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.

The figure, which depicts 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 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 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

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, 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

In spatial geometry, when solving problems with prisms, there is often a problem with calculating the area of ​​the sides or faces that form these three-dimensional figures. This article is devoted to the issue of determining the area of ​​the base of the prism and its lateral surface.

Figure prism

Before proceeding to the consideration of the formulas for the area of ​​​​the base and the surface of a prism of one kind or another, it is necessary to understand what kind of figure we are talking about.

A prism in geometry is a spatial figure consisting of two parallel polygons that are equal to each other, and several quadrangles or parallelograms. The number of the latter is always equal to the number of vertices of one polygon. For example, if the figure is formed by two parallel n-gons, then the number of parallelograms will be n.

The connecting n-gons of the parallelogram are called the sides of the prism, and their total area is the area of ​​the side surface of the figure. The n-gons themselves are called bases.

The figure above shows an example of a paper prism. The yellow rectangle is its upper base. On the second base of the same figure stands. The red and green rectangles are the side faces.

What are the prisms?

There are several types of prisms. All of them differ from each other in just two parameters:

• the type of n-gon forming the bases;
• angle between the n-gon and the side faces.

For example, if the bases are triangles, then the prism is called triangular, if quadrilaterals, as in the previous figure, then the figure is called a quadrangular prism, and so on. In addition, the n-gon can be convex or concave, then this property is also added to the name of the prism.

The angle between the side faces and the base can be either straight or acute or obtuse. In the first case, they talk about a rectangular prism, in the second - about an inclined or oblique.

Regular prisms are distinguished into a special type of figure. They have the highest symmetry among the other prisms. It will be correct only if it is rectangular and its base is a regular n-gon. The figure below shows a set of regular prisms, in which the number of sides of the n-gon varies from three to eight.

Prism surface

Under the surface of the considered figure of an arbitrary type is understood the totality of all points that belong to the faces of the prism. It is convenient to study the surface of a prism by considering its development. Below is an example of such a sweep for a triangular prism.

It can be seen that the entire surface is formed by two triangles and three rectangles.

In the case of a general type prism, its surface will consist of two n-gonal bases and n quadrilaterals.

Let us consider in more detail the issue of calculating the surface area of ​​prisms of different types.

Base area of ​​a prism

Perhaps the easiest task when working with prisms is the problem of finding the base area of ​​a regular figure. Since it is formed by an n-gon, in which all angles and side lengths are the same, it is always possible to divide it into identical triangles, for which angles and sides are known. The total area of ​​the triangles will be the area of ​​the n-gon.

Another way to determine the portion of the surface area of ​​a prism (base) is to use a well-known formula. It looks like this:

S n = n/4*a 2 *ctg(pi/n)

That is, the area S n of an n-gon is uniquely determined based on the knowledge of the length of its side a. Some difficulty in calculating the formula can be the calculation of the cotangent, especially when n>4 (for n≤4, the values ​​of the cotangent are tabular data). To determine this trigonometric function, it is recommended to use a calculator.

When setting a geometric problem, you should be careful, because you may need to find the area of ​​​​the bases of the prism. Then the value obtained by the formula should be multiplied by two.

Base area of ​​a triangular prism

Using the example of a triangular prism, consider how you can find the area of ​​\u200b\u200bthe base of this figure.

First, consider a simple case - a regular prism. The area of ​​\u200b\u200bthe base is calculated according to the formula given in the paragraph above, you need to substitute n \u003d 3 into it. We get:

S 3 = 3/4*a 2 *ctg(pi/3) = 3/4*a 2 *1/√3 = √3/4*a 2

It remains to substitute in the expression the specific values ​​of the length of the side a of an equilateral triangle to get the area of ​​\u200b\u200bone base.

Now suppose we have a prism whose base is an arbitrary triangle. Its two sides a and b and the angle between them α are known. This figure is shown below.

How to find the area of ​​the base of a triangular prism in this case? It must be remembered that the area of ​​any triangle is equal to half the product of the side and the height lowered to this side. The figure shows the height h to side b. The length h corresponds to the product of the sine of the angle alpha and the length of the side a. Then the area of ​​the entire triangle is:

S = 1/2*b*h = 1/2*b*a*sin(α)

This is the base area of ​​the depicted triangular prism.

Side surface

We figured out how to find the area of ​​​​the base of a prism. The lateral surface of this figure always consists of parallelograms. For straight prisms, parallelograms become rectangles, so it's easy to calculate their total area:

S = ∑ i=1 n (a i *b)

Here b is the length of the side edge, and i is the length of the side of the i-th rectangle, which coincides with the length of the side of the n-gon. In the case of a regular n-gonal prism, we get a simple expression:

If the prism is inclined, then to determine the area of ​​its lateral surface, a perpendicular cut should be made, its perimeter P sr calculated and multiplied by the length of the lateral rib.

The figure above shows how this cut should be made for an oblique pentagonal prism.

Prism elements

Name	Definition	Designations on the drawing	Drawing
Foundations	Two faces that are congruent polygons lying in parallel planes.	ABCDE , KLMNP
Side faces	All faces except bases. Each side face is necessarily a parallelogram.	ABLK , BCML , CDNM , DEPN , EAKP
Side surface	Merging side faces.
Full surface	Union of bases and lateral surface.
Side ribs	Common sides of the side faces.	AK , BL , CM , DN , EP
Height	A segment connecting the bases of a prism and perpendicular to them.	KR
Diagonal	A segment connecting two vertices of a prism that do not belong to the same face.	BP
Diagonal plane	The plane passing through the lateral edge of the prism and the diagonal of the base.
Diagonal section	The intersection of a prism and a diagonal plane. A parallelogram is formed in the section, including its special cases - a rhombus, a rectangle, a square.	EBLP
Perpendicular Section	The intersection of a prism and a plane perpendicular to its side edge.

Prism Properties

• 1. The bases of the prism are equal polygons.
• 2. The side faces of the prism are parallelograms.
• 3. The side edges of the prism are parallel and equal.
• 4. Prism volume equal to the product of its height and the area of ​​​​the base:

• 5. The area of ​​the total surface of the prism is equal to the sum of the area of ​​its lateral surface and twice the area of ​​the base.

Prism types

Prisms are straight and oblique.

straight prism- a prism in which all lateral edges are perpendicular to the base.

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

tilted prism- a prism in which at least one lateral edge is not perpendicular to the base.

Lateral surface area of an inclined prism is equal to the product of the perimeter of the perpendicular section and the length of the side rib. Volume of an inclined prism is equal to the product of the area of ​​the perpendicular section and the side edge.

Correct prism is a right prism whose base is a regular polygon.

Properties of a regular prism

• 1. The bases of a regular prism are regular polygons.
• 2. The side faces of a regular prism are equal rectangles.
• 3. The side edges of a regular prism are equal.

see also

Links

Polyhedra
Correct (Platonic Solids)
Regular non-convex	Stellated polyhedron (Stellated octahedron, Stellated dodecahedron, Stellated icosahedron, Stellated icosidodecahedron)
convex
Formulas, theorems, theories
Other

Wikimedia Foundation. 2010 .

See what "Prism (mathematics)" is in other dictionaries:

- (beginning) "Mathematics in nine books" (Chinese traditional 九章算術 ... Wikipedia

A branch of mathematics that studies the properties of various shapes (points, lines, angles, two-dimensional and three-dimensional objects), their size and relative position. For the convenience of teaching, geometry is divided into planimetry and solid geometry. V… … Collier Encyclopedia

Zemlyakov, Alexander Nikolaevich File: Zemlyakov.jpg Alexander Nikolaevich Zemlyakov (April 17, 1950 (19500417), Bologoe January 1, 2005, Chernogolovka) mathematician, outstanding Soviet and Russian teacher, author of educational pedagogical ... ... Wikipedia

Alexander Nikolaevich Zemlyakov (April 17, 1950 (19500417), Bologoe January 1, 2005, Chernogolovka) mathematician, outstanding Soviet and Russian teacher, author of educational and pedagogical literature. Biography Graduated in 1967 with a gold medal ... ... Wikipedia

Dodecahedron A regular polyhedron or Platonic solid is a convex polyhedron consisting of identical regular polygons and having spatial symmetry ... Wikipedia

This term has other meanings, see Pyramidatsu (meanings). The reliability of this section of the article has been questioned. It is necessary to verify the accuracy of the facts stated in this section. There may be explanations on the talk page ... Wikipedia

The video course "Get an A" includes all the topics necessary for the successful passing of the exam in mathematics by 60-65 points. Completely all tasks 1-13 of the Profile USE in mathematics. Also suitable for passing the Basic USE in mathematics. If you want to pass the exam with 90-100 points, you need to solve part 1 in 30 minutes and without mistakes!

Preparation course for the exam for grades 10-11, as well as for teachers. Everything you need to solve part 1 of the exam in mathematics (the first 12 problems) and problem 13 (trigonometry). And this is more than 70 points on the Unified State Examination, and neither a hundred-point student nor a humanist can do without them.

All the necessary theory. Quick solutions, traps and secrets of the exam. All relevant tasks of part 1 from the Bank of FIPI tasks have been analyzed. The course fully complies with the requirements of the USE-2018.

The course contains 5 large topics, 2.5 hours each. Each topic is given from scratch, simply and clearly.

Hundreds of exam tasks. Text problems and probability theory. Simple and easy to remember problem solving algorithms. Geometry. Theory, reference material, analysis of all types of USE tasks. Stereometry. Cunning tricks for solving, useful cheat sheets, development of spatial imagination. Trigonometry from scratch - to task 13. Understanding instead of cramming. Visual explanation of complex concepts. Algebra. Roots, powers and logarithms, function and derivative. Base for solving complex problems of the 2nd part of the exam.

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.