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1. Smallest number The tetrahedron has 6 edges.
2. A prism has n faces. What polygon lies at its base?
(n - 2) - square.
3. Is a prism straight if its two adjacent side faces are perpendicular to the plane of the base?
Yes, it is.
4. In which prism are the lateral edges parallel to its height?
In a straight prism.
5. Is a prism regular if all its edges are equal to each other?
No, it may not be direct.
6. Can the height of one of the side faces of an inclined prism also be the height of the prism?
Yes, if this face is perpendicular to the base.
7. Is there a prism in which: a) the side edge is perpendicular to only one edge of the base; b) only one side face is perpendicular to the base?
a) yes. b) no.
8. A regular triangular prism is divided into two prisms by a plane passing through the midlines of the bases. What is the ratio of the lateral surface areas of these prisms?
By theorem 27 we find that the lateral surfaces are in the ratio 5: 3
9. Will the pyramid be regular if its side faces are regular triangles?
10. How many faces perpendicular to the plane of the base can a pyramid have?
11. Is there a quadrangular pyramid whose opposite side faces are perpendicular to the base?
No, otherwise there would be at least two straight lines passing through the top of the pyramid, perpendicular to the bases.
12. Can all the faces of a triangular pyramid be right triangles?
Yes (Figure 183).
Polyhedra
The main object of study of stereometry is spatial bodies. Body represents a part of space limited by a certain surface.
Polyhedron called a body whose surface consists of finite number flat polygons. A polyhedron is called convex if it is located on one side of the plane of every plane polygon on its surface. General part such a plane and the surface of a polyhedron is called edge. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called edges of the polyhedron, and the vertices are vertices of the polyhedron.
For example, a cube consists of six squares, which are its faces. It contains 12 edges (the sides of the squares) and 8 vertices (the tops of the squares).
The simplest polyhedra are prisms and pyramids, which we will study further.
Prism
Definition and properties of a prism
Prism is a polyhedron consisting of two flat polygons lying in parallel planes compatible parallel transfer, and all segments connecting the corresponding points of these polygons. Polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are lateral edges of the prism.
Prism height is called the distance between the planes of its bases (). A segment connecting two vertices of a prism that do not belong to the same face is called prism diagonal(). The prism is called n-carbon, if its base is an n-gon.
Any prism has the following properties, resulting from the fact that the bases of the prism are combined by parallel translation:
1. The bases of the prism are equal.
2. The lateral edges of the prism are parallel and equal.
The surface of the prism consists of bases and lateral surface. The lateral surface of the prism consists of parallelograms (this follows from the properties of the prism). The area of the lateral surface of a prism is the sum of the areas of the lateral faces.
Straight prism
The prism is called direct, if its lateral edges are perpendicular to the bases. Otherwise the prism is called inclined.
The faces of a right prism are rectangles. The height of a straight prism is equal to its side faces.
Full prism surface is called the sum of the lateral surface area and the areas of the bases.
With the right prism called a right prism with a regular polygon at its base.
Theorem 13.1. The area of the lateral surface of a straight prism is equal to the product of the perimeter and the height of the prism (or, which is the same, by the lateral edge).
Proof. The lateral faces of a right prism are rectangles, the bases of which are the sides of the polygons at the bases of the prism, and the heights are the lateral edges of the prism. Then, by definition, the lateral surface area is:
,
where is the perimeter of the base of a straight prism.
Parallelepiped
If parallelograms lie at the bases of a prism, then it is called parallelepiped. All faces of a parallelepiped are parallelograms. At the same time opposite faces parallelepipeds are parallel and equal.
Theorem 13.2. The diagonals of a parallelepiped intersect at one point and are divided in half by the intersection point.
Proof. Consider two arbitrary diagonals, for example, and . Because the faces of a parallelepiped are parallelograms, then and , which means according to To there are two straight lines parallel to the third. In addition, this means that straight lines and lie in the same plane (plane). This plane intersects parallel planes and along parallel lines and . Thus, a quadrilateral is a parallelogram, and by the property of a parallelogram, its diagonals intersect and are divided in half by the intersection point, which was what needed to be proven.
A right parallelepiped whose base is a rectangle is called rectangular parallelepiped. All faces of a rectangular parallelepiped are rectangles. The lengths of the non-parallel edges of a rectangular parallelepiped are called its linear dimensions (dimensions). There are three such sizes (width, height, length).
Theorem 13.3. In a rectangular parallelepiped, the square of any diagonal equal to the sum squares of its three dimensions 
(proven by applying Pythagorean T twice).
Rectangular parallelepiped, which has all edges equal, is called cube.
Tasks
13.1 How many diagonals does it have? n-carbon prism
13.2 In an inclined triangular prism, the distances between the side edges are 37, 13 and 40. Find the distance between the larger side edge and the opposite side edge.
13.3 Through the side of the lower base of the correct triangular prism a plane is drawn intersecting side faces along segments, the angle between which is . Find the angle of inclination of this plane to the base of the prism.
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 prism ABED is a flat figure - a rectangle lying in 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 x12 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 lateral rib 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 height of this prism.
It follows from this: 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 in 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 with dashed lines, as if they were 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 the development full surface prisms. 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 isometric image prisms and determining the locations of points F and K should follow the same sequence.
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 rib CE.
I, c. We design the upper base of the prism as a polygon equal and parallel to the lower base, we get complex drawing prisms.
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 vertical line connection, 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 vertices of the bases) draw auxiliary straight lines perpendicular to the projections of the edges;
b) with radius R (equal to the side of the base CD), we make a notch at point D on the auxiliary straight line drawn from point D 2 ; 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 (
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 polygon and with respectively parallel sides, and all edges not lying 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 prisms .
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 the order of traversal, 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, it stands out private view: correct prisms.
A straight prism is called correct, if its bases are regular polygons.
Parallelepiped
Parallelepiped- This 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 known properties parallelogram. A rectangular parallelepiped having equal measurements, are called cube
.A cube has all 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;
- children's 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 whyS 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
Prism volume equal to the product base area to height.
