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 |
*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 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 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.
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 quadrangles, 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 hexahedron, 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
Because the 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 quadrilaterals, 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 on 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
Prism. Parallelepiped
prism is called a polyhedron whose two faces are equal n-gons (grounds) , lying in parallel planes, and the remaining n faces are parallelograms (side edges) . Side rib prism is the side of the lateral face that does not belong to the base.
A prism whose lateral edges are perpendicular to the planes of the bases is called straight prism (Fig. 1). If the side edges are not perpendicular to the planes of the bases, then the prism is called oblique . Correct A prism is a straight prism whose bases are regular polygons.
Height prism is called 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 A section of a prism by a plane passing through two side edges that do not belong to the same face is called. Perpendicular section called the section of the prism by a plane perpendicular to the lateral edge of the prism.
Side surface area prism is the sum of the areas of all side faces. Full surface area the sum of the areas of all the faces of the prism is called (i.e., the sum of the areas of the side faces and the areas of the bases).
For an arbitrary prism, the formulas are true:
where l is the length of the side rib;
H- height;
P
Q
S side
S full
S main is the area of the bases;
V is the volume of the prism.
For a straight prism, the following formulas are true:
where p- the perimeter of the base;
l is the length of the side rib;
H- height.
Parallelepiped A prism whose base is a parallelogram is called. 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 oblique . A right parallelepiped whose base is a rectangle is called rectangular. A rectangular parallelepiped in which all edges are 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 the box is a prism, its main elements are defined in the same way as they are defined for prisms.
Theorems.
1. The diagonals of the 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 true:
where l is the length of the side rib;
H- height;
P is the perimeter of the perpendicular section;
Q– Area of perpendicular section;
S side is the lateral surface area;
S full is the total surface area;
S main is the area of the bases;
V is the volume of the prism.
For a right parallelepiped, the following formulas are true:
where p- the perimeter of the base;
l is the length of the side rib;
H is the height of the right parallelepiped.
For a rectangular parallelepiped, the following formulas are true:
(3)
where p- the perimeter of the base;
H- height;
d- diagonal;
a,b,c– measurements of a parallelepiped.
The correct formulas for a cube are:
where a is the length of the rib;
d is the diagonal of the cube.
Example 1 The diagonal of a rectangular cuboid is 33 dm, and its measurements are related as 2:6:9. Find the measurements of the cuboid.
Solution. To find the dimensions of the parallelepiped, we use formula (3), i.e. the fact that the square of the hypotenuse of a cuboid is equal to the sum of the squares of its dimensions. Denote by k coefficient of proportionality. Then the dimensions of the parallelepiped will be equal to 2 k, 6k and 9 k. We write formula (3) for the problem data:
Solving this equation for k, we get:
Hence, 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 whose base is an equilateral triangle with a side of 8 cm, if the lateral edge is equal to the side of the base and is 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 \u200b\u200bits base and height. The area of the base of this prism is the area of an 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 BUT 1 of the upper base we lower the perpendicular to the plane of the lower base BUT 1 D. Its length will be the height of the prism. Consider D BUT 1 AD: since this is the angle of inclination of the side rib BUT 1 BUT to the base plane BUT 1 BUT= 8 cm. From this triangle we find BUT 1 D:
Now we calculate the volume using formula (1):
Answer: 192 cm3.
Example 3 The lateral edge of a regular hexagonal prism is 14 cm. The area of \u200b\u200bthe 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 AA 1 DD 1 , since the diagonal AD regular hexagon ABCDEF is the largest. In order to calculate the lateral surface area of a prism, it is necessary to know the side of the base and the length of the lateral rib.
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:
Find the area of the lateral surface of the prism:
The area of a regular hexagon with a side of 6 cm is:
Find the total surface area of the prism:
Answer: 
Example 4 The base of a right parallelepiped is a rhombus. The areas of diagonal sections are 300 cm 2 and 875 cm 2. Find the area of the side surface of the parallelepiped.
Solution. Let's make a drawing (Fig. 5).
Denote the side of the rhombus by a, the diagonals of the rhombus d 1 and d 2 , the height of the box h. To find the lateral surface area of a straight 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.
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 get the equality We get the following.
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√ .
A 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
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
A task
Find the total 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.
