A right parallelepiped with a square base. Rectangular parallelepiped

Rectangular parallelepiped

A rectangular parallelepiped is a right parallelepiped whose all faces are rectangles.

It is enough to look around us, and we will see that the objects around us have a shape similar to a parallelepiped. They can be distinguished by color, have a lot of additional details, but if these subtleties are discarded, then we can say that, for example, a cabinet, box, etc., have approximately the same shape.

We come across the concept of a rectangular parallelepiped almost every day! Look around and tell me where you see rectangular parallelepipeds? Look at the book, it's exactly the same shape! A brick, a matchbox, a wooden block have the same shape, and even right now you are inside a rectangular parallelepiped, because the classroom is the brightest interpretation of this geometric figure.

Exercise: What examples of parallelepiped can you name?

Let's take a closer look at the cuboid. And what do we see?

First, we see that this figure is formed from six rectangles, which are the faces of a cuboid;

Secondly, a cuboid has eight vertices and twelve edges. The edges of a cuboid are the sides of its faces, and the vertices of the cuboid are the vertices of the faces.

Exercise:

1. What is the name of each of the faces of a rectangular parallelepiped? 2. Thanks to what parameters can a parallelogram be measured? 3. Define opposite faces.

Types of parallelepipeds

But parallelepipeds are not only rectangular, but they can also be straight and inclined, and straight lines are divided into rectangular, non-rectangular and cubes.

Assignment: Look at the picture and say what parallelepipeds are shown on it. How does a rectangular parallelepiped differ from a cube?

Properties of a rectangular parallelepiped

A rectangular parallelepiped has a number of important properties:

Firstly, the square of the diagonal of this geometric figure is equal to the sum of the squares of its three main parameters: height, width and length.

Secondly, all four of its diagonals are absolutely identical.

Thirdly, if all three parameters of a parallelepiped are the same, that is, the length, width and height are equal, then such a parallelepiped is called a cube, and all its faces will be equal to the same square.

Exercise

1. Does a rectangular parallelepiped have equal sides? If there are any, then show them in the figure. 2. What geometric shapes do the faces of a rectangular parallelepiped consist of? 3. What is the arrangement of equal edges in relation to each other? 4. Name the number of pairs of equal faces of this figure. 5. Find the edges in a rectangular parallelepiped that indicate its length, width, height. How many did you count?

Task

To beautifully decorate a birthday present for her mother, Tanya took a box in the shape of a rectangular parallelepiped. The size of this box is 25cm*35cm*45cm. To make this packaging beautiful, Tanya decided to cover it beautiful paper, the cost of which is 3 hryvnia per 1 dm2. How much money should you spend on wrapping paper?

Do you know that the famous illusionist David Blaine spent 44 days in a glass parallelepiped suspended over the Thames as part of an experiment. For these 44 days he did not eat, but only drank water. In his voluntary prison, David took only writing materials, a pillow and mattress, and handkerchiefs.

When you were little and played with cubes, you may have made the shapes shown in Figure 154. These figures give an idea of rectangular parallelepiped. For example, a box of chocolates, a brick, a matchbox, a packaging box, and a juice box have the shape of a rectangular parallelepiped.

Figure 155 shows a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1.

A rectangular parallelepiped is limited by six edges. Each face is a rectangle, i.e. The surface of a rectangular parallelepiped consists of six rectangles.

The sides of the faces are called edges of a rectangular parallelepiped, vertices of faces − vertices of a rectangular parallelepiped. For example, segments AB, BC, A 1 B 1 are edges, and points B, A 1, C 1 are vertices of the parallelepiped ABCDA 1 B 1 C 1 D 1 (Fig. 155).

A rectangular parallelepiped has 8 vertices and 12 edges.

The faces AA 1 B 1 B and DD 1 C 1 C do not have common vertices. Such edges are called opposite. In the parallelepiped ABCDA 1 B 1 C 1 D 1 there are two more pairs of opposite faces: rectangles ABCD and A 1 B 1 C 1 D 1, as well as rectangles AA 1 D 1 D and BB 1 C 1 C.

Opposite faces of a rectangular parallelepiped are equal.

In Figure 155, the face ABCD is called basis rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 .

The surface area of ​​a parallelepiped is the sum of the areas of all its faces.

To have an idea of ​​the dimensions of a rectangular parallelepiped, it is enough to consider any three edges that have a common vertex. The lengths of these edges are called measurements rectangular parallelepiped. To distinguish them, they use names: length, width, height(Fig. 156).

A rectangular parallelepiped in which all dimensions are equal is called cube(Fig. 157). The surface of the cube consists of six equal squares.

If a box in the shape of a rectangular parallelepiped is opened (Fig. 158) and cut along four vertical edges (Fig. 159), and then unfolded, we get a figure consisting of six rectangles (Fig. 160). This figure is called development of a rectangular parallelepiped.

Figure 161 shows a figure consisting of six equal squares. It is the development of a cube.

Using a development, you can make a model of a rectangular parallelepiped.

This can be done, for example, like this. Draw its outline on paper. Cut it out, bend it along the segments corresponding to the edges of the rectangular parallelepiped (see Fig. 159), and glue it together.

A rectangular parallelepiped is a type of polyhedron - a figure whose surface consists of polygons. Figure 162 shows polyhedra.

One type of polyhedron is pyramid.

This figure is not new to you. Studying the course Ancient world, you got acquainted with one of the seven wonders of the world - the Egyptian pyramids.

Figure 163 shows the pyramids MABC, MABCD, MABCDE. The surface of the pyramid consists of side faces− triangles having a common vertex, and grounds(Fig. 164). The common vertex of the lateral faces is called edges of the base of the pyramid, and the sides of the side faces that do not belong to the base are lateral edges of the pyramid.

Pyramids can be classified according to the number of sides of the base: triangular, quadrangular, pentagonal (see Fig. 163), etc.

The surface of a triangular pyramid consists of four triangles. Any of these triangles can serve as the base of a pyramid. This base is a type of pyramid, any face of which can serve as its base.

Figure 165 shows a figure that can serve development of a quadrangular pyramid. It consists of a square and four equal isosceles triangles.

Figure 166 shows a figure consisting of four equal equilateral triangles. Using this figure, you can make a model of a triangular pyramid, all of whose faces are equilateral triangles.

Polyhedra are examples geometric bodies.

Figure 167 shows familiar ones geometric bodies, which are not polyhedra. You will learn more about these bodies in 6th grade.

In this lesson, everyone will be able to study the topic “Rectangular parallelepiped”. At the beginning of the lesson, we will repeat what arbitrary and straight parallelepipeds are, remember the properties of their opposite faces and diagonals of the parallelepiped. Then we'll look at what a cuboid is and discuss its basic properties.

Topic: Perpendicularity of lines and planes

Lesson: Cuboid

A surface composed of two equal parallelograms ABCD and A 1 B 1 C 1 D 1 and four parallelograms ABV 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1 is called parallelepiped(Fig. 1).

Rice. 1 Parallelepiped

That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (bases), they lie in parallel planes so that the side edges AA 1, BB 1, DD 1, CC 1 are parallel. Thus, a surface composed of parallelograms is called parallelepiped.

Thus, the surface of a parallelepiped is the sum of all the parallelograms that make up the parallelepiped.

1. The opposite faces of a parallelepiped are parallel and equal.

(the shapes are equal, that is, they can be combined by overlapping)

For example:

ABCD = A 1 B 1 C 1 D 1 (equal parallelograms by definition),

AA 1 B 1 B = DD 1 C 1 C (since AA 1 B 1 B and DD 1 C 1 C are opposite faces of the parallelepiped),

AA 1 D 1 D = BB 1 C 1 C (since AA 1 D 1 D and BB 1 C 1 C are opposite faces of the parallelepiped).

2. The diagonals of a parallelepiped intersect at one point and are bisected by this point.

The diagonals of the parallelepiped AC 1, B 1 D, A 1 C, D 1 B intersect at one point O, and each diagonal is divided in half by this point (Fig. 2).

Rice. 2 The diagonals of a parallelepiped intersect and are divided in half by the intersection point.

3. There are three quadruples of equal and parallel edges of a parallelepiped: 1 - AB, A 1 B 1, D 1 C 1, DC, 2 - AD, A 1 D 1, B 1 C 1, BC, 3 - AA 1, BB 1, CC 1, DD 1.

Definition. A parallelepiped is called straight if its lateral edges are perpendicular to the bases.

Let the side edge AA 1 be perpendicular to the base (Fig. 3). This means that straight line AA 1 is perpendicular to straight lines AD and AB, which lie in the plane of the base. This means that the side faces contain rectangles. And the bases contain arbitrary parallelograms. Let us denote ∠BAD = φ, the angle φ can be any.

Rice. 3 Right parallelepiped

So, a right parallelepiped is a parallelepiped in which the side edges are perpendicular to the bases of the parallelepiped.

Definition. The parallelepiped is called rectangular, if its lateral edges are perpendicular to the base. The bases are rectangles.

The parallelepiped ABCDA 1 B 1 C 1 D 1 is rectangular (Fig. 4), if:

1. AA 1 ⊥ ABCD (lateral edge perpendicular to the plane of the base, that is, a straight parallelepiped).

2. ∠BAD = 90°, i.e. the base is a rectangle.

Rice. 4 Rectangular parallelepiped

A rectangular parallelepiped has all the properties of an arbitrary parallelepiped. But there are additional properties that are derived from the definition of a cuboid.

So, cuboid is a parallelepiped whose side edges are perpendicular to the base. The base of a cuboid is a rectangle.

1. In a rectangular parallelepiped, all six faces are rectangles.

ABCD and A 1 B 1 C 1 D 1 are rectangles by definition.

2. Lateral ribs are perpendicular to the base. This means that all the lateral faces of a rectangular parallelepiped are rectangles.

3. All dihedral angles of a rectangular parallelepiped are right.

Let us consider, for example, the dihedral angle of a rectangular parallelepiped with edge AB, i.e., the dihedral angle between planes ABC 1 and ABC.

AB is an edge, point A 1 lies in one plane - in the plane ABB 1, and point D in the other - in the plane A 1 B 1 C 1 D 1. Then the dihedral angle under consideration can also be denoted as follows: ∠A 1 ABD.

Let's take point A on edge AB. AA 1 is perpendicular to edge AB in the plane АВВ-1, AD is perpendicular to edge AB in the plane ABC. So, ∠A 1 AD - linear angle given dihedral angle. ∠A 1 AD = 90°, which means that the dihedral angle at edge AB is 90°.

∠(ABB 1, ABC) = ∠(AB) = ∠A 1 ABD= ∠A 1 AD = 90°.

Similarly, it is proved that any dihedral angles of a rectangular parallelepiped are right.

Square diagonal of a cuboid equal to the sum squares of its three dimensions.

Note. The lengths of the three edges emanating from one vertex of a cuboid are the measurements of the cuboid. They are sometimes called length, width, height.

Given: ABCDA 1 B 1 C 1 D 1 - rectangular parallelepiped (Fig. 5).

Prove: .

Rice. 5 Rectangular parallelepiped

Proof:

Straight line CC 1 is perpendicular to plane ABC, and therefore to straight line AC. This means that the triangle CC 1 A is right-angled. According to the Pythagorean theorem:

Let's consider right triangle ABC. According to the Pythagorean theorem:

But BC and AD are opposite sides of the rectangle. So BC = AD. Then:

Because , A , That. Since CC 1 = AA 1, this is what needed to be proven.

The diagonals of a rectangular parallelepiped are equal.

Let us denote the dimensions of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =

The prism is called parallelepiped, if its bases are parallelograms. Cm. Fig.1.

Properties of a parallelepiped:

The opposite faces of a parallelepiped are parallel (that is, they lie in parallel planes) and equal.

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

Adjacent faces of a parallelepiped– two faces that have a common edge.

Opposite faces of a parallelepiped– faces that do not have common edges.

Opposite vertices of a parallelepiped– two vertices that do not belong to the same face.

Diagonal of a parallelepiped– a segment that connects opposite vertices.

If the lateral edges are perpendicular to the planes of the bases, then the parallelepiped is called direct.

A right parallelepiped whose bases are rectangles is called rectangular. A prism, all of whose faces are squares, is called cube.

Parallelepiped- a prism whose bases are parallelograms.

Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the plane of the base.

Rectangular parallelepiped is a right parallelepiped whose bases are rectangles.

Cube– a rectangular parallelepiped with equal edges.

parallelepiped called a prism whose base is a parallelogram; Thus, a parallelepiped has six faces and all of them are parallelograms.

Opposite faces are pairwise equal and parallel. The parallelepiped has four diagonals; they all intersect at one point and are divided in half at it. Any face can be taken as a base; volume equal to the product base area per height: V = Sh.

A parallelepiped whose four lateral faces are rectangles is called a straight parallelepiped.

A right parallelepiped whose six faces are rectangles is called rectangular. Cm. Fig.2.

The volume (V) of a right parallelepiped is equal to the product of the base area (S) and the height (h): V = Sh .

For a rectangular parallelepiped, in addition, the formula holds V=abc, where a,b,c are edges.

The diagonal (d) of a rectangular parallelepiped is related to its edges by the relation d 2 = a 2 + b 2 + c 2 .

Rectangular parallelepiped- a parallelepiped whose side edges are perpendicular to the bases, and the bases are rectangles.

Properties of a rectangular parallelepiped:

In a rectangular parallelepiped, all six faces are rectangles.

All dihedral angles of a rectangular parallelepiped are right.

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions (the lengths of three edges that have a common vertex).

The diagonals of a rectangular parallelepiped are equal.

A rectangular parallelepiped, all of whose faces are squares, is called a cube. All edges of the cube are equal; the volume (V) of a cube is expressed by the formula V=a 3, where a is the edge of the cube.