The area of ​​the lateral surface of the prism. Hello! In this publication, we will analyze a group of tasks on stereometry. Consider a combination of bodies - a prism and a cylinder. At the moment, this article completes the entire series of articles related to the consideration of types of tasks in stereometry.

If new tasks 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 so that you can learn how to solve all problems with a short answer as part of the exam. The material will be enough for years to come (the program in mathematics is static).

The presented tasks are related to the calculation of the area of ​​the 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. In a straight prism, the side faces are rectangles.

The lateral surface area 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 in 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 expressed as follows:

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

The lateral surface of this prism consists of four rectangles equal in 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), so the area of ​​the side face is:

Side surface area:

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

The area of ​​the lateral surface of this prism is equal to the sum of the areas of the three lateral 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. Find the length of the edge of the base. Consider the projection (top view):

We have a regular triangle in 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:

So AD \u003d 2AC \u003d 1.2. Thus, the area of ​​\u200b\u200bthe lateral surface is equal to:

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

The desired area is equal to the sum of the areas of all side faces. For a regular hexagonal prism, the side faces 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.

Find the length of the edge of the base. Consider the projection (top view):

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

Consider a 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 the tangent in a right triangle:

AC \u003d 2AB, since OB is a median, that is, it divides AC in half, which means AC \u003d 10.

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

76485. Find the area of ​​the lateral surface of a regular triangular prism inscribed in a cylinder whose base radius is 8√3 and whose 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), then we have a regular triangle inscribed in a circle. The side of this triangle is expressed in terms of the radius as:

Details of this relationship. So it will be equal

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

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

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.

Any polygon can lie at the base of the prism - a triangle, a quadrilateral, etc. Both bases are exactly the same, and accordingly, by which the angles of parallel faces are connected to each other, they are always parallel. At the base of a regular prism lies a regular polygon, that is, one in which all sides are equal. In a straight prism, the edges between the side faces are perpendicular to the base. In this case, a polygon with any number of angles can lie at the base of a straight prism. A prism whose base is a parallelogram is called a parallelepiped. A rectangle is a special case of a parallelogram. If this figure lies at the base, and the side faces are located at right angles to the base, the parallelepiped is called rectangular. The second name of this geometric body is rectangular.

How she looks like

There are quite a lot of rectangular prisms in the environment of modern man. This, for example, is the usual cardboard from under shoes, computer components, etc. Look around. Even in a room, you will surely see many rectangular prisms. This is a computer case, and a bookcase, and a refrigerator, and a cabinet, and many other items. The form is extremely popular mainly because it allows you to use the space as efficiently as possible, whether you are decorating the interior or packing things in cardboard before moving.

Properties of a rectangular prism

A rectangular prism has a number of specific properties. Any pair of faces can serve as its, since all adjacent faces are located at the same angle to each other, and this angle is 90 °. The volume and surface area of ​​a rectangular prism is easier to calculate than any other. Take any object that has the shape of a rectangular prism. Measure its length, width and height. To find the volume, it is enough to multiply these measurements. That is, the formula looks like this: V \u003d a * b * h, where V is the volume, a and b are the sides of the base, h is the height that coincides with the side edge of this geometric body. The base area is calculated by the formula S1=a*b. To get the side surface, you must first calculate the perimeter of the base using the formula P=2(a+b) and then multiply it by the height. It turns out the formula S2=P*h=2(a+b)*h. To calculate the total surface area of ​​a rectangular prism, add twice the area of ​​the base and the area of ​​the side surface. The formula is S=2S1+S2=2*a*b+2*(a+b)*h=2

Polyhedra

The main object of study of stereometry are three-dimensional bodies. Body is a part of space bounded by some surface.

polyhedron A body whose surface consists of a finite number of plane polygons is called. A polyhedron is called convex if it lies on one side of the plane of every flat polygon on its surface. The common part of such a plane and the surface of a polyhedron is called edge. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called edges of the polyhedron, and the vertices vertices of the polyhedron.

For example, a cube consists of six squares that are its faces. It contains 12 edges (sides of squares) and 8 vertices (vertices of squares).

The simplest polyhedra are prisms and pyramids, which we will study further.

Prism

Definition and properties of a prism

prism is called a polyhedron consisting of two flat polygons lying in parallel planes combined by parallel translation, and all segments connecting the corresponding points of these polygons. The polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are side edges of the prism.

Prism height called the distance between the planes of its bases (). A segment connecting two vertices of a prism that do not belong to the same face is called prism diagonal(). The prism is called n-coal if its base is an n-gon.

Any prism has the following properties, which follow from the fact that the bases of the prism are combined by parallel translation:

1. The bases of the prism are equal.

2. The side edges of the prism are parallel and equal.

The surface of a prism is made up of bases and lateral surface. The lateral surface of the prism consists of parallelograms (this follows from the properties of the prism). The area of ​​the lateral surface of a prism is the sum of the areas of the lateral faces.

straight prism

The prism is called straight if its side edges are perpendicular to the bases. Otherwise, the prism is called oblique.

The faces of a straight prism are rectangles. The height of a straight prism is equal to its side faces.

full prism surface is the sum of the lateral surface area and the areas of the bases.

Correct prism is called a right prism with a regular polygon at the base.

Theorem 13.1. The area of ​​the lateral surface of a straight prism is equal to the product of the perimeter and the height of the prism (or, equivalently, to the lateral edge).

Proof. The side faces of a straight prism are rectangles whose bases are the sides of the polygons at the bases of the prism, and the heights are the side edges of the prism. Then, by definition, the lateral surface area is:

,

where is the perimeter of the base of a straight prism.

Parallelepiped

If parallelograms lie at the bases of a prism, then it is called parallelepiped. All the faces of a parallelepiped are parallelograms. In this case, the opposite faces of the parallelepiped are parallel and equal.

Theorem 13.2. The diagonals of the parallelepiped intersect at one point and the intersection point is divided in half.

Proof. Consider two arbitrary diagonals, for example, and . Because the faces of the parallelepiped are parallelograms, then and , which means that according to T about two straight lines parallel to the third . In addition, this means that the lines and lie in the same plane (the plane). This plane intersects parallel planes and along parallel lines and . Thus, a quadrilateral is a parallelogram, and by the property of a parallelogram, its diagonals and intersect and the intersection point is divided in half, which was required to be proved.

A right parallelepiped whose base is a rectangle is called cuboid. All faces of a cuboid are rectangles. The lengths of non-parallel edges of a rectangular parallelepiped are called its linear dimensions (measurements). There are three sizes (width, height, length).

Theorem 13.3. In a cuboid, the square of any diagonal is equal to the sum of the squares of its three dimensions (proved by applying Pythagorean T twice).

A rectangular parallelepiped in which all edges are equal is called cube.

Tasks

13.1 How many diagonals does n- carbon prism

13.2 In an inclined triangular prism, the distances between the side edges are 37, 13, and 40. Find the distance between the larger side face and the opposite side edge.

13.3 Through the side of the lower base of a regular triangular prism, a plane is drawn that intersects the side faces along segments, the angle between which is . Find the angle of inclination of this plane to the base of the 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. Direct 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. Dimensions 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 a polyhedron is called, in which 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 bases are called lateral.
The sides of the side faces and bases are called prism edges, the ends of the edges are called the tops of the prism. Lateral ribs called edges that do not belong to the bases. The union of side faces is called side surface of the prism, and the union of all faces is called 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. straight prism called a prism, in which the 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 ^ - perimeter of the perpendicular section;
S b - side surface area;
V - volume;
S p - area of ​​the total surface of the prism.

V=SH S p \u003d 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 consecutive 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 verify 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 holds for the 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 A "C") as the lines of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example, AC equals A"C") as 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 a prismatic surface - side faces; edges of the prismatic surface - side edges of the prism. By virtue of the previous theorem, the bases of the prism are equal polygons. All side faces of the prism parallelograms; all side edges are equal to each other.
It is obvious that if the base of the prism ABCDE and one of the edges AA" are given in magnitude and direction, then it is possible to construct a prism by drawing the edges BB", CC", .., equal and parallel to the 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 a straight line if its bases are perpendicular sections of a prismatic surface. In this case, the height of the prism is, of course, its side rib; side edges will rectangles.
Prisms can be classified by the number of side faces, equal to the number of 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 the given prism and abcde be its perpendicular section, so that the segments ab, bc, .. are perpendicular to its side edges. Face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that matches ab; the area of ​​\u200b\u200bthe face BCV "C" is equal to the product of the base BB" by the height bc, etc. Therefore, the side 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.