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Definition of prism. Theorem on the lateral surface area of ​​a straight prism

Definition 1. Prismatic surface
Theorem 1. On parallel sections of a prismatic surface
Definition 2. Perpendicular section of a prismatic surface
Definition 3. Prism
Definition 4. Prism height
Definition 5. Right prism
Theorem 2. The area of ​​the lateral surface of the prism

Parallelepiped:
Definition 6. Parallelepiped
Theorem 3. On the intersection of the diagonals of a parallelepiped
Definition 7. Right parallelepiped
Definition 8. Rectangular parallelepiped
Definition 9. Measurements of a parallelepiped
Definition 10. Cube
Definition 11. Rhombohedron
Theorem 4. On the diagonals of a rectangular parallelepiped
Theorem 5. Volume of a prism
Theorem 6. Volume of a straight prism
Theorem 7. Volume of a rectangular parallelepiped

Prism is a polyhedron whose two faces (bases) lie in parallel planes, and the edges that do not lie in these faces are parallel to each other.
Faces other than the bases are called lateral.
The sides of the side faces and bases are called prism ribs, the ends of the edges are called the vertices of the prism. Lateral ribs edges that do not belong to the bases are called. The union of lateral faces is called lateral surface of the prism, and the union of all faces is called the full surface of the prism. Prism height called the perpendicular dropped from the point of the upper base to the plane of the lower base or the length of this perpendicular. Direct prism called a prism whose side edges are perpendicular to the planes of the bases. Correct called a straight prism (Fig. 3), at the base of which lies a regular polygon.

Designations:
l - side rib;
P - base perimeter;
S o - base area;
H - height;
P^ - perpendicular section perimeter;
S b - lateral surface area;
V - volume;
S p - area full surface prisms.

V=SH
S p = S b + 2S o
S b = P ^ l
Definition 1 . A prismatic surface is a figure formed by parts of several planes parallel to one straight line, limited by those straight lines along which these planes successively intersect one another*; these lines are parallel to each other and are called edges of the prismatic surface.
*It is assumed that every two successive planes intersect and that the last plane intersects the first

Theorem 1 . Sections of a prismatic surface by planes parallel to each other (but not parallel to its edges) are equal polygons.
Let ABCDE and A"B"C"D"E" be sections of a prismatic surface by two parallel planes. To make sure that these two polygons are equal, it is enough to show that triangles ABC and A"B"C" are equal and have the same direction of rotation and that the same is true for triangles ABD and A"B"D", ABE and A"B"E". But the corresponding sides of these triangles are parallel (for example, AC is parallel to AC) like the line of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example, AC is equal to A"C"), like opposite sides of a parallelogram, and that the angles formed by these sides are equal and have the same direction.

Definition 2 . A perpendicular section of a prismatic surface is a section of this surface by a plane perpendicular to its edges. Based on the previous theorem, all perpendicular sections of the same prismatic surface will be equal polygons.

Definition 3 . A prism is a polyhedron bounded by a prismatic surface and two planes parallel to each other (but not parallel to the edges of the prismatic surface)
The faces lying in these last planes are called prism bases; faces belonging to the prismatic surface - side faces; edges of the prismatic surface - side ribs of the prism. By virtue of the previous theorem, the base of the prism is equal polygons. All side faces prisms - parallelograms; all side ribs are equal to each other.
Obviously, if the base of the prism ABCDE and one of the edges AA" in size and direction are given, then it is possible to construct a prism by drawing edges BB", CC", ... equal and parallel to edge AA".

Definition 4 . The height of a prism is the distance between the planes of its bases (HH").

Definition 5 . A prism is called straight if its bases are perpendicular sections of the prismatic surface. In this case, the height of the prism is, of course, its side rib; the side edges will be rectangles.
Prisms can be classified according to the number of side faces, equal number sides of the polygon that serves as its base. Thus, prisms can be triangular, quadrangular, pentagonal, etc.

Theorem 2 . The area of ​​the lateral surface of the prism is equal to the product of the lateral edge and the perimeter of the perpendicular section.
Let ABCDEA"B"C"D"E" be a given prism and abcde its perpendicular section, so that the segments ab, bc, .. are perpendicular to its lateral edges. The face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that coincides with ab; the area of ​​the face ВСВ "С" is equal to the product of the base ВВ" by the height bc, etc. Consequently, lateral surface(i.e., the sum of the areas of the side faces) is equal to the product of the side edge, in other words, the total length of the segments AA", BB", .., by the sum ab+bc+cd+de+ea.

Definition. Prism- this is a polyhedron, all of whose vertices are located in two parallel planes, and in these same two planes lie two faces of the prism, which are equal polygons with, respectively parallel sides, and all edges not lying 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 lateral surface of the prism .

All lateral faces of the prism are parallelograms .

The edges that do not lie at the bases are called the lateral edges of the prism ( AA 1, BB 1, CC 1, DD 1, EE 1).

Prism diagonal is a segment whose ends are two vertices of a prism that do not lie on the same face (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 order of traversal, the vertices of one base are indicated, and then, in the same order, the vertices of another; the ends of each side edge are designated by the same letters, only the vertices lying in one base are designated 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 there is a pentagon at the base, so the prism is called pentagonal prism. But because such a prism has 7 faces, then it heptahedron(2 faces - the bases of the prism, 5 faces - parallelograms, - its side faces)

Among straight prisms, a particular type stands out: regular prisms.

A straight prism is called correct, if its bases are regular polygons.

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

Parallelepiped

Parallelepiped is a quadrangular prism, at the base of which lies a parallelogram (an inclined parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.

Rectangular parallelepiped- a right parallelepiped whose base is a rectangle.

Properties and theorems:


Some properties of a parallelepiped are similar to the known properties of a parallelogram. A rectangular parallelepiped having equal dimensions is called cube .A cube has all equal squares.Diagonal square, equal to the sum squares of its three dimensions

,

where d is the diagonal of the square;
a is the side of the square.

An idea of ​​a prism is given by:





The area of ​​the total and lateral surface 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 lateral faces. The bases of the prism are equal polygons, then their areas are equal. That's why

S full = S side + 2S main,

Where S full- total surface area, S side-lateral surface area, S base- base area

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

S side= P basic * h,

Where S side-area of ​​the lateral surface of a straight prism,

P main - perimeter of the base of a straight prism,

h is the height of the straight prism, equal to the side edge.

Prism volume

The volume of a prism is equal to the product of the area of ​​the base and the height.

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 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 ​​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 can be seen 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 = 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 the total surface area of ​​the 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 or side rib length: 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.

Sand is poured into a box shaped like a regular quadrangular prism. 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















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 bisect it.

2. In a rectangular parallelepiped, the square of the length of the diagonal is equal to the sum of the 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. Side rib correct hexagonal prism equal to 14 cm. The area of ​​the largest diagonal section is equal to 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 H = 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.

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.