How does she look
Rectangular prisms surrounded modern man quite a bit of. This, for example, is the usual cardboard from under shoes, computer components, etc. Look around. Even in a room, you will surely see many rectangular prisms. This is a computer case, and a bookcase, and a refrigerator, and a cabinet, and many other items. The form is extremely popular mainly because it allows you to use the space as efficiently as possible, whether you are decorating the interior or packing things in cardboard before moving.Properties of a rectangular prism
A rectangular prism has a number of specific properties. Any pair of faces can serve as its, since all adjacent faces are located at the same angle to each other, and this angle is 90 °. The volume and surface area of a rectangular prism is easier to calculate than any other. Take any object that has the shape of a rectangular prism. Measure its length, width and height. To find the volume, it is enough to multiply these measurements. That is, the formula looks like this: V=a*b*h, where V is the volume, a and b are the sides of the base, h is the height that coincides with the side edge of this geometric body. The base area is calculated by the formula S1=a*b. To get the side surface, you must first calculate the perimeter of the base using the formula P=2(a+b), and then multiply it by the height. It turns out the formula S2=P*h=2(a+b)*h. To calculate the total surface area of a rectangular prism, add twice the area of the base and the area of the side surface. The formula is S=2S1+S2=2*a*b+2*(a+b)*h=2Lecture: Prism, its bases, side edges, height, side surface; straight prism; right prism
Prism
If you have learned with us flat figures from past questions, it means that they are completely ready to study three-dimensional figures. The first solid that we will learn will be a prism.
Prism- This is a three-dimensional body that has a large number of faces.
This figure has two polygons at the bases, which are located in parallel planes, and all side faces are in the form of a parallelogram.
Fig 1. Fig. 2
So, let's figure out what a prism consists of. To do this, pay attention to Fig.1
As mentioned earlier, the prism has two bases that are parallel to each other - these are the pentagons ABCEF and GMNJK. Moreover, these polygons are equal to each other.
All other faces of the prism are called side faces - they consist of parallelograms. For example, BMNC, AGKF, FKJE, etc.
The common surface of all side faces is called side surface.
Each pair of adjacent faces has a common side. Such a common side is called an edge. For example, MB, CE, AB, etc.
If the upper and lower bases of the prism are connected by a perpendicular, then it will be called the height of the prism. In the figure, the height is marked as a straight line OO 1.
There are two main types of prism: oblique and straight.
If side ribs prisms are not perpendicular to the bases, then such a prism is called oblique.
If all the edges of a prism are perpendicular to the bases, then such a prism is called straight.
If the bases of a prism are regular polygons (those with equal sides), then such a prism is called correct.
If the bases of the prism are not parallel to each other, then such a prism will be called truncated.
You can see it in Fig.2
Formulas for finding volume, area of a prism
There are three basic formulas for finding volume. They differ from each other in their application:
Similar formulas for finding the surface area of a prism:
Polyhedra
The main object of study of stereometry are three-dimensional bodies. Body is a part of space bounded by some surface.
polyhedron A body whose surface consists of a finite number of plane polygons is called. A polyhedron is called convex if it lies on one side of the plane of every flat polygon on its surface. The common part of 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 vertices of the polyhedron.
For example, a cube consists of six squares that are its faces. It contains 12 edges (sides of squares) and 8 vertices (vertices of squares).
The simplest polyhedra are prisms and pyramids, which we will study further.
Prism
Definition and properties of a prism
prism is called a polyhedron consisting of two flat polygons lying in parallel planes combined by parallel translation, and all segments connecting the corresponding points of these polygons. The polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are side edges of the prism.
Prism height 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-coal if its base is an n-gon.
Any prism has the following properties, which follow from the fact that the bases of the prism are combined by parallel translation:
1. The bases of the prism are equal.
2. The side edges of the prism are parallel and equal.
The surface of a prism is made up 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 straight if its side edges are perpendicular to the bases. Otherwise, the prism is called oblique.
The faces of a straight prism are rectangles. The height of a straight prism is equal to its side faces.
full prism surface is the sum of the lateral surface area and the areas of the bases.
Correct prism called a straight prism regular polygon at the 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, equivalently, to the lateral edge).
Proof. The side faces of a straight prism are rectangles whose bases are the sides of the polygons at the bases of the prism, and the heights are the side 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 the faces of a parallelepiped are parallelograms. In this case, the opposite faces of the parallelepiped are parallel and equal.
Theorem 13.2. The diagonals of the parallelepiped intersect at one point and the intersection point is divided in half.
Proof. Consider two arbitrary diagonals, for example, and . Because the faces of the parallelepiped are parallelograms, then and , which means that according to T about two straight lines parallel to the third . In addition, this means that the lines and lie in the same plane (the 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 and intersect and the intersection point is divided in half, which was to be proved.
A right parallelepiped whose base is a rectangle is called cuboid. At cuboid all faces are rectangles. The lengths of non-parallel edges of a rectangular parallelepiped are called its linear dimensions (measurements). There are three sizes (width, height, length).
Theorem 13.3. In a cuboid, the square of any diagonal is equal to the sum of the squares of its three dimensions 
(proved by applying Pythagorean T twice).
A rectangular parallelepiped in which all edges are equal is called cube.
Tasks
13.1 How many diagonals does 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 face and the opposite side edge.
13.3Through the side of the lower base of the correct triangular prism a plane is drawn that intersects the side faces along the segments, the angle between which is . Find the angle of inclination of this plane to the base of the prism.
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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. Direct 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. Dimensions 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 a polyhedron is called, in which 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 bases are called lateral.
The sides of the side faces and bases are called prism edges, the ends of the edges are called the tops of the prism. Lateral ribs called edges that do not belong to the bases. The union of side faces is called side surface of the prism, and the union of all faces is called 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. 
straight prism called a prism, in which the side edges 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 ^ - perimeter of the perpendicular section;
S b - side surface area;
V - volume;
S p - area of the total surface of the prism.
|
V=SH |
*It is assumed that every two consecutive 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 verify 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 the 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 A "C") as the lines of intersection of a certain plane with two parallel planes; it follows that these sides are equal (e.g. AC equals A"C") as opposite sides 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 a prismatic surface - side faces; edges of the prismatic surface - side edges of the prism. By virtue of the previous theorem, the bases of the prism are equal polygons. All side faces of the prism parallelograms; all side edges are equal to each other.
It is obvious that if the base of the prism ABCDE and one of the edges AA" are given in magnitude and direction, then it is possible to construct a prism by drawing the edges BB", CC", .., equal and parallel to the 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 a straight line if its bases are perpendicular sections of a prismatic surface. In this case, the height of the prism is, of course, its side rib; side edges will rectangles.
Prisms can be classified according to the number of side faces, equal number 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 the given prism and abcde be its perpendicular section, so that the segments ab, bc, .. are perpendicular to its side edges. Face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that matches ab; the area of \u200b\u200bthe face BCV "C" is equal to the product of the base BB" by the height bc, etc. Therefore, the side surface (i.e., the sum of the areas of the side faces) is equal to the product of the side edge, in other words, the total length of the segments AA", BB", .., by the sum ab+bc+cd+de+ea.
