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 straight prisms, a particular type stands out: 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.

Different prisms are different from each other. At the same time, they have a lot in common. To find the area of ​​\u200b\u200bthe base of a prism, you need to figure out what kind it looks like.

General theory

A prism is any polyhedron whose sides have the form of a parallelogram. Moreover, any polyhedron can be at its base - from a triangle to an n-gon. Moreover, the bases of the prism are always equal to each other. What does not apply to the side faces - they can vary significantly in size.

When solving problems, it is not only the area of ​​\u200b\u200bthe base of the prism that is encountered. It may be necessary to know the lateral surface, that is, all faces that are not bases. The full surface will already be the union of all the faces that make up the prism.

Sometimes heights appear in tasks. It is perpendicular to the bases. The diagonal of a polyhedron is a segment that connects in pairs any two vertices that do not belong to the same face.

It should be noted that the area of ​​the base of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures in the upper and lower faces, then their areas will be equal.

triangular prism

It has at the base a figure with three vertices, that is, a triangle. It is known to be different. If then it is enough to recall that its area is determined by half the product of the legs.

Mathematical notation looks like this: S = ½ av.

To find out the area of ​​\u200b\u200bthe base in a general form, the formulas are useful: Heron and the one in which half of the side is taken to the height drawn to it.

The first formula should be written like this: S \u003d √ (p (p-a) (p-in) (p-s)). This entry contains a semi-perimeter (p), that is, the sum of three sides divided by two.

Second: S = ½ n a * a.

If you want to know the area of ​​\u200b\u200bthe base of a triangular prism, which is regular, then the triangle turns out to be equilateral. It has its own formula: S = ¼ a 2 * √3.

quadrangular prism

Its base is any of the known quadrilaterals. It can be a rectangle or a square, a parallelepiped or a rhombus. In each case, in order to calculate the area of ​​\u200b\u200bthe base of the prism, you will need your own formula.

If the base is a rectangle, then its area is determined as follows: S = av, where a, b are the sides of the rectangle.

When it comes to a quadrangular prism, the base area of ​​a regular prism is calculated using the formula for a square. Because it is he who lies at the base. S \u003d a 2.

In the case when the base is a parallelepiped, the following equality will be needed: S \u003d a * n a. It happens that a side of a parallelepiped and one of the angles are given. Then, to calculate the height, you will need to use an additional formula: na \u003d b * sin A. Moreover, the angle A is adjacent to the side "b", and the height is na opposite to this angle.

If a rhombus lies at the base of the prism, then the same formula will be needed to determine its area as for a parallelogram (since it is a special case of it). But you can also use this one: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.

Regular pentagonal prism

This case involves splitting the polygon into triangles, the areas of which are easier to find out. Although it happens that the figures can be with a different number of vertices.

Since the base of the prism is a regular pentagon, it can be divided into five equilateral triangles. Then the area of ​​\u200b\u200bthe base of the prism is equal to the area of ​​​​one such triangle (the formula can be seen above), multiplied by five.

Regular hexagonal prism

According to the principle described for a pentagonal prism, it is possible to divide the base hexagon into 6 equilateral triangles. The formula for the area of ​​​​the base of such a prism is similar to the previous one. Only in it should be multiplied by six.

The formula will look like this: S = 3/2 and 2 * √3.

Tasks

No. 1. A regular line is given. Its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of ​​\u200b\u200bthe base of the prism and the entire surface.

Solution. The base of a prism is a square, but its side is not known. You can find its value from the diagonal of the square (x), which is related to the diagonal of the prism (d) and its height (h). x 2 \u003d d 2 - n 2. On the other hand, this segment "x" is the hypotenuse in a triangle whose legs are equal to the side of the square. That is, x 2 \u003d a 2 + a 2. Thus, it turns out that a 2 \u003d (d 2 - n 2) / 2.

Substitute the number 22 instead of d, and replace “n” with its value - 14, it turns out that the side of the square is 12 cm. Now it’s easy to find out the base area: 12 * 12 \u003d 144 cm 2.

To find out the area of ​​\u200b\u200bthe entire surface, you need to add twice the value of the base area and quadruple the side. The latter is easy to find by the formula for a rectangle: multiply the height of the polyhedron and the side of the base. That is, 14 and 12, this number will be equal to 168 cm 2. The total surface area of ​​the prism is found to be 960 cm 2 .

Answer. The base area of ​​the prism is 144 cm2. The entire surface - 960 cm 2 .

No. 2. Dana At the base lies a triangle with a side of 6 cm. In this case, the diagonal of the side face is 10 cm. Calculate the areas: the base and the side surface.

Solution. Since the prism is regular, its base is an equilateral triangle. Therefore, its area turns out to be equal to 6 squared times ¼ and the square root of 3. A simple calculation leads to the result: 9√3 cm 2. This is the area of ​​one base of the prism.

All side faces are the same and are rectangles with sides of 6 and 10 cm. To calculate their areas, it is enough to multiply these numbers. Then multiply them by three, because the prism has exactly so many side faces. Then the area of ​​the side surface is wound 180 cm 2 .

Answer. Areas: base - 9√3 cm 2, side surface of the prism - 180 cm 2.

"Lesson of the Pythagorean theorem" - The Pythagorean theorem. Determine the type of quadrilateral KMNP. Warm up. Introduction to the theorem. Determine the type of triangle: Lesson plan: Historical digression. Solving simple problems. And find a ladder 125 feet long. Calculate the height CF of trapezoid ABCD. Proof. Showing pictures. Proof of the theorem.

"Volume of a prism" - The concept of a prism. direct prism. The volume of the original prism is equal to the product S · h. How to find the volume of a straight prism? The prism can be divided into straight triangular prisms with height h. Draw the altitude of triangle ABC. The solution of the problem. Lesson goals. Basic steps in proving the direct prism theorem? Study of the prism volume theorem.

"Prism polyhedra" - Define a polyhedron. DABC is a tetrahedron, a convex polyhedron. The use of prisms. Where are prisms used? ABCDMP is an octahedron, made up of eight triangles. ABCDA1B1C1D1 is a parallelepiped, a convex polyhedron. Convex polyhedron. The concept of a polyhedron. Polyhedron A1A2..AnB1B2..Bn is a prism.

"Prism class 10" - A prism is a polyhedron whose faces are in parallel planes. The use of a prism in everyday life. Sside = Pbased. + h For a straight prism: Sp.p = Pmain. h + 2Smain. Inclined. Correct. Straight. Prism. Formulas for finding the area. The use of prism in architecture. Sp.p \u003d S side + 2 S based.

"Proof of the Pythagorean theorem" - Geometric proof. The meaning of the Pythagorean theorem. Pythagorean theorem. Euclid's proof. "In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs." Proofs of the theorem. The significance of the theorem is that most of the theorems of geometry can be deduced from it or with its help.

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. The diagonal section of a regular 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 denote the action of extracting a square root in solving problems, the symbol is used√ .

A task.

The diagonal of a regular prism forms a right triangle with the diagonal of the base and the height of the prism. 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

A task

Solution.
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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