Which rectangular parallelepiped is called a cube. Rectangular parallelepiped. Cube

A prism is a polyhedron that consists of two flat polygons, combined by parallel translation, and all the segments connecting the corresponding points of these polygons. The polygons are called the bases of the prism, and the segments connecting the corresponding vertices are called the lateral edges of the prism (Fig. 146).

Since parallel translation is motion, the bases of the prism are equal. Since during parallel translation the plane goes into a parallel plane (or into itself), then

The bases of the prism lie in parallel planes. Since during parallel translation the points are shifted along parallel (or coinciding) lines by the same distance, then the side edges of the prism are parallel and equal.

Figure 147, a shows a quadrangular prism. Flat polygons are combined by corresponding parallel translation and are the bases of the prism, and the segments are the lateral edges of the prism. The bases of the prism are equal (parallel translation is a movement and transforms a figure into an equal figure, paragraph 79). The side ribs are parallel and equal.

The surface of the prism consists of the base and the side surface. The lateral surface consists of parallelograms. In each of these parallelograms, two sides are the corresponding sides of the bases, and the other two are adjacent lateral edges of the prism.

In Figure 147, the lateral surface of the prism consists of parallelograms. The full surface consists of the bases and the above parallelograms.

The height of a prism is the distance between the planes of its bases. A segment that connects two vertices that do not belong to the same face is called a prism diagonal. The diagonal section of a prism is the section of its plane passing through two lateral edges that do not belong to the same face.

Figure 147a shows a prism - its height, - one of its diagonals. The section is one of the diagonal sections of this prism.

A prism is called straight if its side edges are perpendicular to the bases. Otherwise the prism is called

inclined A right prism is called regular if its bases are regular polygons.

Figure 147, a shows an inclined prism, and Figure 147, b - a straight one, here the edge is perpendicular to the bases of the prism. Figure 148 shows regular prisms; their bases are, respectively, a regular triangle, a square, and a regular hexagon.

If the bases of the prism are parallelograms, then it is called a parallelepiped. All faces of a parallelepiped are parallelograms. Figure 147, a shows an inclined parallelepiped, and Figure 147, b - a straight parallelepiped.

The faces of a parallelepiped that do not have common vertices are called opposite. In Figure 147, and the faces are opposite.

It is possible to prove some properties of a parallelepiped.

T.3.1. A parallelepiped's opposite faces are parallel and equal.

T.3.2. The diagonals of a parallelepiped intersect at one point and are divided in half by the intersection point.

The point of intersection of the diagonals of a parallelepiped is its center of symmetry.

A right parallelepiped whose base is a rectangle is called a cuboid. All faces of a rectangular parallelepiped are rectangles.

A rectangular parallelepiped with all edges equal is called a cube.

The lengths of the non-parallel edges of a rectangular parallelepiped are called its linear dimensions or dimensions. A rectangular parallelepiped has three linear dimensions.

For a rectangular parallelepiped the following theorem is true:

T.3.3. In a rectangular parallelepiped, the square of any diagonal is equal to the sum of the squares of its three linear dimensions.

For example, in a cube (Fig. 149) with edge a the diagonals are equal:

Topic: general lesson on the topic: “Rectangular parallelepiped.”

Goals:

To consolidate and generalize the ability to convert one unit of area measurement into another and use this knowledge when solving problems, the ability to calculate the surface area and volume of a rectangular parallelepiped and the ability to solve problems using the acquired concepts.
Develop logical thinking and speech.
Foster a love for the subject in the classroom.

I. Organizational moment2 minutes

Good afternoon, dear travelers to the land of knowledge!

To start today's lesson, I would like to know if you are ready for the lesson, what is your mood, do you have a desire to learn something new in today's lesson?

As the ancient Greek philosopher Saadi said: “A student who learns without desire is a bird without wings.”

And I would like you to have a desire to learn, to learn something new, unknown, not only in today’s lesson, but always and only in this case will you “fly” higher and higher with your “wings”.

And I also really want to turn to the words of the famous Russian mathematician A.I. Mordkovich: “Whoever studies mathematics from childhood develops attention, trains his brain, his will, and cultivates perseverance and perseverance in achieving the goal.”

This is what we will need in today's lesson: attention, perseverance and perseverance in order to achieve our goals.

II. Graphic dictation.5min (slide)

Is the statement true?

The surface of the cube consists of 6 equal squares.
The surface area of a rectangular parallelepiped is calculated using the formula:

S =av +sun +ac

3. Every cube is a rectangular parallelepiped.

4.Each rectangular parallelepiped has edges. These are segments.

5. Any rectangular parallelepiped consists of faces. He has 8 of them.

6. A rectangular parallelepiped has three dimensions - length, width, height.

7.The volume of a rectangular parallelepiped is calculated using the formula: V = A X V X c.

Count verbally(slide)

Find the area of the shaded figures (Figure 1). Draw three different shapes with an area of 8 square meters. units.

2. a) Take the necessary measurements and calculate the area of the rectangle (Figure 3). What is its perimeter?

3.Calculate the volume of a cube with a side of 5 cm.

III. A little historical background5 minutes(slide)

We are surrounded by many objects. They differ in shape, size, material from which they are made, color,…. People are interested in the different qualities of these objects. Mathematicians are interested in their shape and size.

The balls you have played with many times are spherical in shape, although they are all different sizes. Many celestial bodies have a shape close to the shape of a ball, including our planet. The glass and pencil are shaped like a cylinder.

Please note that the shapes of objects are very diverse and not every shape has a special name.

Since mathematicians study not the objects themselves, but their forms, instead of objects she considers geometric bodies: cylinder, ball, cube, etc. (samples of figures on the teacher’s desk). The names of many geometric bodies come from ancient times, and they originated from corresponding objects. For example, the terms “cone” (an object used to plug a barrel), “pyramid” (fire, bonfire), “cylinder” (roller), “rectangular parallelepiped” (rectangular planes) came from Ancient Greece.

Among the many different geometric bodies there is a large group of polyhedra. These figures (the teacher shows the figures) are polyhedra. And we will answer the question: “Why are these bodies called polyhedra?” during our lesson.

IV. Industrial gymnastics.2 minutes.

1) Extend your arms forward. Then slowly rotate your palms 10 times - this will help relieve tension from your hands.

2) Slowly lower your chin to your chest. Then raise your head again. Repeat the exercise 5 times. Good for the neck and shoulders.

3) While sitting, raise your left leg 40 cm from the floor. Move your toes up and down. Then do the same with your right leg.

V. Main part. 20 minutes.(slide)

Teacher’s explanation: “Guys, today we will remember how to find the area of a rectangle and the volume of a parallelepiped on the model of our school. Here is a scaled-down model of the school. Our class is the accounting chamber, each group is a department, and each of you is: chief engineer, engineer, foreman, foreman, accountant. Each department will receive its own task: find the area and volume of the floor.

Each group is given parallelepipeds. Students must measure the length, width, height of the parallelepiped, then find the area and volume of the floor. Lined up on the board and on the desks in each compartment table, in which you need to record and calculate all the data.

	Length	Width	Height	S	V
1 department
2 department
3 department
4 department
5 department

VI. Lesson summary. 1 min.

Teacher: “We did a good job today: we found out what the area and volume of each floor in our school is, and you will come for your salary tomorrow, when I check your independent work, you will find out who received how much.”

VI. Homework. No. 843, 845. (slide)

	Length cm	Width cm	Height cm	S cm 2	V
1 department

Please note that in accordance with Federal Law N 273-FZ “On Education in the Russian Federation”, organizations engaged in educational activities organize the training and education of students with disabilities, both together with other students and in separate classes or groups.

Pedagogical activities in accordance with the new Federal State Educational Standard require the teacher to have a system of special knowledge in the field of anatomy, physiology, special psychology, defectology and social work.

Only now you can take distance learning directly on the Infourok website with 40% discount according to the advanced training course (72 hours). Upon completion of the course, you will receive a printed certificate of advanced training of the established form (delivery of the certificate is free).

Parallelepiped

Library
materials

Topic: general lesson on the topic: “Rectangular parallelepiped.”

Goals:

To consolidate and generalize the ability to convert one unit of area measurement into another and use this knowledge when solving problems, the ability to calculate the surface area and volume of a rectangular parallelepiped and the ability to solve problems using the acquired concepts.

Develop logical thinking and speech.

Foster a love for the subject in the classroom.

I. Organizational moment2 minutes

Good afternoon, dear travelers to the land of knowledge!

To start today's lesson, I would like to know if you are ready for the lesson, what is your mood, do you have a desire to learn something new in today's lesson?

As the ancient Greek philosopher Saadi said: “A student who learns without desire is a bird without wings.”

This is what we will need in today's lesson: attention, perseverance and perseverance in order to achieve our goals.

II. Graphic dictation.5min (slide)

Is the statement true?

The surface of the cube consists of 6 equal squares.

The surface area of a rectangular parallelepiped is calculated using the formula:

S =av +sun +ac

3. Every cube is a rectangular parallelepiped.

4.Each rectangular parallelepiped has edges. These are segments.

5. Any rectangular parallelepiped consists of faces. He has 8 of them.

6. A rectangular parallelepiped has three dimensions - length, width, height.

7.The volume of a rectangular parallelepiped is calculated using the formula: V = A X V X c .

Count verbally(slide)

Find the area of the shaded figures (Figure 1). Draw three different shapes with an area of 8 square meters. units.

2. a) Take the necessary measurements and calculate the area of the rectangle (Figure 3). What is its perimeter?

3.Calculate the volume of a cube with a side of 5 cm.

I II. A little historical background5 minutes(slide)

Please note that the shapes of objects are very diverse and not every shape has a special name.

IV . Industrial gymnastics.2 minutes.

1) Extend your arms forward. Then slowly rotate your palms 10 times - this will help relieve tension from your hands.

2) Slowly lower your chin to your chest. Then raise your head again. Repeat the exercise 5 times. Good for the neck and shoulders.

3) While sitting, raise your left leg 40 cm from the floor. Move your toes up and down. Then do the same with your right leg.

V. Main part. 20 minutes.(slide)

Length

Width

Height

1 department

2 department

3 department

4 department

5 department

VI . Lesson summary. 1 min.

VI. Homework. No. 843, 845. (slide)

Length

Width

Height

1 department

Length

Width

Height

Brief description of the document:

Topic: summary lesson

Project

My dream house

Completed by: Last name, first name, class

Head: Tatyana Vladimirovna Lobacheva, mathematics teacher, MAOU “Cadet School 82”

Naberezhnye Chelny

1. Introduction

2. House sketch

3. Geometric shapes

3.1 Triangle

3.2 Rectangle

4. Volumetric geometric shapes

4.1 Pyramid

4.2 Parallelepiped

4.4 Cylinder

4.6 Prism

5 Conclusion

6 Literature

Introduction

Everyone noticed how many figures there were around. People have long been interested in their diversity, structure and properties. The science of geometry appeared, which makes it possible to study and measure figures. Many professions use this knowledge (from a simple carpenter to architects and designers creating spacecraft).

The question arises, what attracts people to architectural objects? We believe that the construction is unusual but strong, as well as the correct proportions and beautiful color scheme.

How can mathematics help in planning and creating an architectural object?

During construction, the geometric problem of dividing polyhedra into parts is most often solved. The concept of scale must be used. He depicts an object from a mathematical point of view, representing it in the form of a figure that could be seen when looking at it from above from the right and left sides. Various calculations are performed to calculate the amount of material required.

In our work, we would like to present a model of a residential building in which mathematical knowledge, skills and abilities were useful to us. We limit ourselves to studying the geometric shapes that we will need to create the model.

Fundamental question: How do we see the home of our future?

Hypothesis: Is it possible to create a layout of a residential property using models of geometric shapes.

The purpose of this work: study geometric shapes and three-dimensional geometric shapes.

Job objectives:

1. Identify the main geometric shapes, consider their elements and properties.

2. Consider the development of geometric shapes.

3. Create an architectural object using models of geometric shapes.

Research methods: collecting information, studying literature, observing everyday life, analysis.

Practical significance: this work contributes to the formation of an idea of the connection between mathematics and architecture and construction.

Volumetric geometric shapes.

In construction and architecture, several spatial geometric figures and their combination are usually used at once. In the modern world, we are surrounded by many buildings consisting of complex geometric shapes, most of which are polyhedra. There are a lot of examples of this, just look around and we will notice that the buildings in which we live, the shops we go to, schools and kindergartens, etc. presented in the form of polyhedra.

Polyhedron is a body whose surface consists of a finite number of flat polygons.

Let's look at the shapes that we will use when creating the layout.

Rectangular parallelepiped. Cube

A cuboid is a parallelepiped whose all faces are rectangles.

A rectangular parallelepiped has six faces, twelve edges and eight vertices.

The three legs of a rectangular parallelepiped that meet at one vertex are called its length, width and height. The rectangular parallelepiped is limited by its faces from below, above and from the sides. Each face is a rectangle. The base of a rectangular parallelepiped is the lower and upper faces.

Side faces are all edges except the bottom and top. The faces intersect along segments - the edges of a rectangular parallelepiped. The points at which the edges intersect are called the vertices of a rectangular parallelepiped.

To make any geometric body in a model, you need to draw its development. The development of the surface of a geometric body is a flat figure, which is obtained by combining all the faces or all surfaces that bound the body with one plane.

Development of a rectangular parallelepiped.

A rectangular parallelepiped in which all the hips are equal is called a cube. The surface of the cube consists of 6 equal squares.

Development of a cube.

Pyramid.

A polyhedron, one of whose faces is a polygon and the remaining faces are triangles with a common vertex, is called a pyramid.

The common vertex of all lateral faces is called the vertex of the pyramid.

The height of a pyramid is the perpendicular drawn from the top of the pyramid to its base.

Development of a quadrangular pyramid.

Prism.

A straight prism is one of the simplest polyhedra. Each face (polygon, bounding polyhedron) of the polyhedron is located in its own plane. The intersection of the faces of a polyhedron passes along the line of its edges.

In Fig. - pentagonal rectangular prism (at the base of the prism is a pentagon). It has 10 peaks; 5 side faces; 2 bases (upper and lower). The lateral faces of a rectangular prism are rectangles.

Development of a triangular prism.

Cylinder.

In the reality around us there are many objects that have the shape of a cylinder, for example a bucket, a tin can, a pencil case, a piece of round wire, etc.

A cylinder can be formed by rotating a rectangle around one of its sides (Fig. 344, 345).

The cylinder has two bases, which are circles, and a side surface, which is called a cylindrical surface (Fig. 346).

Cylinder development.

If the side surface of the cylinder is expanded and placed on a plane, we obtain a rectangle (Fig. 347).

The development of the full surface of the cylinder consists of a rectangle, the length of which is equal to the circumference of the base of the cylinder, and the height is equal to the height of the cylinder and two circles (Fig. 348).

Cone.

If you rotate a right triangle about one of its legs, you get a geometric body called a cone (Fig. 435, 436, 437).

The base of the cone is a circle.

The hypotenuse of a right triangle, the movement of which forms the lateral surface of the cone, is called the generator of the cone. The height of a right circular cone, lowered from its apex to the base, passes through the center of the base.

Cone development.

Conclusion

The purpose of this work was to highlight the main geometric shapes.

To achieve this goal:

1. The main geometric shapes were identified.

2. The main three-dimensional geometric shapes were identified.

3. Residential objects were observed to determine their geometric shape.

4. A plan for a residential property has been drawn up.

5. Calculations of the sizes of geometric shapes were carried out.

6. Developments of geometric figures were constructed.

7. A model of the house has been made.

As a result of the project, the following conclusions can be drawn:

1. In the course of our work, we confirmed the hypothesis that it is possible to create a model of a house using models of geometric shapes.

2. Man is gradually reducing the number of geometric shapes used, particularly in architecture, in favor of rectilinear ones (cubes and parallelepipeds), thereby impoverishing the world around him.

3. Planning and building houses is a difficult but interesting task. We have to make a lot of calculations and measurements.

Literature

1. Mathematics: 5th grade: textbook for students of general education institutions A.G. Merzlyak, V.B. Polonsky, M.S. Yakir – M.: Ventana-Graf, 2012.

2. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook: a manual for students in grades 5-6 of secondary school. - M.: "Enlightenment", 1989.

3. Sharygin I.F., Erganzhieva L.N. Visual geometry: a textbook for students in grades 5-6. - M.: "Miros", 1995.

4. http://oldskola1.narod.ru/Nikitin/