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 correct prism can be reflected like this:

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.

Lateral surface 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.

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 its length lateral rib. 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 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.

Prism. Parallelepiped

Prism is a polyhedron whose two faces are equal n-gons (bases) , lying in parallel planes, and the remaining n faces are parallelograms (side faces) . Lateral rib The side of a prism that does not belong to the base is called the side of the prism.

A prism whose lateral edges are perpendicular to the planes of the bases is called direct prism (Fig. 1). If the side edges are not perpendicular to the planes of the bases, then the prism is called inclined . Correct A prism is a right prism whose bases are regular polygons.

Height prism is the distance between the planes of the bases. Diagonal A prism is a segment connecting two vertices that do not belong to the same face. Diagonal section is called a section of a prism by a plane passing through two lateral edges that do not belong to the same face. Perpendicular section is called a section of a prism by a plane perpendicular to the side edge of the prism.

Lateral surface area of a prism is the sum of the areas of all lateral faces. Total surface area is called the sum of the areas of all faces of the prism (i.e. the sum of the areas of the side faces and the areas of the bases).

For an arbitrary prism the following formulas are true::

Where l– length of the side rib;

H- height;

P

Q

S side

S full

S base– area of ​​the bases;

V– volume of the prism.

For a straight prism the following formulas are correct:

Where p– base perimeter;

l– length of the side rib;

H- height.

parallelepiped called a prism whose base is a parallelogram. A parallelepiped whose lateral edges are perpendicular to the bases is called direct (Fig. 2). If the side edges are not perpendicular to the bases, then the parallelepiped is called inclined . A right parallelepiped whose base is a rectangle is called rectangular. A rectangular parallelepiped with all edges equal is called cube

The faces of a parallelepiped that do not have common vertices are called opposite . The lengths of edges emanating from one vertex are called measurements parallelepiped. Since a parallelepiped is a prism, its main elements are defined in the same way as they are defined for prisms.

Theorems.

1. The diagonals of a parallelepiped intersect at one point and are bisected by it.

2. In a rectangular parallelepiped, the square of the length of the diagonal equal to the sum squares of its three dimensions:

3. All four diagonals of a rectangular parallelepiped are equal to each other.

For an arbitrary parallelepiped the following formulas are valid:

Where l– length of the side rib;

H- height;

P– perpendicular section perimeter;

Q– Perpendicular cross-sectional area;

S side– lateral surface area;

S full– total surface area;

S base– area of ​​the bases;

V– volume of the prism.

For a right parallelepiped the following formulas are correct:

Where p– base perimeter;

l– length of the side rib;

H– height of a right parallelepiped.

For a rectangular parallelepiped the following formulas are correct:

(3)

Where p– base perimeter;

H- height;

d– diagonal;

a,b,c– measurements of a parallelepiped.

The following formulas are correct for a cube:

Where a– rib length;

d- diagonal of the cube.

Example 1. The diagonal of a rectangular parallelepiped is 33 dm, and its dimensions are in the ratio 2: 6: 9. Find the dimensions of the parallelepiped.

Solution. To find the dimensions of the parallelepiped, we use formula (3), i.e. by the fact that the square of the hypotenuse of a cuboid is equal to the sum of the squares of its dimensions. Let us denote by k proportionality factor. Then the dimensions of the parallelepiped will be equal to 2 k, 6k and 9 k. Let us write formula (3) for the problem data:

Solving this equation for k, we get:

This means that the dimensions of the parallelepiped are 6 dm, 18 dm and 27 dm.

Answer: 6 dm, 18 dm, 27 dm.

Example 2. Find the volume of an inclined triangular prism, the base of which is an equilateral triangle with a side of 8 cm, if the side edge is equal to the side of the base and inclined at an angle of 60º to the base.

Solution . Let's make a drawing (Fig. 3).

In order to find the volume of an inclined prism, you need to know the area of ​​its base and height. The area of ​​the base of a given prism is the area equilateral triangle with a side of 8 cm. Let's calculate it:

The height of a prism is the distance between its bases. From the top A 1 of the upper base, lower the perpendicular to the plane of the lower base A 1 D. Its length will be the height of the prism. Consider D A 1 AD: since this is the angle of inclination of the side edge A 1 A to the base plane, A 1 A= 8 cm. From this triangle we find A 1 D:

Now we calculate the volume using formula (1):

Answer: 192 cm 3.

Example 3. The lateral edge of a regular hexagonal prism is 14 cm. The area of ​​the largest diagonal section is 168 cm 2. Find the total surface area of ​​the prism.

Solution. Let's make a drawing (Fig. 4)

The largest diagonal section is a rectangle A.A. 1 DD 1 since diagonal AD regular hexagon ABCDEF is the largest. In order to calculate the lateral surface area of ​​the prism, it is necessary to know the side of the base and the length of the side edge.

Knowing the area of ​​the diagonal section (rectangle), we find the diagonal of the base.

Since then

Since then AB= 6 cm.

Then the perimeter of the base is:

Let us find the area of ​​the lateral surface of the prism:

The area of ​​a regular hexagon with side 6 cm is:

Find the total surface area of ​​the prism:

Answer:

Example 4. The base of a right parallelepiped is a rhombus. The diagonal cross-sectional areas are 300 cm2 and 875 cm2. Find the area of ​​the lateral surface of the parallelepiped.

Solution. Let's make a drawing (Fig. 5).

Let us denote the side of the rhombus by A, diagonals of a rhombus d 1 and d 2, parallelepiped height h. To find the area of ​​the lateral surface of a right parallelepiped, it is necessary to multiply the perimeter of the base by the height: (formula (2)). Base perimeter p = AB + BC + CD + DA = 4AB = 4a, because ABCD- rhombus N = AA 1 = h. That. Need to find A And h.

Let's consider diagonal sections. AA 1 SS 1 – a rectangle, one side of which is the diagonal of a rhombus AC = d 1, second – side edge AA 1 = h, Then

Similarly for the section BB 1 DD 1 we get:

Using the property of a parallelogram such that the sum of the squares of the diagonals is equal to the sum of the squares of all its sides, we obtain the equality We obtain the following.

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

Prism figure

Before moving on to considering formulas for the base area and surface of a prism of one type or another, you should 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 a figure is formed by two parallel n-gons, then the number of parallelograms will be n.

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

The picture above shows an example of a prism made from paper. The yellow rectangle is its top base. The figure stands on a second similar base. The red and green rectangles are the side faces.

What types of prisms are there?

There are several types of prisms. They all differ from each other in only two parameters:

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

For example, if the bases are triangles, then the prism is called triangular, if it is quadrilateral, as in the previous figure, then the figure is called a quadrangular prism, and so on. In addition, an 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, acute or obtuse. In the first case they speak of a rectangular prism, in the second - of an inclined or oblique one.

Regular prisms are classified as a special type of figures. They have the highest symmetry among other prisms. It will be regular 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 an n-gon varies from three to eight.

Prism surface

The surface of the figure of arbitrary type under consideration is understood as the set of all points that belong to the faces of the prism. It is convenient to study the surface of a prism by examining its development. Below is an example of such a development 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 prism general type its surface will consist of two n-gonal bases and n quadrilaterals.

Let's take a closer look at the issue of calculating the surface area of ​​prisms different types.

The base area of ​​a regular prism

Perhaps the simplest problem when working with prisms is the problem of finding the base area the right figure. Since it is formed by an n-gon whose angles and side lengths are all the same, it can always be divided into identical triangles whose 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 knowledge of the length of its side a. Some difficulty when calculating using the formula can be the calculation of the cotangent, especially when n>4 (for n≤4 the cotangent values ​​are tabular data). To determine this trigonometric function It is recommended to use a calculator.

When posing a geometric problem, you should be careful, since you may need to find the area of ​​the base of the prism. Then the value obtained from the formula should be multiplied by two.

Base area of ​​a triangular prism

Using the example of a triangular prism, let's look at how you can find the area of ​​the base of this figure.

Let's first consider a simple case - a regular prism. The area of ​​the base is calculated using the formula given in the paragraph above; you need to substitute n=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 the specific values ​​of the length of side a of an equilateral triangle into the expression to obtain the area of ​​one base.

Now suppose that there is 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 in this case to find the area of ​​the base of a triangular prism? 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. In the figure, height h is drawn 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 triangular prism shown.

Lateral surface

We looked at 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 their total area is easy to calculate:

S = ∑ i=1 n (a i *b)

Here b is the length of the side edge, a 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 obtain a simple expression:

If the prism is inclined, then to determine the area of ​​its lateral surface, one should make a perpendicular cut, calculate its perimeter P sr and multiply it by the length of the lateral edge.

The picture above shows how this cut should be made for an inclined pentagonal prism.

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