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Which prism is called a parallelepiped. Rectangular parallelepiped - Knowledge Hypermarket. Examples of solving typical Unified State Exam tasks

10.09.2024

Definition

Polyhedron we will call a closed surface composed of polygons and bounding a certain part of space.

The segments that are the sides of these polygons are called ribs polyhedron, and the polygons themselves are edges. The vertices of polygons are called polyhedron vertices.

We will consider only convex polyhedra (this is a polyhedron that is located on one side of each plane containing its face).

The polygons that make up a polyhedron form its surface. The part of space that is bounded by a given polyhedron is called its interior.

Definition: prism

Consider two equal polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) located in parallel planes so that the segments \(A_1B_1, \A_2B_2, ..., A_nB_n\) parallel. A polyhedron formed by the polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) , as well as parallelograms \(A_1B_1B_2A_2, \A_2B_2B_3A_3, ...\), is called (\(n\)-gonal) prism.

Polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) are called prism bases, parallelograms \(A_1B_1B_2A_2, \A_2B_2B_3A_3, ...\)– side faces, segments \(A_1B_1, \ A_2B_2, \ ..., A_nB_n\)- lateral ribs.
Thus, the lateral edges of the prism are parallel and equal to each other.

Let's look at an example - a prism \(A_1A_2A_3A_4A_5B_1B_2B_3B_4B_5\), at the base of which lies a convex pentagon.

Height prisms are a perpendicular dropped from any point of one base to the plane of another base.

If the side edges are not perpendicular to the base, then such a prism is called inclined(Fig. 1), otherwise – direct. In a straight prism, the side edges are heights, and the side faces are equal rectangles.

If a regular polygon lies at the base of a straight prism, then the prism is called correct.

Definition: concept of volume

The unit of volume measurement is a unit cube (a cube measuring \(1\times1\times1\) units\(^3\), where unit is a certain unit of measurement).

We can say that the volume of a polyhedron is the amount of space that this polyhedron limits. Otherwise: this is a quantity whose numerical value shows how many times a unit cube and its parts fit into a given polyhedron.

Volume has the same properties as area:

1. The volumes of equal figures are equal.

2. If a polyhedron is composed of several non-intersecting polyhedra, then its volume is equal to the sum of the volumes of these polyhedra.

3. Volume is a non-negative quantity.

4. Volume is measured in cm\(^3\) (cubic centimeters), m\(^3\) (cubic meters), etc.

Theorem

1. The area of ​​the lateral surface of the prism is equal to the product of the perimeter of the base and the height of the prism.
The lateral surface area is the sum of the areas of the lateral faces of the prism.

2. The volume of the prism is equal to the product of the base area and the height of the prism: \

Definition: parallelepiped

Parallelepiped is a prism with a parallelogram at its base.

All faces of the parallelepiped (there are \(6\) : \(4\) side faces and \(2\) bases) are parallelograms, and the opposite faces (parallel to each other) are equal parallelograms (Fig. 2).


Diagonal of a parallelepiped is a segment connecting two vertices of a parallelepiped that do not lie on the same face (there are \(8\) of them: \(AC_1,\A_1C,\BD_1,\B_1D\) etc.).

Rectangular parallelepiped is a right parallelepiped with a rectangle at its base.
Because Since this is a right parallelepiped, the side faces are rectangles. This means that in general all the faces of a rectangular parallelepiped are rectangles.

All diagonals of a rectangular parallelepiped are equal (this follows from the equality of triangles \(\triangle ACC_1=\triangle AA_1C=\triangle BDD_1=\triangle BB_1D\) etc.).

Comment

Thus, a parallelepiped has all the properties of a prism.

Theorem

The lateral surface area of ​​a rectangular parallelepiped is \

The total surface area of ​​a rectangular parallelepiped is \

Theorem

The volume of a cuboid is equal to the product of its three edges emerging from one vertex (three dimensions of the cuboid): \


Proof

Because In a rectangular parallelepiped, the lateral edges are perpendicular to the base, then they are also its heights, that is, \(h=AA_1=c\) Because the base is a rectangle, then \(S_(\text(main))=AB\cdot AD=ab\). This is where this formula comes from.

Theorem

The diagonal \(d\) of a rectangular parallelepiped is found using the formula (where \(a,b,c\) are the dimensions of the parallelepiped) \

Proof

Let's look at Fig. 3. Because the base is a rectangle, then \(\triangle ABD\) is rectangular, therefore, by the Pythagorean theorem \(BD^2=AB^2+AD^2=a^2+b^2\) .

Because all lateral edges are perpendicular to the bases, then \(BB_1\perp (ABC) \Rightarrow BB_1\) perpendicular to any straight line in this plane, i.e. \(BB_1\perp BD\) . This means that \(\triangle BB_1D\) is rectangular. Then, by the Pythagorean theorem \(B_1D=BB_1^2+BD^2=a^2+b^2+c^2\), thd.

Definition: cube

Cube is a rectangular parallelepiped, all of whose faces are equal squares.


Thus, the three dimensions are equal to each other: \(a=b=c\) . So the following are true

Theorems

1. The volume of a cube with edge \(a\) is equal to \(V_(\text(cube))=a^3\) .

2. The diagonal of the cube is found using the formula \(d=a\sqrt3\) .

3. Total surface area of ​​a cube \(S_(\text(full cube))=6a^2\).

In geometry, the key concepts are plane, point, straight line and angle. Using these terms, you can describe any geometric figure. Polyhedra are usually described in terms of simpler figures that lie in the same plane, such as a circle, triangle, square, rectangle, etc. In this article we will look at what a parallelepiped is, describe the types of parallelepipeds, its properties, what elements it consists of, and also give the basic formulas for calculating the area and volume for each type of parallelepiped.

Definition

A parallelepiped in three-dimensional space is a prism, all sides of which are parallelograms. Accordingly, it can only have three pairs of parallel parallelograms or six faces.

To visualize a parallelepiped, imagine an ordinary standard brick. A brick is a good example of a rectangular parallelepiped that even a child can imagine. Other examples include multi-storey panel houses, cabinets, food storage containers of appropriate shape, etc.

Varieties of figure

There are only two types of parallelepipeds:

  1. Rectangular, all side faces of which are at an angle of 90° to the base and are rectangles.
  2. Sloping, the side edges of which are located at a certain angle to the base.

What elements can this figure be divided into?

  • As in any other geometric figure, in a parallelepiped any 2 faces with a common edge are called adjacent, and those that do not have it are parallel (based on the property of a parallelogram, which has pairs of parallel opposite sides).
  • The vertices of a parallelepiped that do not lie on the same face are called opposite.
  • The segment connecting such vertices is a diagonal.
  • The lengths of the three edges of a cuboid that meet at one vertex are its dimensions (namely, its length, width and height).

Shape Properties

  1. It is always built symmetrically with respect to the middle of the diagonal.
  2. The intersection point of all diagonals divides each diagonal into two equal segments.
  3. Opposite faces are equal in length and lie on parallel lines.
  4. If you add the squares of all dimensions of a parallelepiped, the resulting value will be equal to the square of the length of the diagonal.

Calculation formulas

The formulas for each particular case of a parallelepiped will be different.

For an arbitrary parallelepiped, it is true that its volume is equal to the absolute value of the triple scalar product of the vectors of three sides emanating from one vertex. However, there is no formula for calculating the volume of an arbitrary parallelepiped.

For a rectangular parallelepiped the following formulas apply:

  • V=a*b*c;
  • Sb=2*c*(a+b);
  • Sp=2*(a*b+b*c+a*c).
  • V - volume of the figure;
  • Sb - lateral surface area;
  • Sp - total surface area;
  • a - length;
  • b - width;
  • c - height.

Another special case of a parallelepiped in which all sides are squares is a cube. If any of the sides of the square is designated by the letter a, then the following formulas can be used for the surface area and volume of this figure:

  • S=6*a*2;
  • V=3*a.
  • S - area of ​​the figure,
  • V is the volume of the figure,
  • a is the length of the figure's face.

The last type of parallelepiped we are considering is a straight parallelepiped. What is the difference between a right parallelepiped and a cuboid, you ask. The fact is that the base of a rectangular parallelepiped can be any parallelogram, but the base of a straight parallelepiped can only be a rectangle. If we denote the perimeter of the base, equal to the sum of the lengths of all sides, as Po, and denote the height by the letter h, we have the right to use the following formulas to calculate the volume and areas of the total and lateral surfaces.

Translated from Greek, parallelogram means plane. A parallelepiped is a prism with a parallelogram at its base. There are five types of parallelogram: oblique, straight and cuboid. The cube and rhombohedron also belong to the parallelepiped and are its variety.

Before moving on to the basic concepts, let's give some definitions:

  • The diagonal of a parallelepiped is a segment that unites the vertices of the parallelepiped that are opposite each other.
  • If two faces have a common edge, then we can call them adjacent edges. If there is no common edge, then the faces are called opposite.
  • Two vertices that do not lie on the same face are called opposite.

What properties does a parallelepiped have?

  1. The faces of a parallelepiped lying on opposite sides are parallel to each other and equal to each other.
  2. If you draw diagonals from one vertex to another, then the intersection point of these diagonals will divide them in half.
  3. The sides of the parallelepiped lying at the same angle to the base will be equal. In other words, the angles of the co-directed sides will be equal to each other.

What types of parallelepiped are there?

Now let's figure out what kind of parallelepipeds there are. As mentioned above, there are several types of this figure: straight, rectangular, inclined parallelepiped, as well as cube and rhombohedron. How do they differ from each other? It's all about the planes that form them and the angles they form.

Let's look in more detail at each of the listed types of parallelepiped.

  • As is already clear from the name, an inclined parallelepiped has inclined faces, namely those faces that are not at an angle of 90 degrees in relation to the base.
  • But for a right parallelepiped, the angle between the base and the edge is exactly ninety degrees. It is for this reason that this type of parallelepiped has such a name.
  • If all the faces of the parallelepiped are identical squares, then this figure can be considered a cube.
  • A rectangular parallelepiped received this name because of the planes that form it. If they are all rectangles (including the base), then this is a cuboid. This type of parallelepiped is not found very often. Translated from Greek, rhombohedron means face or base. This is the name given to a three-dimensional figure whose faces are rhombuses.



Basic formulas for a parallelepiped

The volume of a parallelepiped is equal to the product of the area of ​​the base and its height perpendicular to the base.

The area of ​​the lateral surface will be equal to the product of the perimeter of the base and the height.
Knowing the basic definitions and formulas, you can calculate the base area and volume. The base can be chosen at your discretion. However, as a rule, a rectangle is used as the base.

CHAPTER THREE

POLYhedra

1. PARALLELEPIPED AND PYRAMID

Properties of faces and diagonals of a parallelepiped

72. Theorem. In a parallelepiped:

1)opposite sides are equal and parallel;

2) all four diagonals intersect at one point and bisect there.

1) The faces (Fig. 80) BB 1 C 1 C and AA 1 D 1 D are parallel, because two intersecting straight lines BB 1 and B 1 C 1 of one face are parallel to two intersecting straight lines AA 1 and A 1 D 1 of the other (§ 15 ); these faces are equal, since B 1 C 1 = A 1 D 1, B 1 B = A 1 A (like opposite sides of parallelograms) and / BB 1 C 1 = / AA 1 D 1 .

2) Take (Fig. 81) some two diagonals, for example AC 1 and ВD 1, and draw auxiliary lines AD 1 and ВС 1.

Since the edges AB and D 1 C 1 are respectively equal and parallel to the edge DC, they are equal and parallel to each other; As a result, the figure AD 1 C 1 B is a parallelogram in which the straight lines C 1 A and BD 1 are diagonals, and in a parallelogram the diagonals are divided in half at the point of intersection.

Let us now take one of these diagonals, for example AC 1, with a third diagonal, let us say, with B 1 D. In exactly the same way we can prove that they are divided in half at the point of intersection. Consequently, diagonals B 1 D and AC 1 and diagonals AC 1 and BD 1 (which we took earlier) intersect at the same point, precisely in the middle of the diagonal
AC 1. Finally, taking the same diagonal AC 1 with the fourth diagonal A 1 C, we also prove that they are bisected. This means that the point of intersection of this pair of diagonals lies in the middle of the diagonal AC 1. Thus, all four diagonals of the parallelepiped intersect at the same point and are bisected by this point.

73. Theorem. In a rectangular parallelepiped, the square of any diagonal (AS 1, drawing 82) equal to the sum of the squares of its three dimensions .

Drawing the diagonal of the base AC, we obtain triangles AC 1 C and ACB. Both of them are rectangular: the first because the parallelepiped is straight and, therefore, edge CC 1 is perpendicular to the base; the second because the parallelepiped is rectangular and, therefore, a rectangle lies at its base. From these triangles we find:

AC 1 2 = AC 2 + CC 1 2 and AC 2 = AB 2 + BC 2

Hence,

AC 1 2 = AB 2 + BC 2 + CC 1 2 = AB 2 + AD 2 + AA 1 2.

Consequence.In a rectangular parallelepiped, all diagonals are equal.