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The side surface area is correct. Prism

IN school curriculum In a stereometry course, the study of three-dimensional figures usually begins with a simple geometric body - the polyhedron of a prism. 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 are perpendicular sides shaped like parallelograms (or rectangles if the prism is not inclined).

What does a prism look like?

A regular quadrangular prism is a hexagon, the bases of which are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure- 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 geometric body . These include:

Sometimes in geometry problems you can come across the concept of a section. The definition will sound like this: a section is all the points of a volumetric body belonging to a cutting plane. The section can be perpendicular (intersects the edges of the figure at an angle of 90 degrees). For a rectangular prism, a diagonal section is also considered ( maximum quantity sections that can be constructed - 2), passing through 2 edges and 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.

To find the reduced prismatic elements, various relations and formulas are used. Some of them are known from the planimetry course (for example, to find the area of ​​the base of a prism, it is enough to remember 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 ​​its base and height:

V = Sbas h

Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in 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 development.

From the drawing it is clear that lateral surface 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 = Posn h

Taking into account that the perimeter of the square is equal to P = 4a, the formula takes the form:

Sside = 4a h

For cube:

Sside = 4a²

To calculate area full surface of a prism, you need to add 2 base areas to the lateral area:

Sfull = Sside + 2Smain

In relation to a quadrangular regular prism, the formula looks like:

Stotal = 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, the formulas can be derived:

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

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

Sdiag = ah√2

To calculate the diagonal of a prism, use the formula:

dprize = √(2a² + h²)

To understand how to apply the given relationships, you can practice and solve several simple tasks.

Examples of problems with solutions

Here are some tasks found on state final exams in mathematics.

Task 1.

In a box that has the right shape quadrangular prism, sand is poured. The height of its level is 10 cm. What will the sand level be if you move it into a container of the same shape, but with a base twice as long?

It should be reasoned 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 denote the length of the base by 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

Since V₁ = V₂, we can equate the expressions:

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 correct 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 at the base there is a square with a diagonal of 6√2. The diagonal of the side face has the same size, 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 a known diagonal:

a = d / √2 = 6√2 / √2 = 6

The total surface area is found using the formula for a 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, i.e. 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 area 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 involving a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and rectangle, as well as to know the formulas for finding the volume and surface area.

How to find the area of ​​a cube















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The lateral surface area of ​​the prism. Hello! In this publication we will analyze a group of problems in stereometry. Let's consider a combination of bodies - a prism and a cylinder. On at the moment This article completes the entire series of articles related to the consideration of types of tasks in stereometry.

If new ones appear in the task bank, then, of course, there will be additions to the blog in the future. But what is already there is quite enough for you to learn how to solve all the problems with a short answer as part of the exam. There will be enough material for years to come (the mathematics program is static).

The presented tasks involve calculating the area of ​​a prism. I note that below we consider a straight prism (and, accordingly, a straight cylinder).

Without knowing any formulas, we understand that the lateral surface of a prism is all its lateral faces. A straight prism has rectangular side faces.

The area of ​​the lateral surface of such a prism is equal to the sum of the areas of all its lateral faces (that is, rectangles). If we are talking about a regular prism into which a cylinder is inscribed, then it is clear that all the faces of this prism are EQUAL rectangles.

Formally, the lateral surface area of ​​a regular prism can be reflected as follows:


27064. A regular quadrangular prism is circumscribed about a cylinder whose base radius and height are equal to 1. Find the lateral surface area of ​​the prism.

The lateral surface of this prism consists of four rectangles of equal area. The height of the face is 1, the edge of the base of the prism is 2 (these are two radii of the cylinder), therefore the area of ​​the side face is equal to:

Side surface area:

73023. Find the lateral surface area of ​​a regular triangular prism circumscribed about a cylinder whose base radius is √0.12 and height is 3.

The lateral surface area of ​​this prism is equal to the sum three squares side faces (rectangles). To find the area of ​​the side face, you need to know its height and the length of the base edge. The height is three. Let's find the length of the base edge. Consider the projection (top view):

We have regular triangle into which a circle with radius √0.12 is inscribed. From the right triangle AOC we can find AC. And then AD (AD=2AC). By definition of tangent:

This means AD = 2AC = 1.2. Thus, the lateral surface area is equal to:

27066. Find the lateral surface area of ​​a regular hexagonal prism circumscribed about a cylinder whose base radius is √75 and height is 1.

The required area is equal to the sum of the areas of all side faces. A regular hexagonal prism has lateral faces that are equal rectangles.

To find the area of ​​a face, you need to know its height and the length of the base edge. The height is known, it is equal to 1.

Let's find the length of the base edge. Consider the projection (top view):

We have a regular hexagon, into which a circle of radius √75 is inscribed.

Let's consider right triangle ABO. We know the leg OB (this is the radius of the cylinder). We can also determine the angle AOB, it is equal to 300 (triangle AOC is equilateral, OB is a bisector).

Let's use the definition of tangent in a right triangle:

AC = 2AB, since OB is the median, that is, it divides AC in half, which means AC = 10.

Thus, the area of ​​the side face is 1∙10=10 and the area of ​​the side surface is:

76485. Find the lateral surface area of ​​a regular triangular prism inscribed in a cylinder whose base radius is 8√3 and height is 6.

The area of ​​the lateral surface of the specified prism of three equal-sized faces (rectangles). To find the area, you need to know the length of the edge of the base of the prism (we know the height). If we consider the projection (top view), we have a regular triangle inscribed in a circle. The side of this triangle is expressed in terms of radius as:

Details of this relationship. So it will be equal

Then the area of ​​the side face is: 24∙6=144. And the required area:

245354. A regular quadrangular prism is circumscribed about a cylinder whose base radius is 2. The lateral surface area of ​​the prism is 48. Find the height of the cylinder.

It's simple. We have four side faces of equal area, therefore the area of ​​one face is 48:4=12. Since the radius of the base of the cylinder is 2, the edge of the base of the prism will be early 4 - it is equal to the diameter of the cylinder (these are two radii). We know the area of ​​the face and one edge, the second being the height will be equal to 12:4=3.

27065. Find the lateral surface area of ​​a regular triangular prism circumscribed about a cylinder whose base radius is √3 and height is 2.

Best regards, Alexander.

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- this is a 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 lateral edges

Diagonal section- the boundaries of the intersection of the prism and the diagonal plane. The diagonal cross 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 lateral edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are indicated by the corresponding letters:

  • The 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 lateral faces of the prism
  • Total surface - the sum of the areas of all bases and side faces (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 side faces are rectangles
  • The side edges are equal to each other
  • The side faces are perpendicular to the bases
  • The lateral ribs are parallel to each other and equal
  • Perpendicular section perpendicular to all side ribs and parallel to the bases
  • Angles of perpendicular section - straight
  • The diagonal cross 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" means 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 properties of a regular quadrangular prism above) Note. This is part of a lesson with geometry problems (section stereometry - prism). Here are problems that are difficult to solve. If you need to solve a geometry problem that is not here, write about it in the forum. To indicate the action of retrieving square root the symbol is used in solving problems√ .

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

√144 = 12 cm.
From where 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

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

Solution.
Since the base of a regular quadrangular prism is a square, we find the side of the base (denoted as a) using 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 = 4 2
h 2 + 12.5 = 16
h 2 = 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 = 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.

These are the most common three-dimensional figures among other similar ones that are found in everyday life and nature. Stereometry, or spatial geometry, studies their properties. In this article we will discuss the question of how you can find the lateral surface area of ​​a regular triangular prism, as well as a quadrangular and hexagonal one.

What is a prism?

Before calculating the lateral surface area of ​​a regular triangular prism and other types of this figure, you should understand what they are. Then we will learn to determine the quantities of interest.

A prism, from the point of view of geometry, is a volumetric body that is bounded by two arbitrary identical polygons and n parallelograms, where n is the number of sides of one polygon. It’s easy to draw such a figure; to do this, you should draw some kind of polygon. Then draw a segment from each of its vertices that will be equal in length and parallel to all the others. Then you need to connect the ends of these lines together so that you get another polygon equal to the original one.

Above you can see that the figure is limited by two pentagons (they are called the lower and upper bases of the figure) and five parallelograms, which correspond to rectangles in the figure.

All prisms differ from each other in two main parameters:

  • the type of polygon underlying the figure;
  • angles between parallelograms and bases.

The number of sides of a rectangle gives the name to a prism. From here we get the above-mentioned triangular, hexagonal and quadrangular figures.

They also differ in the amount of slope. As for the marked angles, if they are equal to 90 o, then such a prism is called straight or rectangular (the angle of inclination is zero). If some of the angles are not right, then the figure is called oblique. The difference between them is clear at first glance. The picture below shows these varieties.

As you can see, the height h coincides with the length of its side edge. In the case of an oblique angle, this parameter is always smaller.

Which prism is called correct?

Since we must answer the question of how to find the lateral surface area of ​​a regular prism (triangular, quadrangular, and so on), we need to define this type of volumetric figure. Let's analyze the material in more detail.

A regular prism is a rectangular figure in which a regular polygon forms identical bases. This figure can be an equilateral triangle, a square, or others. Any n-gon whose side lengths and angles are all the same will be regular.

A number of such prisms are shown schematically in the figure below.

Lateral surface of the prism

As was said in this figure consists of n + 2 planes, which, intersecting, form n + 2 faces. Two of them belong to the bases, the rest are formed by parallelograms. The area of ​​the entire surface consists of the sum of the areas of the indicated faces. If we do not include the values ​​of the two bases, then we get the answer to the question of how to find the lateral surface area of ​​a prism. So, you can determine its meaning and bases separately from each other.

Below is given for which the lateral surface is formed by three quadrangles.

Let's consider the calculation process further. Obviously, the area of ​​the lateral surface of the prism is equal to the sum of the n areas of the corresponding parallelograms. Here n is the number of sides of the polygon forming the base of the figure. The area of ​​each parallelogram can be found by multiplying the length of its side by its height. This applies to the general case.

If the prism under study is straight, then the procedure for determining the area of ​​its lateral surface S b is greatly simplified, since such a surface consists of rectangles. In this case, you can use the following formula:

Where h is the height of the figure, P o is the perimeter of its base

Regular prism and its lateral surface

In the case of such a figure, the formula given in the paragraph above takes on a very specific form. Since the perimeter of an n-gon is equal to the product of the number of its sides and the length of one, the following formula is obtained:

Where a is the side length of the corresponding n-gon.

Lateral surface area of ​​quadrangular and hexagonal

Let's use the formula above to determine the required values ​​for the three types of shapes noted. The calculations will look like this:

For a triangular formula will take the form:

For example, the side of a triangle is 10 cm, and the height of the figure is 7 cm, then:

S 3 b = 3*10*7 = 210 cm 2

In the case of a quadrangular prism, the desired expression takes the form:

If we take the same length values ​​as in the previous example, then we get:

S 4 b = 4*10*7 = 280 cm 2

The lateral surface area of ​​a hexagonal prism is calculated by the formula:

Substituting the same numbers as in the previous cases, we have:

S 6 b = 6*10*7 = 420 cm 2

Note that in the case of a regular prism of any type, its lateral surface is formed by identical rectangles. In the examples above, the area of ​​each of them was a*h = 70 cm 2.

Calculation for an oblique prism

Determining the value of the lateral surface area for a given figure is somewhat more difficult than for a rectangular one. Nevertheless, the above formula remains the same, only instead of the base perimeter, the perpendicular cut perimeter should be taken, and instead of the height, the length of the side edge should be taken.

The picture above shows a quadrangular oblique prism. The shaded parallelogram is the perpendicular slice whose perimeter P sr must be calculated. The length of the side edge in the figure is indicated by the letter C. Then we get the formula:

The perimeter of the cut can be found if the angles of the parallelograms forming the lateral surface are known.