Definition. Prism- this is a polyhedron, all the vertices of which are located in two parallel planes, and in the same two planes there are two faces of the prism, which are equal polygons with respectively 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 side surface of the prism .

All side faces of a prism are parallelograms .

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

Prism Diagonal a segment is called, the ends of which are two vertices of the prism that do not lie on one of its faces (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 the bypass, the vertices of one base are indicated, and then, in the same order, the vertices of the other; the ends of each side edge are designated by the same letters, only the vertices lying in one base are indicated 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, the base is a pentagon, so the prism is called pentagonal prism. But since such a prism has 7 faces, then it heptahedron(2 faces are the bases of the prism, 5 faces are parallelograms, are its side faces)

Among the straight prisms stands out private view: regular prisms.

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

Parallelepiped

cuboid- a right parallelepiped whose base is a rectangle.

Properties and theorems:

,

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

The idea of ​​a prism is given by:

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

Total and lateral surface area of ​​the prism

S full \u003d S side + 2S main,

where S full- total surface area, S side- side surface area, S main- base area

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

S side\u003d P main * h,

where S side is the area of ​​the lateral surface of a straight prism,

P main - the 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.

Definition.

This is a hexagon, the bases of which are two equal squares, and the side faces are equal rectangles.

Side rib is the common side of two adjacent side faces

Prism Height is a line segment perpendicular to the bases of the prism

Prism Diagonal- a segment connecting two vertices of the bases that do not belong to the same face

Diagonal plane- a plane that passes through the diagonal of the prism and its side edges

Diagonal section- the boundaries of the intersection of the prism and the diagonal plane. Diagonal section of the correct quadrangular prism is a rectangle

Perpendicular section (orthogonal section)- this is the intersection of a prism and a plane drawn perpendicular to its side edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are marked with the corresponding letters:

• Bases ABCD and A 1 B 1 C 1 D 1 are equal and parallel to each other
• Side faces AA 1 D 1 D, AA 1 B 1 B, BB 1 C 1 C and CC 1 D 1 D, each of which is a rectangle
• Lateral surface - the sum of the areas of all the side faces of the prism
• Total surface - the sum of the areas of all bases and side faces (the sum of the area of ​​the side surface and bases)
• Side ribs AA 1 , BB 1 , CC 1 and DD 1 .
• Diagonal B 1 D
• Base diagonal BD
• Diagonal section BB 1 D 1 D
• Perpendicular section A 2 B 2 C 2 D 2 .

Properties of a regular quadrangular prism

• The bases are two equal squares
• The bases are parallel to each other
• The sides are rectangles.
• Side faces are equal to each other
• Side faces are perpendicular to the bases
• Lateral ribs are parallel to each other and equal
• Perpendicular section perpendicular to all side ribs and parallel to the bases
• Perpendicular Section Angles - Right
• The diagonal section of a regular quadrangular prism is a rectangle
• Perpendicular (orthogonal section) parallel to the bases

Formulas for a regular quadrangular prism

Instructions for solving problems

Correct prism- a prism at the base of which lies a regular polygon, and the side edges are perpendicular to the planes of the base. That is, a regular quadrangular prism contains at its base square. (see above the properties of a regular quadrangular prism) Note. This is part of the lesson with tasks in geometry (section solid geometry - prism). Here are the tasks that cause difficulties in solving. If you need to solve a problem in geometry, which is not here - write about it in the forum. To indicate the action of extracting square root symbol is used in problem solving√ .

Task.

The diagonal of a regular prism forms with the diagonal of the base and the height of the prism right triangle. Accordingly, according to the Pythagorean theorem, the diagonal of a given regular quadrangular prism will be equal to:
√((12√2) 2 + 14 2 ) = 22 cm

Answer: 22 cm

Task

Decision.
Since the base of a regular quadrangular prism is a square, then the side of the base (denoted as a) is found by the Pythagorean theorem:

A 2 + a 2 = 5 2
2a 2 = 25
a = √12.5

The height of the side face (denoted as h) will then be equal to:

H 2 + 12.5 \u003d 4 2
h 2 + 12.5 = 16
h 2 \u003d 3.5
h = √3.5

The total surface area will be equal to the sum of the lateral surface area and twice the base area

S = 2a 2 + 4ah
S = 25 + 4√12.5 * √3.5
S = 25 + 4√43.75
S = 25 + 4√(175/4)
S = 25 + 4√(7*25/4)
S \u003d 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.

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Prism. Parallelepiped

prism is called a polyhedron whose two faces are equal n-gons (grounds) , lying in parallel planes, and the remaining n faces are parallelograms (side edges) . Side rib prism is the side of the lateral face that does not belong to the base.

A prism whose lateral edges are perpendicular to the planes of the bases is called straight prism (Fig. 1). If the side edges are not perpendicular to the planes of the bases, then the prism is called oblique . correct A prism is a straight prism whose bases are regular polygons.

Height prism is called the distance between the planes of the bases. Diagonal A prism is a segment connecting two vertices that do not belong to the same face. diagonal section A section of a prism by a plane passing through two side edges that do not belong to the same face is called. Perpendicular section called the section of the prism by a plane perpendicular to the lateral edge of the prism.

Side surface area prism is the sum of the areas of all side faces. Full surface area the sum of the areas of all the faces of the prism is called (i.e., the sum of the areas of the side faces and the areas of the bases).

For an arbitrary prism, the formulas are true:

where l is the length of the side rib;

H- height;

P

Q

S side

S full

S main is the area of ​​the bases;

V is the volume of the prism.

For a straight prism, the following formulas are true:

where p- the perimeter of the base;

l is the length of the side rib;

H- height.

Parallelepiped A prism whose base is a parallelogram is called. A parallelepiped whose lateral edges are perpendicular to the bases is called direct (Fig. 2). If the side edges are not perpendicular to the bases, then the parallelepiped is called oblique . A right parallelepiped whose base is a rectangle is called rectangular. A rectangular parallelepiped in which all edges are equal is called cube.

The faces of a parallelepiped that do not have common vertices are called opposite . The lengths of edges emanating from one vertex are called measurements parallelepiped. Since the box is a prism, its main elements are defined in the same way as they are defined for prisms.

Theorems.

1. The diagonals of the parallelepiped intersect at one point and bisect it.

2. In a rectangular parallelepiped, the square of the length of the diagonal is equal to the sum of the squares of its three dimensions:

3. All four diagonals of a rectangular parallelepiped are equal to each other.

For an arbitrary parallelepiped, the following formulas are true:

where l is the length of the side rib;

H- height;

P is the perimeter of the perpendicular section;

Q– Area of ​​perpendicular section;

S side is the lateral surface area;

S full is the total surface area;

S main is the area of ​​the bases;

V is the volume of the prism.

For a right parallelepiped, the following formulas are true:

where p- the perimeter of the base;

l is the length of the side rib;

H is the height of the right parallelepiped.

For a rectangular parallelepiped, the following formulas are true:

(3)

where p- the perimeter of the base;

H- height;

d- diagonal;

a,b,c– measurements of a parallelepiped.

The correct formulas for a cube are:

where a is the length of the rib;

d is the diagonal of the cube.

Example 1 The diagonal of a rectangular cuboid is 33 dm, and its measurements are related as 2:6:9. Find the measurements of the cuboid.

Decision. To find the dimensions of the parallelepiped, we use formula (3), i.e. the fact that the square of the hypotenuse of a cuboid is equal to the sum of the squares of its dimensions. Denote by k coefficient of proportionality. Then the dimensions of the parallelepiped will be equal to 2 k, 6k and 9 k. We write formula (3) for the problem data:

Solving this equation for k, we get:

Hence, the dimensions of the parallelepiped are 6 dm, 18 dm and 27 dm.

Answer: 6 dm, 18 dm, 27 dm.

Example 2 Find the volume of an inclined triangular prism whose base is an equilateral triangle with a side of 8 cm, if the lateral edge is equal to the side of the base and is inclined at an angle of 60º to the base.

Decision . Let's make a drawing (Fig. 3).

In order to find the volume of an inclined prism, you need to know the area of ​​\u200b\u200bits base and height. The area of ​​the base of this prism is the area of ​​an equilateral triangle with a side of 8 cm. Let's calculate it:

The height of a prism is the distance between its bases. From the top BUT 1 of the upper base we lower the perpendicular to the plane of the lower base BUT 1 D. Its length will be the height of the prism. Consider D BUT 1 AD: since this is the angle of inclination of the side rib BUT 1 BUT to the base plane BUT 1 BUT= 8 cm. From this triangle we find BUT 1 D:

Now we calculate the volume using formula (1):

Answer: 192 cm3.

Example 3 The lateral edge of a regular hexagonal prism is 14 cm. The area of ​​\u200b\u200bthe largest diagonal section is 168 cm 2. Find the total surface area of ​​the prism.

Decision. Let's make a drawing (Fig. 4)

The largest diagonal section is a rectangle AA 1 DD 1 , since the diagonal AD regular hexagon ABCDEF is the largest. In order to calculate the lateral surface area of ​​a prism, it is necessary to know the side of the base and the length of the lateral rib.

Knowing the area of ​​the diagonal section (rectangle), we find the diagonal of the base.

Because , then

Since then AB= 6 cm.

Then the perimeter of the base is:

Find the area of ​​the lateral surface of the prism:

The area of ​​a regular hexagon with a side of 6 cm is:

Find the total surface area of ​​the prism:

Answer:

Example 4 The base of a right parallelepiped is a rhombus. The areas of diagonal sections are 300 cm 2 and 875 cm 2. Find the area of ​​the side surface of the parallelepiped.

Decision. Let's make a drawing (Fig. 5).

Denote the side of the rhombus by a, the diagonals of the rhombus d 1 and d 2 , the height of the box h. To find the lateral surface area of ​​a straight parallelepiped, it is necessary to multiply the perimeter of the base by the height: (formula (2)). Base perimeter p = AB + BC + CD + DA = 4AB = 4a, as ABCD- rhombus. H = AA 1 = h. That. Need to find a and h.

Consider diagonal sections. AA 1 SS 1 - a rectangle, one side of which is the diagonal of a rhombus AC = d 1 , second - side edge AA 1 = h, then

Similarly for the section BB 1 DD 1 we get:

Using the property of a parallelogram such that the sum of the squares of the diagonals is equal to the sum of the squares of all its sides, we get the equality We get the following.

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 surface prisms is the sum of the lateral surface area and the areas of the bases.

Correct prism is called a right prism with a 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 required to be proved.

A right parallelepiped whose base is a rectangle is called cuboid. All faces of a cuboid are rectangles. The lengths of non-parallel edges of a cuboid 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.

13.3 Through the side of the lower base of a regular triangular prism, a plane is drawn that intersects the side faces along segments, the angle between which is . Find the angle of inclination of this plane to the base of the prism.