Prism characteristic. Prism base area: from triangular to polygonal

Definition 1. Prismatic surface
Theorem 1. On parallel sections of a prismatic surface
Definition 2. Perpendicular section of a prismatic surface
Definition 3. Prism
Definition 4. Prism height
Definition 5. Right prism
Theorem 2. The area of ​​the lateral surface of the prism

Parallelepiped:
Definition 6. Parallelepiped
Theorem 3. On the intersection of the diagonals of a parallelepiped
Definition 7. Right parallelepiped
Definition 8. Rectangular parallelepiped
Definition 9. Measurements of a parallelepiped
Definition 10. Cube
Definition 11. Rhombohedron
Theorem 4. On the diagonals of a rectangular parallelepiped
Theorem 5. Volume of a prism
Theorem 6. Volume of a straight prism
Theorem 7. Volume of a rectangular parallelepiped

Prism is a polyhedron whose two faces (bases) lie in parallel planes, and the edges that do not lie in these faces are parallel to each other.
Faces other than the bases are called lateral.
The sides of the side faces and bases are called prism ribs, the ends of the edges are called the vertices of the prism. Lateral ribs edges that do not belong to the bases are called. The union of lateral faces is called lateral surface of the prism, and the union of all faces is called the full surface of the prism. Prism height called the perpendicular dropped from the point of the upper base to the plane of the lower base or the length of this perpendicular. Direct prism called a prism whose side ribs are perpendicular to the planes of the bases. Correct called a straight prism (Fig. 3), at the base of which lies a regular polygon.

Designations:
l - side rib;
P - base perimeter;
S o - base area;
H - height;
P^ - perpendicular section perimeter;
S b - lateral surface area;
V - volume;
S p is the area of ​​the total surface of the prism.

V=SH
S p = S b + 2S o
S b = P ^ l
Definition 1 . A prismatic surface is a figure formed by parts of several planes parallel to one straight line, limited by those straight lines along which these planes successively intersect one another*; these lines are parallel to each other and are called edges of the prismatic surface.
*It is assumed that every two successive planes intersect and that the last plane intersects the first

Theorem 1 . Sections of a prismatic surface by planes parallel to each other (but not parallel to its edges) are equal polygons.
Let ABCDE and A"B"C"D"E" be sections of a prismatic surface by two parallel planes. To make sure that these two polygons are equal, it is enough to show that triangles ABC and A"B"C" are equal and have the same direction of rotation and that the same holds for triangles ABD and A"B"D", ABE and A"B"E". But the corresponding sides of these triangles are parallel (for example, AC is parallel to AC) like the line of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example, AC is equal to A"C"), like opposite sides of a parallelogram, and that the angles formed by these sides are equal and have the same direction.

Definition 2 . A perpendicular section of a prismatic surface is a section of this surface by a plane perpendicular to its edges. Based on the previous theorem, all perpendicular sections of the same prismatic surface will be equal polygons.

Definition 3 . A prism is a polyhedron bounded by a prismatic surface and two planes parallel to each other (but not parallel to the edges of the prismatic surface)
The faces lying in these last planes are called prism bases; faces belonging to the prismatic surface - side faces; edges of the prismatic surface - side ribs of the prism. By virtue of the previous theorem, the base of the prism is equal polygons. All lateral faces of the prism - parallelograms; all side ribs are equal to each other.
Obviously, if the base of the prism ABCDE and one of the edges AA" in size and direction are given, then it is possible to construct a prism by drawing edges BB", CC", ... equal and parallel to edge AA".

Definition 4 . The height of a prism is the distance between the planes of its bases (HH").

Definition 5 . A prism is called straight if its bases are perpendicular sections of the prismatic surface. In this case, the height of the prism is, of course, its side rib; the side edges will be rectangles.
Prisms can be classified according to the number of lateral faces equal to the number of sides of the polygon that serves as its base. Thus, prisms can be triangular, quadrangular, pentagonal, etc.

Theorem 2 . The area of ​​the lateral surface of the prism is equal to the product of the lateral edge and the perimeter of the perpendicular section.
Let ABCDEA"B"C"D"E" be a given prism and abcde its perpendicular section, so that the segments ab, bc, .. are perpendicular to its lateral edges. The face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that coincides with ab; the area of ​​the face ВСВ "С" is equal to the product of the base ВВ" by the height bc, etc. Consequently, the lateral surface (i.e. the sum of the areas of the lateral faces) is equal to the product of the lateral edge, in other words, total length segments AA", BB", .., for the amount ab+bc+cd+de+ea.

Definition. Prism is a polyhedron, all of whose vertices are located in two parallel planes, and in these same two planes lie two faces of the prism, which are equal polygons with correspondingly parallel sides, and all edges that do not lie in these planes are parallel.

Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).

All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).

All side faces form lateral surface of the prism .

All lateral faces of the prism are parallelograms .

The edges that do not lie at the bases are called the lateral edges of the prism ( AA 1, BB 1, CC 1, DD 1, EE 1).

Prism diagonal is a segment whose ends are two vertices of a prism that do not lie on the same face (AD 1).

The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .

Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in traversal order, the vertices of one base are indicated, and then, in the same order, the vertices of another; the ends of each side edge are designated by the same letters, only the vertices lying in one base are designated by letters without an index, and in the other - with an index)

The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1 there is a pentagon at the base, so the prism is called pentagonal prism. But because such a prism has 7 faces, then it heptahedron(2 faces - the bases of the prism, 5 faces - parallelograms, - its side faces)

Among straight prisms, a particular type stands out: regular prisms.

A straight prism is called correct, if its bases are regular polygons.

U correct prism all side faces are equal rectangles. A special case of a prism is a parallelepiped.

Parallelepiped

Parallelepiped is a quadrangular prism, at the base of which lies a parallelogram (an inclined parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.

Rectangular parallelepiped- a right parallelepiped whose base is a rectangle.

Properties and theorems:


Some properties of a parallelepiped are similar to the known properties of a parallelogram. A rectangular parallelepiped having equal dimensions is called cube .All faces of a cube are equal squares. The square of the diagonal is equal to the sum of the squares of its three dimensions

,

where d is the diagonal of the square;
a is the side of the square.

An idea of ​​a prism is given by:

  • various architectural structures;
  • Kids toys;
  • packaging boxes;
  • designer items, etc.





The area of ​​the total and lateral surface of the prism

Total surface area of ​​the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its lateral faces. The bases of the prism are equal polygons, then their areas are equal. That's why

S full = S side + 2S main,

Where S full- total surface area, S side-lateral surface area, S base- base area

The lateral surface area of ​​a straight prism is equal to the product of the perimeter of the base and the height of the prism.

S side= P basic * h,

Where S side-area of ​​the lateral surface of a straight prism,

P main - perimeter of the base of a straight prism,

h is the height of the straight prism, equal to the side edge.

Prism volume

The volume of a prism is equal to the product of the area of ​​the base and the height.

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General information about straight prism

The lateral surface of a prism (more precisely, the lateral surface area) is called sum areas of the side faces. The total surface of the prism is equal to the sum of the lateral surface and the areas of the bases.

Theorem 19.1. The lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism, i.e., the length of the side edge.

Proof. The lateral faces of a straight prism are rectangles. The bases of these rectangles are the sides of the polygon lying at the base of the prism, and the heights are equal to the length of the side edges. It follows that the lateral surface of the prism is equal to

S = a 1 l + a 2 l + ... + a n l = pl,

where a 1 and n are the lengths of the base edges, p is the perimeter of the base of the prism, and I is the length of the side edges. The theorem has been proven.

Practical task

Problem (22) . In an inclined prism it is carried out section, perpendicular to the side ribs and intersecting all the side ribs. Find the lateral surface of the prism if the perimeter of the section is equal to p and the side edges are equal to l.

Solution. The plane of the drawn section divides the prism into two parts (Fig. 411). Let us subject one of them to parallel translation, combining the bases of the prism. In this case, we obtain a straight prism, the base of which is the cross-section of the original prism, and the side edges are equal to l. This prism has the same lateral surface as the original one. Thus, the lateral surface of the original prism is equal to pl.

Summary of the covered topic

Now let’s try to summarize the topic we covered about prisms and remember what properties a prism has.


Prism properties

Firstly, a prism has all its bases as equal polygons;
Secondly, in a prism all its lateral faces are parallelograms;
Thirdly, in such a multifaceted figure as a prism, all lateral edges are equal;

Also, it should be remembered that polyhedra such as prisms can be straight or inclined.

Which prism is called a straight prism?

If the side edge of a prism is located perpendicular to the plane of its base, then such a prism is called a straight one.

It would not be superfluous to recall that the lateral faces of a straight prism are rectangles.

What type of prism is called oblique?

But if the side edge of a prism is not located perpendicular to the plane of its base, then we can safely say that it is an inclined prism.

Which prism is called correct?



If a regular polygon lies at the base of a straight prism, then such a prism is regular.

Now let us remember the properties that a regular prism has.

Properties of a regular prism

Firstly, regular polygons always serve as the bases of a regular prism;
Secondly, if we consider the side faces of a regular prism, they are always equal rectangles;
Thirdly, if you compare the sizes of the side ribs, then in a regular prism they are always equal.
Fourthly, a correct prism is always straight;
Fifthly, if in a regular prism the lateral faces have the shape of squares, then such a figure is usually called a semi-regular polygon.

Prism cross section

Now let's look at the cross section of the prism:



Homework

Now let's try to consolidate the topic we've learned by solving problems.

Let's draw an inclined triangular prism, the distance between its edges will be equal to: 3 cm, 4 cm and 5 cm, and the lateral surface of this prism will be equal to 60 cm2. Having these parameters, find the side edge of this prism.

Do you know that geometric figures constantly surround us not only in geometry lessons, but also in Everyday life There are objects that resemble one or another geometric figure.



Every home, school or work has a computer whose system unit is shaped like a straight prism.

If you pick up a simple pencil, you will see that the main part of the pencil is a prism.

Walking along the central street of the city, we see that under our feet lies a tile that has the shape of a hexagonal prism.

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions

Polygons ABCDE and FHKMP lying in parallel planes are called the bases of the prism, the perpendicular OO 1 lowered from any point of the base to the plane of another is called the height of the prism. Parallelograms ABHF, BCKH, etc. are called the lateral faces of the prism, and their sides SC, DM, etc., connecting the corresponding vertices of the bases, are called lateral edges. In a prism, all lateral edges are equal to each other as segments of parallel straight lines enclosed between parallel planes.
A prism is called a straight line ( Fig. 282, b) or oblique ( Fig.282,c) depending on whether its side ribs are perpendicular or inclined to the bases. A straight prism has rectangular side faces. The lateral edge can be taken as the height of such a prism.
A right prism is called regular if its bases are regular polygons. In such a prism, all side faces are equal rectangles.
To depict a prism in a complex drawing, you need to know and be able to depict the elements of which it consists (a point, a straight line, a flat figure).
and their image in the complex drawing (Fig. 283, a - i)

a) Complex drawing of a prism. The base of the prism is located on the projection plane P 1; one of the side faces of the prism is parallel to the projection plane P 2.
b) Near the base of the prism DEF - flat figure - regular triangle, located in the plane P 1; the side of the triangle DE is parallel to the x-axis 12 - The horizontal projection merges with the given base and, therefore, is equal to its natural size; The frontal projection merges with the x 12 axis and is equal to the side of the base of the prism.
c) The upper base of the prism ABC is a flat figure - a triangle located in horizontal plane. The horizontal projection merges with the projection of the lower base and covers it, since the prism is straight; frontal projection - straight, parallel to the x 12 axis, at a distance of the height of the prism.
d) The lateral face of the ABED prism is a flat figure - a rectangle lying in the frontal plane. Frontal projection - a rectangle equal to the natural size of the face; horizontal projection is a straight line equal to the side of the base of the prism.
e) and f) The lateral faces of the ACFD and CBEF prisms are flat figures - rectangles lying in horizontal projecting planes located at an angle of 60° to the projection plane P 2. Horizontal projections are straight lines, located to the x 12 axis at an angle of 60°, and are equal to the natural size of the sides of the base of the prism; frontal projections are rectangles, the image of which is smaller than life-size: two sides of each rectangle are equal to the height of the prism.
g) Edge AD of the prism is a straight line, perpendicular to the projection plane P 1. Horizontal projection - point; frontal - straight, perpendicular to the x 12 axis, equal to the side edge of the prism (prism height).
h) Side AB of the upper base is straight, parallel to planes P 1 and P 2. Horizontal and frontal projections are straight, parallel to the x 12 axis and equal to the side of the given base of the prism. The frontal projection is spaced from the x-axis 12 at a distance equal to the height of the prism.
i) The vertices of the prism. Point E - the top of the lower base is located on the plane P 1. The horizontal projection coincides with the point itself; frontal - lies on the x 12 axis. Point C - the top of the upper base - is located in space. Horizontal projection has depth; frontal - height equal to the height of this prism.
This implies: When designing any polyhedron, you need to mentally divide it into its component elements and determine the order of their representation, consisting of successive graphic operations. Figures 284 and 285 show examples of sequential graphic operations when performing a complex drawing and visual representation (axonometry) of prisms.
(Fig. 284).

Given:
1. The base is located on the projection plane P 1.
2. Neither side of the base is parallel to the x-axis 12.
I. Complex drawing.
I, a. We design the lower base - a polygon, which, by condition, lies in the plane P1.
I, b. We design the upper base - a polygon equal to the lower base with sides correspondingly parallel to the lower base, spaced from the lower base by the height H of the given prism.
I, c. We design the side edges of the prism - segments located parallel; their horizontal projections are points merging with the projections of the vertices of the bases; frontal - segments (parallel) obtained from connecting with straight lines the projections of the vertices of the bases of the same name. The frontal projections of the ribs, drawn from the projections of vertices B and C of the lower base, are depicted by dashed lines as invisible.
I, g. Given: horizontal projection F 1 of point F on the upper base and frontal projection K 2 of point K on the side face. It is required to determine the locations of their second projections.
For point F. The second (frontal) projection F 2 of point F will coincide with the projection of the upper base, as a point lying in the plane of this base; its place is determined by the vertical communication line.
For point K - The second (horizontal) projection K 1 of point K will coincide with the horizontal projection of the side face, as a point lying in the plane of the face; its place is determined by the vertical communication line.
II. Prism surface development- a flat figure made up of side faces - rectangles, in which two sides are equal to the height of the prism, and the other two are equal to the corresponding sides of the base, and from two bases equal to each other - irregular polygons.
The natural dimensions of the bases and sides of the faces necessary for constructing the development are revealed on the projections; we build on them; On a straight line we sequentially plot the sides AB, BC, CD, DE and EA of the polygon - the bases of the prism, taken from the horizontal projection. On the perpendiculars drawn from points A, B, C, D, E and A, we plot the height H of this prism taken from the frontal projection and draw a straight line through the marks. As a result, we obtain a scan of the side faces of the prism.
If we attach the bases of the prism to this development, we obtain a development of the full surface of the prism. The bases of the prism should be attached to the corresponding side face using the triangulation method.
On the upper base of the prism, using radii R and R 1, we determine the location of point F, and on the side face, using radius R 3 and H 1, we determine point K.
III. A visual representation of a prism in dimetry.
III, a. We depict the lower base of the prism according to the coordinates of points A, B, C, D and E (Fig. 284 I, a).
III, b. We depict the upper base parallel to the lower one, spaced from it by the height H of the prism.
III, c. We depict the side edges by connecting the corresponding vertices of the bases with straight lines. We determine the visible and invisible elements of the prism and outline them with the corresponding lines,
III, d. We determine points F and K on the surface of the prism - Point F - on the upper base is determined using dimensions i and e; point K - on the side face using i 1 and H" .
For an isometric image of the prism and determining the locations of points F and K, the same sequence should be followed.
Fig.285).

Given:
1. The base is located on the plane P 1.
2. The side ribs are parallel to the P 2 plane.
3. Neither side of the base is parallel to the x 12 axis
I. Complex drawing.
I, a. We design according to this condition: the bottom base is a polygon lying in the plane P1, and the side edge is a segment parallel to the plane P2 and inclined to the plane P1.
I, b. We design the remaining side edges - segments equal and parallel to the first edge SE.
I, c. We design the upper base of the prism as a polygon, equal and parallel to the lower base, and obtain a complex drawing of the prism.
We identify invisible elements on projections. The frontal projection of the edge of the VM and the horizontal projection of the side of the base CD are depicted by dashed lines as invisible.
I, g. Given the frontal projection Q 2 of point Q on the projection A 2 K 2 F 2 D 2 of the side face; you need to find its horizontal projection. To do this, draw an auxiliary line through point Q 2 in the projection A 2 K 2 F 2 D 2 of the prism face, parallel to the side edges of this face. We find the horizontal projection of the auxiliary line and on it, using a vertical connection line, we determine the location of the desired horizontal projection Q 1 of point Q.
II. Prism surface development.
Having the natural dimensions of the sides of the base on the horizontal projection, and the dimensions of the ribs on the frontal projection, it is possible to construct a complete development of the surface of a given prism.
We will roll the prism, rotating it each time around the side edge, then each side face of the prism on the plane will leave a trace (parallelogram) equal to its natural size. We will construct the side scan in the following order:
a) from points A 2, B 2, D 2. . . E 2 (frontal projections of the vertices of the bases) we draw auxiliary straight lines perpendicular to the projections of the ribs;
b) radius R ( equal to side base CD) we make a notch at point D on the auxiliary line drawn from point D2; by connecting straight points C 2 and D and drawing straight lines parallel to E 2 C 2 and C 2 D, we obtain the side face CEFD;
c) then, by similarly arranging the following side faces, we obtain a development of the side faces of the prism. To obtain a complete development of the surface of this prism, we attach it to the corresponding faces of the base.
III. A visual representation of a prism in isometry.
III, a. We depict the lower base of the prism and the edge CE, using coordinates according to (

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