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Properties of the diagonals of a regular quadrangular prism. Prism base area: triangular to polygonal

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

When solving problems on the topic " regular quadrangular prism" implies that:

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√ .

Task.

In a regular quadrangular prism, the base area is 144 cm 2 and the height is 14 cm. Find the diagonal of the prism and the total surface area.

Solution.
A regular quadrilateral is a square.
Accordingly, the side of the base will be equal to

√144 = 12 cm.
Whence the diagonal of the base of a regular rectangular prism will be equal to
√(12 2 + 12 2 ) = √288 = 12√2

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

Task

Find the total surface area of a regular quadrangular prism if its diagonal is 5 cm and the diagonal of the side face is 4 cm.

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.

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 the base of a triangular prism, which is regular, then the triangle is 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 straight 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 (n). 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.

A prism is a fairly simple geometric three-dimensional figure. Nevertheless, some schoolchildren have problems in determining its main properties, the cause of which, as a rule, is associated with incorrectly used terminology. In this article, we will consider what prisms are, what they are called, and also describe in detail the regular quadrangular prism.

Prism in geometry

The study of volumetric figures is a task of stereometry - an important part of spatial geometry. In stereometry, a prism is understood as such a figure, which is formed by the parallel translation of an arbitrary flat polygon at a certain distance in space. Parallel translation involves such a movement in which rotation around an axis perpendicular to the plane of the polygon is completely excluded.

Prism elements and Euler's theorem

Since the three-dimensional figure under consideration is a polyhedron, that is, it is formed by a set of intersecting planes, it is characterized by a certain number of vertices, edges and faces. All of them are elements of a prism.

In the middle of the 18th century, the Swiss mathematician Leonhard Euler established a connection between the number of basic elements of a polyhedron. This relationship is written with the following simple formula:

Number of edges = number of vertices + number of faces - 2

For any prism, this equality is true. Let's give an example of its use. Suppose there is a regular quadrangular prism. It is shown in the figure below.

It can be seen that the number of vertices for it is 8 (4 for each quadrangular base). The number of sides or faces is 6 (2 bases and 4 side rectangles). Then the number of edges for it will be equal to:

Number of ribs = 8 + 6 - 2 = 12

Complete classification of prisms

It is important to understand this classification so that you do not get confused in the terminology later and use the correct formulas to calculate, for example, the surface area or volume of figures.

For any prism of arbitrary shape, 4 features can be distinguished that will characterize it. Let's list them:

By the number of corners of the polygon at the base: triangular, pentagonal, octagonal, and so on.
Polygon type. It may be right or wrong. For example, a right triangle is irregular, but an equilateral triangle is correct.
According to the type of convexity of the polygon. It can be concave or convex. The most common are convex prisms.
At the angles between the bases and side parallelograms. If all these angles are equal to 90o, then they speak of a straight prism, if not all of them are straight, then such a figure is called an oblique.

Of all these points, I would like to dwell on the last one. A straight prism is also called a rectangular prism. This is due to the fact that for it parallelograms are rectangles in the general case (in some cases they can be squares).

For example, the figure above shows a pentagonal concave rectangular, or straight figure.

The base of this prism is a regular quadrilateral, that is, a square. The figure above has already shown what this prism looks like. In addition to the two squares that bound it at the top and bottom, it also includes 4 rectangles.

Let us denote the side of the base of a regular quadrangular prism by the letter a, and the length of its lateral edge by the letter c. This length is also the height of the figure. Then the area of the entire surface of this prism is expressed by the formula:

S = 2*a2 + 4*a*c = 2*a*(a + 2*c)

Here the first term reflects the contribution of the bases to the total area, the second term is the area of the lateral surface.

Taking into account the introduced notation for the lengths of the sides, we write the formula for the volume of the figure in question:

That is, the volume is calculated as the product of the area of the square base and the length of the side rib.

figure cube

Everyone knows this ideal three-dimensional figure, but few people thought that it is a regular quadrangular prism, the side of which is equal to the length of the side of the square base, that is, c \u003d a.

For a cube, the formulas for the total surface area and volume take the form:

Since a cube is a prism consisting of 6 identical squares, any parallel pair of them can be considered a base.

The cube is a highly symmetrical figure, which in nature is realized in the form of crystal lattices of many metallic materials and ionic crystals. For example, the lattices of gold, silver, copper and table salt are cubic.

In the school curriculum for 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 quadrilaterals, 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 hexagon, at the bases of which there are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure is a straight parallelepiped.

A drawing showing a quadrangular prism is shown below.

You can also see in the picture the most important elements that make up a 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 lateral 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 the individual elements of a 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

Because the V₁ = V₂, the expressions can be equated:

10a² = 4ha²

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

As a result, the new sand level 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 quadrangles, 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 for 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

A prism is a geometric three-dimensional figure, the characteristics and properties of which are studied in high school. As a rule, when studying it, such quantities as volume and surface area are considered. In the same article, we will reveal a slightly different question: we will give a method for determining the length of the diagonals of a prism using the example of a quadrangular figure.

What shape is called a prism?

In geometry, the following definition of a prism is given: it is a three-dimensional figure bounded by two polygonal identical sides that are parallel to each other, and a certain number of parallelograms. The figure below shows an example of a prism that fits this definition.

We see that the two red pentagons are equal to each other and are in two parallel planes. Five pink parallelograms connect these pentagons into a single object - a prism. The two pentagons are called the bases of the figure, and its parallelograms are the side faces.

Prisms are straight and inclined, which are also called rectangular and oblique. The difference between them lies in the angles between the base and the side faces. For a rectangular prism, all these angles are 90 o .

By the number of sides or vertices of the polygon at the base, they speak of triangular, pentagonal, quadrangular prisms, and so on. Moreover, if this polygon is regular, and the prism itself is straight, then such a figure is called regular.

The prism shown in the previous figure is a pentagonal oblique. Below is a pentagonal straight prism, which is correct.

All calculations, including the method for determining the diagonals of a prism, are conveniently performed precisely for correct figures.

What elements characterize a prism?

The elements of a figure are the parts that make it up. Specifically for a prism, three main types of elements can be distinguished:

tops;
edges or sides;
ribs.

Faces are bases and side planes, which are parallelograms in the general case. In a prism, each side always belongs to one of two types: either it is a polygon or a parallelogram.

The edges of a prism are those segments that bound each side of the figure. Like faces, edges also come in two types: those belonging to the base and the side surface, or those belonging only to the side surface. The former are always twice as many as the latter, regardless of the type of prism.

The vertices are the intersection points of the three edges of the prism, two of which lie in the plane of the base, and the third belongs to the two side faces. All vertices of the prism are in the planes of the bases of the figure.

The numbers of the described elements are connected in a single equality, which has the following form:

P \u003d B + C - 2.

Here P is the number of edges, B - vertices, C - sides. This equality is called Euler's polyhedron theorem.

The figure shows a triangular regular prism. Everyone can count that it has 6 vertices, 5 sides and 9 edges. These figures are consistent with Euler's theorem.

Prism diagonals

After such properties as volume and surface area, in geometry problems one often encounters information about the length of one or another diagonal of the figure under consideration, which is either given or needs to be found from other known parameters. Consider what are the diagonals of a prism.

All diagonals can be divided into two types:

Lying in the plane of the faces. They connect non-adjacent vertices of either the polygon at the base of the prism, or the side surface parallelogram. The value of the lengths of such diagonals is determined based on the knowledge of the lengths of the corresponding edges and the angles between them. To determine the diagonals of parallelograms, the properties of triangles are always used.
Prisms lying inside the volume. These diagonals connect non-similar vertices of two bases. These diagonals are completely inside the figure. Their lengths are somewhat more difficult to calculate than for the previous type. The calculation method involves taking into account the lengths of the edges and the base, and parallelograms. For straight and regular prisms, the calculation is relatively simple, since it is carried out using the Pythagorean theorem and the properties of trigonometric functions.

Diagonals of the sides of a quadrangular right prism

The figure above shows four identical straight prisms, and the parameters of their edges are given. Diagonal A, Diagonal B, and Diagonal C prisms show the diagonals of three different faces with a dashed red line. Since the prism is a straight line with a height of 5 cm, and its base is a rectangle with sides of 3 cm and 2 cm, it is not difficult to find the marked diagonals. To do this, you need to use the Pythagorean theorem.

The length of the diagonal of the base of the prism (Diagonal A) is:

D A \u003d √ (3 2 +2 2) \u003d √13 ≈ 3.606 cm.

For the side face of a prism, the diagonal is (see Diagonal B):

D B \u003d √ (3 2 +5 2) \u003d √34 ≈ 5.831 cm.

Finally, the length of another side diagonal is (see Diagonal C):

D C \u003d √ (2 2 +5 2) \u003d √29 ≈ 5.385 cm.

Length of the inner diagonal

Now let's calculate the length of the diagonal of the quadrangular prism, which is shown in the previous figure (Diagonal D). This is not so difficult to do if you notice that it is the hypotenuse of a triangle in which the legs will be the height of the prism (5 cm) and the diagonal D A shown in the figure at the top left (Diagonal A). Then we get:

D D \u003d √ (D A 2 +5 2) \u003d √ (2 2 +3 2 +5 2) \u003d √38 ≈ 6.164 cm.

Right quadrangular prism

The diagonal of a regular prism whose base is a square is calculated in the same way as in the example above. The corresponding formula looks like:

D = √(2*a 2 +c 2).

Where a and c are the lengths of the side of the base and the side edge, respectively.

Note that in the calculations we used only the Pythagorean theorem. To determine the lengths of the diagonals of regular prisms with a large number of vertices (pentagonal, hexagonal, and so on), it is already necessary to apply trigonometric functions.