What is the difference between a right parallelepiped? Inclined parallelepiped: properties, formulas and tasks for a math tutor

Or (equivalently) a polyhedron, which has six faces and each of them - parallelogram.

Types of parallelepiped

There are several types of parallelepipeds:

• A cuboid is a parallelepiped whose faces are all rectangles.
• A right parallelepiped is a parallelepiped with 4 lateral faces that are rectangles.
• An inclined parallelepiped is a parallelepiped whose side faces are not perpendicular to the bases.

Essential elements

Two faces of a parallelepiped that do not have a common edge are called opposite, and those that have a common edge are called adjacent. Two vertices of a parallelepiped that do not belong to the same face are called opposite. The segment connecting opposite vertices is called the diagonal of the parallelepiped. The lengths of three edges of a rectangular parallelepiped that have a common vertex are called its dimensions.

Properties

• The parallelepiped is symmetrical about the middle of its diagonal.
• Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided in half by it; in particular, all diagonals of a parallelepiped intersect at one point and are bisected by it.
• The opposite faces of a parallelepiped are parallel and equal.
• The square of the diagonal length of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Basic formulas

Right parallelepiped

Lateral surface area S b =P o *h, where P o is the perimeter of the base, h is the height

Total surface area S p =S b +2S o, where S o is the base area

Volume V=S o *h

Rectangular parallelepiped

Lateral surface area S b =2c(a+b), where a, b are the sides of the base, c is the side edge of the rectangular parallelepiped

Total surface area S p =2(ab+bc+ac)

Volume V=abc, where a, b, c are the dimensions of a rectangular parallelepiped.

Cube

Surface area: $S=6a^2$
Volume: $V=a^3$, Where $a$- edge of a cube.

Any parallelepiped

The volume and ratios in an inclined parallelepiped are often determined using vector algebra. The volume of a parallelepiped is equal to the absolute value of the mixed product of three vectors determined by the three sides of the parallelepiped emanating from one vertex. The relationship between the lengths of the sides of the parallelepiped and the angles between them gives the statement that the Gram determinant of the indicated three vectors is equal to the square of their mixed product: 215.

In mathematical analysis

In mathematical analysis under an n-dimensional cuboid $B$ understand many points $x\; =\; (x\_1,\backslash ldots,x\_n)$ kind $B\; =\; \backslash (x|a\_1\backslash leqslant\; x\_1\backslash leqslant\; b\_1,\backslash ldots,a\_n\backslash leqslant\; x\_n\backslash leqslant\; b\_n\backslash )$

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An excerpt characterizing the 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 block of wood 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 with 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.

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 rectangular parallelepiped 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. This means that ∠A 1 AD is the linear angle of a 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.

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the 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:

Consider the 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 =

Lesson objectives:

1. Educational:

Introduce the concept of a parallelepiped and its types;
- formulate (using the analogy with a parallelogram and a rectangle) and prove the properties of a parallelepiped and a cuboid;
- repeat questions related to parallelism and perpendicularity in space.

2. Developmental:

Continue the development of cognitive processes in students such as perception, comprehension, thinking, attention, memory;
- promote the development of elements of creative activity in students as qualities of thinking (intuition, spatial thinking);
- to develop in students the ability to draw conclusions, including by analogy, which helps to understand intra-subject connections in geometry.

3. Educational:

Contribute to the development of organization and habits of systematic work;
- contribute to the formation of aesthetic skills when making notes and making drawings.

Lesson type: lesson-learning new material (2 hours).

Lesson structure:

1. Organizational moment.
2. Updating knowledge.
3. Studying new material.
4. Summing up and setting homework.

Equipment: posters (slides) with evidence, models of various geometric bodies, including all types of parallelepipeds, graphic projector.

During the classes.

1. Organizational moment.

2. Updating knowledge.

Communicating the topic of the lesson, formulating goals and objectives together with students, showing the practical significance of studying the topic, repeating previously studied issues related to this topic.

3. Studying new material.

3.1. Parallelepiped and its types.

Models of parallelepipeds are demonstrated, identifying their features, which help formulate the definition of a parallelepiped using the concept of a prism.

Definition:

parallelepiped called a prism whose base is a parallelogram.

A drawing of a parallelepiped is made (Figure 1), the elements of a parallelepiped as a special case of a prism are listed. Slide 1 is shown.

Schematic notation of the definition:

Conclusions from the definition are formulated:

1) If ABCDA 1 B 1 C 1 D 1 is a prism and ABCD is a parallelogram, then ABCDA 1 B 1 C 1 D 1 – parallelepiped.

2) If ABCDA 1 B 1 C 1 D 1 – parallelepiped, then ABCDA 1 B 1 C 1 D 1 is a prism and ABCD is a parallelogram.

3) If ABCDA 1 B 1 C 1 D 1 is not a prism or ABCD is not a parallelogram, then
ABCDA 1 B 1 C 1 D 1 – not parallelepiped.

4) . If ABCDA 1 B 1 C 1 D 1 – not parallelepiped, then ABCDA 1 B 1 C 1 D 1 is not a prism or ABCD is not a parallelogram.

Next, special cases of a parallelepiped are considered with the construction of a classification scheme (see Fig. 3), models are demonstrated, the characteristic properties of straight and rectangular parallelepipeds are highlighted, and their definitions are formulated.

Definition:

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

Definition:

The parallelepiped is called rectangular, if its side edges are perpendicular to the base, and the base is a rectangle (see Figure 2).

After recording the definitions in a schematic form, conclusions from them are formulated.

3.2. Properties of parallelepipeds.

Search for planimetric figures, the spatial analogues of which are parallelepiped and cuboid (parallelogram and rectangle). In this case, we are dealing with the visual similarity of the figures. Using the inference rule by analogy, the tables are filled in.

Inference rule by analogy:

1. Select from among previously studied figures a figure similar to this one.
2. Formulate the property of the selected figure.
3. Formulate a similar property of the original figure.
4. Prove or disprove the formulated statement.

After formulating the properties, the proof of each of them is carried out according to the following scheme:

• discussion of the proof plan;
• demonstration of a slide with evidence (slides 2 – 6);
• Students completing evidence in their notebooks.

3.3 Cube and its properties.

Definition: A cube is a rectangular parallelepiped in which all three dimensions are equal.

By analogy with a parallelepiped, students independently make a schematic notation of the definition, derive consequences from it and formulate the properties of the cube.

4. Summing up and setting homework.

Homework:

1. Using the lesson notes from the geometry textbook for grades 10-11, L.S. Atanasyan and others, study Chapter 1, §4, paragraph 13, Chapter 2, §3, paragraph 24.
2. Prove or disprove the property of a parallelepiped, item 2 of the table.
3. Answer security questions.

Control questions.

1. It is known that only two side faces of the parallelepiped are perpendicular to the base. What type of parallelepiped?

2. How many side faces of a rectangular shape can a parallelepiped have?

3. Is it possible to have a parallelepiped with only one side face:

1) perpendicular to the base;
2) has the shape of a rectangle.

4. In a right parallelepiped, all diagonals are equal. Is it rectangular?

5. Is it true that in a right parallelepiped the diagonal sections are perpendicular to the planes of the base?

6. State the converse theorem to the theorem about the square of the diagonal of a rectangular parallelepiped.

7. What additional features distinguish a cube from a rectangular parallelepiped?

8. Will a parallelepiped be a cube in which all the edges at one of the vertices are equal?

9. State the theorem on the square of the diagonal of a cuboid for the case of a cube.

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 point of intersection 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.