat II noun; those. wedge, counter wedge; gib; gib arm bar ≈ ...
Large English-Russian Dictionary - EDGE - 1. noun. 1) a) edge, edge; edge, border cutting edge ≈ sharp edge jagged, ragged edge ≈ jagged edge at, ...
Large English-Russian Dictionary - DOWNSTREAM TOE OF DAM - 1. a thrust prism of the downstream slope of a dam; drainage prism of the downstream slope of the dam; bottom tooth of the dam 2. bottom [lower edge] of the bottom slope ...
Large English-Russian Dictionary - ANALYSER - noun 1) analyzer ( electronic device) 2) tester 3) physical. scattering prism ∙ Syn: analyzer analyzer tester (physical) scattering prism ...
Large English-Russian Dictionary - STELLA OCTANGULA - lat. octagonal star; stellated octahedron
- PRISM - 1) prism 2) prism-reflector 3) prismatic 4) prismatic. prism of the second order - prism of the second kind prism over polyhedron - prism over the polyhedron right truncated ...
English-Russian scientific and technical dictionary - OCTAGONAL PRISM - math. octagonal prism
English-Russian scientific and technical dictionary - OCTAGONAL INSERT - octagonal cutting insert
Modern English-Russian dictionary of mechanical engineering and production automation - OPTICAL - devices in which radiation from any region of the spectrum (ultraviolet, visible, infrared) is converted (transmitted, reflected, refracted, polarized). Paying tribute to historical tradition, optical...
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TEXT TRANSCRIPT OF THE LESSON:
Let's remember the definition of a prism.
PRISM - a polyhedron, two faces of which (bases) are equal polygons located in parallel planes, and the other faces (lateral) 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 contain regular polygons.
The lateral faces of the prism are parallelograms.
Let's prove the theorem.
The volume of a straight prism is equal to the product of the area of the base and the height.
First we prove the theorem for a triangular prism, and then for an arbitrary one.
Given: straight prism
Prove: V = Sbas. h.
Proof:
1. ВСDB1C1D1—direct prism. AC BD (we choose the height that divides ΔBCD into two triangles), draw the plane (CAA1) (BCD), we get two prisms, the bases of which are right triangles. Then V1 is the volume of the prism BCAB1C1A1 and is equal to SBCA.h
V2 is the volume of prism ACDA1C1D1 and is equal to SACD.h
Then the volume of the prism ВСDB1C1D1 will be equal to the sum volumes of the prism BCAB1C1A1 and ACDA1C1D1, therefore, V= SBCA.h+ SACD.h we take the common factor out of brackets and find that the volume of the prism will be equal to h (SBCA + SACD)
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 area of the base BCD. Q.E.D.
2. Consider an n-gonal arbitrary 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 - areas of the bases 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 the triangular prisms.
This convex pentagonal prism can be divided into three straight triangular prisms. Let's find the volume of each prism and add these volumes. Let's take the common factor h out of brackets and find that the volume of a 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 a given prism, which means that the volume of a given 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, in which each edge is equal to a, if a) n=3; b) n=4; c) n=6. d) n=8
Regular n-gonal prism,
a-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 regular triangle, the area of which is found by the formula.
Then the volume is equal
b) n=4, that is, the base is a quadrangle, and since the prism is regular, 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, so V=
c) n=6. We find the volume of a regular hexagonal prism using 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 find the volume of a regular octagonal prism using the formula:
We find the area of the base using 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, through the side of the lower base and the opposite vertex of the upper base, a section is drawn that makes an angle of 60 with the plane of the base. Find the volume of the prism if the side is equal to a.
Regular triangular prism with
side a.
Section ABC1 was carried out
Let's construct the SC AB, segment C1K in the section plane AC1B. According to the theorem of three perpendiculars -
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 equal to the square root of three)
Let's consider the triangle ΔSKV, it is rectangular since SK is the height drawn to point K, then by definition of the sine of an acute angle right triangle we have = sin ∠СВК, angle СВК is equal to 60 degrees, since the triangle at the base is regular, which means that all its angles are equal.
CK=ВС sin60°, since ВС=а, 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 equilateral triangle calculated by the formula.
Drawn from the spatial corners of the bases perpendicular to its opposite sides. From the points of their intersection draw a vertical line, which will be the axis prisms. When building trihedral prism it is necessary to choose the right point of view. The subject should be depicted in such a way that it appears three-dimensional, with two visible planes and a front edge slightly offset to the side. Triangular prism with such a rotation it will be the most expressive, voluminous and expedient, provided that the subject is located in the optimal angle.
Great difficulties are experienced when determining the values of segments of faces in foreshortening based on prisms. To avoid errors, it is recommended to use an additional circle ( in plan, top view), on which, in accordance with the visible position of the object, the spatial angles of the base are accurately determined prisms. Thus, for correct prismatic diagrams it is necessary to construct a cylindrical diagram and then construct faceted diagrams in it.
Construction trihedral prism should begin with a horizontal ( it must be carried out strictly horizontally). This makes it possible to correctly determine the position of the surface of the prism bases relative to the axis of the body. After which you should carry out a vertical axial. 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 an 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 edge segments. 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 visible 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, the ellipse should be repeated, after which, connecting the spatial points of the bases with vertical edges, a triangular prism is constructed. On prisms, the circle (ellipse) of the lower base should be slightly wider than the upper one.
When constructing an object on a plane, you should strictly observe and. For greater expressiveness of its volumetric-spatial characteristics, the near edges should be highlighted with more contrast, weakening and softening them as they move away. During a long, many-hour lesson, you can gradually get rid of all auxiliary ones. during the construction process, you should press lightly on , so that as you refine it, you can adjust and delete unnecessary things.
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 corners 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, just as when constructing a trihedral prism, it is necessary to begin work by constructing an ellipse and a circle under it. In accordance with the apparent position of the object from a given point of view, the points of the spatial angles of a regular hexagon should be correctly determined on the 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 place to draw, you need to sit so that the object looks most expressive and three-dimensional, 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 reduced edges, their relationships. In this case, you should also use an auxiliary circle in plan at the lower base of the prism, as shown in Fig. 40. Having constructed 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 lay out equal segments taking into account the rotation of the prism, i.e. from a visible position. Connecting the dots with ease , it is necessary to monitor parallelism opposite sides. Having obtained the points of the spatial angles of the base, just as in the first case, you should transfer them to the lower base of the ellipse. It should be noted that when transferring 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 opposite .
Connecting all points on the foundations, begin to check the work performed. Any errors noticed 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 working on should get rid of auxiliary construction using an eraser.
A trihedral pyramid (Fig. 41) is characterized by three points of spatial angles of the base, a point of the apex and six ribs
For the right pyramids should begin with the construction of its base, which is similar to the construction of a prismatic . Connecting the points of the spatial angles of the basethe height of the full-scale model. Then you should connect the top with the spatial corners of the base.
Subsequence drawing n Iramids
- First stage.The size of the pyramid and its spatial position, the main proportions of the pyramid, and the degree of rotation 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 vertex point and eight edges. The structural axis of the pyramid, similar 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 building a pyramid in a horizontal position, you should pay attention 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 in relation 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 body structure also remains unchanged.
