What lies at the basis of the correct prism. Prism

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A set of tables. Geometry. Grade 10. 14 tables + methodology, . The tables are printed on thick polygraphic cardboard measuring 680 x 980 mm. Brochure with guidelines for the teacher. Study album of 14 sheets.…

Lecture: Prism, its bases, lateral edges, height, lateral surface; straight prism; right prism

Prism

If you have learned flat figures from the previous questions with us, then you are completely ready to study three-dimensional figures. The first solid that we will learn will be a prism.

Prism is a voluminous 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 the side edges of the prism 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:

Any polygon can lie at the base of the prism - a triangle, a quadrilateral, etc. Both bases are exactly the same, and accordingly, by which the angles of parallel faces are connected to each other, they are always parallel. At the base of a regular prism lies a regular polygon, that is, one in which all sides are equal. In a straight prism, the edges between the side faces are perpendicular to the base. In this case, a polygon with any number of angles can lie at the base of a straight prism. A prism whose base is a parallelogram is called a parallelepiped. Rectangle - special case parallelogram. If this figure lies at the base, and the side faces are located at right angles to the base, the parallelepiped is called rectangular. The second name of this geometric body is rectangular.

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 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 \u003d 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 full surface For 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=2

AT school curriculum in the course of solid geometry, the study of three-dimensional figures usually begins with a simple geometric body - a prism polyhedron. The role of its bases is performed by 2 equal polygons lying in parallel planes. A special case is a regular quadrangular prism. Its bases are 2 identical regular quadrangles, to which the sides are perpendicular, having the shape of parallelograms (or rectangles if the prism is not inclined).

What does a prism look like

A regular quadrangular prism is a hexahedron, at the bases of which there are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure- a straight parallelepiped.

The figure, which depicts a quadrangular prism, is shown below.

You can also see in the picture essential elements, which make up the geometric body. They are commonly referred to as:

Sometimes in problems in geometry you can find the concept of a section. The definition will sound like this: a section is all points of a volumetric body that belong to the cutting plane. The section is perpendicular (crosses the edges of the figure at an angle of 90 degrees). For a rectangular prism, a diagonal section is also considered (the maximum number of sections that can be built is 2), passing through 2 edges and the diagonals of the base.

If the section is drawn in such a way that the cutting plane is not parallel to either the bases or the side faces, the result is a truncated prism.

Various ratios and formulas are used to find the reduced prismatic elements. Some of them are known from the course of planimetry (for example, to find the area of the base of a prism, it is enough to recall the formula for the area of a square).

Surface area and volume

To determine the volume of a prism using the formula, you need to know the area of \u200b\u200bits base and height:

V = Sprim h

Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in a more detailed form:

V = a² h

If we are talking about a cube - a regular prism with equal length, width and height, the volume is calculated as follows:

To understand how to find the lateral surface area of a prism, you need to imagine its sweep.

It can be seen from the drawing that the side surface is made up of 4 equal rectangles. Its area is calculated as the product of the perimeter of the base and the height of the figure:

Sside = Pos h

Since the perimeter of a square is P = 4a, the formula takes the form:

Sside = 4a h

For cube:

Sside = 4a²

To calculate the total surface area of a prism, add 2 base areas to the side area:

Sfull = Sside + 2Sbase

As applied to a quadrangular regular prism, the formula has the form:

Sfull = 4a h + 2a²

For the surface area of a cube:

Sfull = 6a²

Knowing the volume or surface area, you can calculate individual elements geometric body.

Finding prism elements

Often there are problems in which the volume is given or the value of the lateral surface area is known, where it is necessary to determine the length of the side of the base or the height. In such cases, formulas can be derived:

base side length: a = Sside / 4h = √(V / h);
height or side rib length: h = Sside / 4a = V / a²;
base area: Sprim = V / h;
side face area: Side gr = Sside / 4.

To determine how much area a diagonal section has, you need to know the length of the diagonal and the height of the figure. For a square d = a√2. Therefore:

Sdiag = ah√2

To calculate the diagonal of the prism, the formula is used:

dprize = √(2a² + h²)

To understand how to apply the above ratios, you can practice and solve a few simple tasks.

Examples of problems with solutions

Here are some of the tasks that appear in the state final exams in mathematics.

Exercise 1.

Sand is poured into a box shaped like a regular quadrangular prism. The height of its level is 10 cm. What will the level of sand be if you move it into a container of the same shape, but with a base length 2 times longer?

It should be argued as follows. The amount of sand in the first and second containers did not change, i.e., its volume in them is the same. You can define the length of the base as a. In this case, for the first box, the volume of the substance will be:

V₁ = ha² = 10a²

For the second box, the length of the base is 2a, but the height of the sand level is unknown:

V₂ = h(2a)² = 4ha²

Insofar as V₁ = V₂, the expressions can be equated:

10a² = 4ha²

After reducing both sides of the equation by a², we get:

As a result new level sand will be h = 10 / 4 = 2.5 cm.

Task 2.

ABCDA₁B₁C₁D₁ is a regular prism. It is known that BD = AB₁ = 6√2. Find the total surface area of the body.

To make it easier to understand which elements are known, you can draw a figure.

Since we are talking about a regular prism, we can conclude that the base is a square with a diagonal of 6√2. The diagonal of the side face has the same value, therefore, the side face also has the shape of a square equal to the base. It turns out that all three dimensions - length, width and height - are equal. We can conclude that ABCDA₁B₁C₁D₁ is a cube.

The length of any edge is determined through the known diagonal:

a = d / √2 = 6√2 / √2 = 6

The total surface area is found by the formula for the cube:

Sfull = 6a² = 6 6² = 216

Task 3.

The room is being renovated. It is known that its floor has the shape of a square with an area of 9 m². The height of the room is 2.5 m. What is the lowest cost of wallpapering a room if 1 m² costs 50 rubles?

Since the floor and ceiling are squares, that is, regular quadrilaterals, and its walls are perpendicular to horizontal surfaces, we can conclude that it is a regular prism. It is necessary to determine the area of its lateral surface.

The length of the room is a = √9 = 3 m.

The square will be covered with wallpaper Sside = 4 3 2.5 = 30 m².

The lowest cost of wallpaper for this room will be 50 30 = 1500 rubles.

Thus, to solve problems on a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and a rectangle, as well as to know the formulas for finding the volume and surface area.

How to find the area of a cube

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 prisms .

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 bypass order, 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.

A regular prism has all side faces equal rectangles. A special case of a prism is a parallelepiped.

Parallelepiped

Parallelepiped- This quadrangular prism, which is based on a parallelogram (oblique parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.

cuboid- 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 faces equal squares. The square of a diagonal is equal to the sum of the squares of its three dimensions

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

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 side faces. the bases of the prism are equal polygons, then their areas are equal. So

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.