The cube has visible edges. §22

All objects that surround us can be mentally fit into simple geometric bodies (cube, ball, cone, cylinder, prism, etc.). By studying the shape of a cube, we learn how to draw, for example, a house, because in a simplified way a house is drawn using the same techniques as a cube. It has vertices, edges and faces, just like a cube. The roof of a house is a multifaceted prism.

Let's draw a cube from life, and then we will use this knowledge on ours to depict more complex subjects such as houses and streets.

A cube is a geometric body formed by the intersection of planes. And, like any three-dimensional object, when depicted on a flat sheet it will undergo changes in accordance with the laws of perspective. The picture shows the horizon line artist's plane of vision. It contains the vanishing points of parallel lines. In our case it is four horizontal lines, tending to the vanishing point on the left and four horizontal lines tending to the vanishing point on the right.

We depict objects in space as our eyes perceive them. (The further away from the viewer, the smaller the object appears, etc.)

The beginning of any painting is composition. We outline our object on the sheet with light lines. There should always be a little more space from the edge at the top than at the bottom. Intuitively determine the scale so that the subject does not look gigantic or too small.

Position the closest vertical edge so that it didn't match with the center of the sheet passing through the intersection of its diagonals. We mark the height with serifs; this is the highest edge in our image, since it is closest to the viewer. By eye we determine the angle of inclination of the ribs lying on the table relative to the horizontal. Train visual memory, remembering the angle. Quickly move your gaze either to the cube or to the drawing.

Let's do the same with the upper ribs. Basic laws explain to us how to transfer space on a sheet linear perspective. All parallel lines merge towards the horizon line into one point. Therefore, to convey that the edge is further from the viewer, we will depict it less and arrange higher. This way, all the edges will be of different heights.

When the far horizontal edges intersected, vertices were formed. The farthest rib, invisible to the eye, passes through them. On initial stage Let's depict the cube as transparent to understand the complete structure of the object.

In order to find out how much the side edges have been reduced, we will use sighting method. Using this method, the outlines of an object are perceived, the artist learns to depict objects proportional and from different angles.

How does he work? Take a pencil at arm's length, close one eye, align the pencil and the image of the edge of the cube in space. The top edge of the pencil should coincide with the top apex of the rib, and with your finger, pinch the point on the pencil that coincides with the bottom apex. Without removing your finger from the pencil, turn it at a right angle and measure the distance between the two edges. Thus, we will see the ratio of the height and width of one face. Remember this ratio and show it in the drawing. This method can also be used to measure and display the ratio of the ribs.

After the linear constructions are completed, we proceed to aerial perspective , and therefore to shading.

The main task of the artist is to convey the three-dimensional forms of objects. We see three faces of our cube, all of them different in tone. The left side is the darkest - this is the object’s own shadow. Thanks to reflected light from surrounding objects or reflexes, we make the shading a little lighter as we move away to the left. The largest edge is made more contrasting than all the others. Thus, they show his proximity to the foreground.

Upper plane darker than the vertical one on the right. The light only glides across it, forming a halftone. Please note that than closer to the light source, the lighter there will be a tone. Hatching can be applied diagonally. Use an eraser to highlight the edge to convey the highlight.

To work on the lightest edge, take hard pencil N or 2H. It will prevent the tone from becoming too dark. Apply the shading vertically, in the direction of the plane.

Falling shadows are always darker than the object's own shadow. The near edge is the line of transition between light and shadow. A falling shadow begins from it. The closer to the subject, the richer the tone. The reflected light from the cube creates a reflex inside the shadow and it brightens a little.

Drawing simple geometric bodies often used on and allows a novice artist to learn how to depict objects in space by applying the laws perspective construction and aerial perspective.

REPEAT THE THEORY

260. Complete the theory.

1) Each face of a rectangular parallelepiped is rectangle.
2) The sides of the faces of a rectangular parallelepiped are called edges, the vertices of the faces are vertices of a rectangular parallelepiped.
3) A parallelepiped has 6 faces, 12 edges, 8 vertices.
4) The faces of a rectangular parallelepiped that do not have common vertices are called opposite.
5) Opposite faces of a rectangular parallelepiped are equal.
6) The surface area of ​​a parallelepiped is called the sum of the areas of its faces.
7) The lengths of three edges of a cuboid having a common vertex are called the dimensions of the cuboid.
8) To distinguish between the dimensions of a rectangular parallelepiped, use the names: length, width and height.
9) A cube is a rectangular parallelepiped with all dimensions are equal.
10) The surface of the cube consists of six equal squares.

SOLVING PROBLEMS

261. The figure shows a rectangular parallelepiped ABCDMKEF. Fill the gaps.

1) Vertex B belongs to the faces AMKV, ABCD, KVSE.
2) The edge EF is equal to the edges KM, AB, CD.
3) The upper face of the parallelepiped is a rectangle MKEF.
4) Edge DF is a common edge of faces AMFD and FECD.
5) The face AMKV is equal to the face FESD.

262. Calculate the surface area of ​​a cube with an edge of 6 cm.

Solution:
The area of ​​one face is equal to
6 2 -6*6 = 36 (cm 2)
The surface area is equal to
6*36 = 216 (cm 2)

Answer: Surface area is 216 cm 2 .

263. The figure shows a rectangular parallelepiped MNKPEFCD, the dimensions of which are 8 cm, 5 cm and 3 cm. Calculate the sum of the lengths of all its edges and the surface area.

Solution:
Sum of edges
4*(8+5+3) = 64 (cm)
The surface area is:
2*(8*3+8*5+5*3) = 158 (cm 2)

Answer: the sum of the lengths of all its edges is 64 cm, the surface area is 158 cm 2.

264. Fill in the blanks.

1) The surface of the pyramid consists of side faces - triangles that have a common top and base.
2) The common vertex of the lateral faces is called the top of the pyramid.
3) The sides of the base of the pyramid are called base ribs, and the sides of the side faces that do not belong to the base - lateral ribs.

265. The figure shows the SABCDE pyramid. Fill the gaps.

1) The figure shows a 5-angle pyramid.
2) The lateral faces of the pyramid are triangles SAB, SBC, SCD, SDE, SEA, and the base is the 5-square, ABCDE.
3) The top of the pyramid is point S.
4) The edges of the base of the pyramid are segments AB, BC, CD, DE, EA, and the side edges are segments SA, SB, SC, SD, SE.

266. The figure shows a pyramid DАВС.yu all faces of which are equilateral triangles with sides of 4 cm. What is the sum of the lengths of all the edges of the pyramid?

Solution:
The sum of the edge lengths is
6*4 = 24 (cm)

Answer: 24 cm.

267. The figure shows a pyramid МАВСD, the lateral faces of which are isosceles triangles with sides of 7 cm, and the base is a square with a side of 8 cm. What is the sum of the lengths of all the edges of the pyramid?

Solution:
The sum of the lengths of the lateral edges is equal to
4*7 = 28 (cm)
The sum of the lengths of the edges of the base is equal to
4*8 = 32 (cm)
Sum of lengths of all edges
28+32 = 60 (cm)

Answer: the sum of the lengths of all the edges of the pyramid is 60 cm.

268. Can it have (yes, no) the shape of a rectangular parallelepiped:
1) apple; 2) box; 3) cake; 4) tree; 5) a piece of cheese; 6) a bar of soap?

Answer: 1) no; 2) yes; 3) yes; 4) no; 5) yes; 6) yes.

269. The figure shows the sequence of steps in the image of a rectangular parallelepiped. Draw a parallelepiped in the same way.

270. The figure shows the sequence of steps of the pyramid image. Draw the same pyramid.

271. What is the size of the edge of a cube if its surface area is 96 cm 2?

Solution:
1) 96:6 = 16 (cm 2) - the area of ​​one face of the cube.
2) 4*4 = 16, which means the edge of the cube is 4 cm.

Answer: 4 cm.

272. Write down the formula for calculating surface area S:

1) a cube whose edge is equal to a;
2) a rectangular parallelepiped whose dimensions are a, b, c.

Answer: 1) S = 6a 2 ; 2) S = 2(аb+ас+bс)

273. To paint the cube shown in the picture on the left, 270 g of paint is required. Part of the cube was cut out. How many grams of paint will be required to paint the part of the surface of the resulting body, highlighted in blue.

Solution:
1) 270:6:9 = 45:9 = 5 (g) - for painting a single face
2) 5*12 = 60 (g) - for painting a blue surface

Answer: you will need 60 g of paint

274. Which of the figures A, B, C, D, D complements the figure E to a parallelepiped?

275. Rectangular parallelepiped and the cube have equal areas surfaces. The height of the parallelepiped is 4 cm, which is 3 times less than its length and 5 cm less than its width. Find the edge of the cube.

Solution:
1) 4*3 = 12 (cm) perellepiped length
2) 4+5 = 9 (cm) width of the parallelepiped
3) 2*(4*12+4*9+12*9) = 384 (cm 2) surface area of ​​the parallelepiped
4) 384:6 = 64 (cm 2) area of ​​the cube face
5) 64 = 8*8 = 8 2, which means the edge of the cube is 8 cm.

Answer: cube edge 8 cm.

276. Trace the visible edges on the image of the cube with a colored pencil so that the cube is visible: 1) from above and to the right; 2) below and to the left.

277. The faces of the cube are numbered from 1 to 6. The figure shows two versions of the development of the same cube, obtained by cutting equally. What number should replace the question mark?

5th grade club

Head Blinkov Alexander Davidovich
2005/2006 academic year

Cube and its development (9.03 and 11.03)

The bodies shown in the figure are made up of cubes. How many cubes are there in each of them?

On the visible faces of the cube there are numbers 1, 2 and 3. And on the scans - two of the above numbers or one. Place the numbers 1, 2, 3, 4, 5, 6 on the sides of the cube so that the sum of the numbers on opposite faces is equal to 7.

The dotted lines in the figure indicate the invisible edges of the cube. Accordingly, solid lines show visible lines. We looked at the cube on the top right. In pictures a, b, c, draw solid lines so that the cube is visible
a) bottom right;
b) top left;
c) bottom left.

Additional tasks

The wooden cube was painted blue on the outside. After that, each rib was divided into 5 parts and this cube was cut into small ones with an rib 5 times smaller. How many small cubes did you get?
A) How many cubes have three sides colored?
B) Two faces?
C) One edge?
D) None?

The segment connecting the two vertices of a cube that are farthest from each other is called its diagonal. How to measure the diagonal of a non-empty cube using a ruler and having three such cubes?

Drawing for problem 2

Problem solving is a practical art, just like swimming and running. It can only be learned through imitation and practice.
D. Polya

Dice are important in the life of a small child. But their role does not decrease during school, in particular, when studying mathematics. We must always remember that a student, a schoolchild, is, first of all, a child. He, like a sponge, absorbs all the information from the world around him. You can always find situations or create conditions that can serve as an impetus for deep reflection and creative and research activities in the student. “In order to push a child to specific ideas, specific means are also needed. It is impossible to count on the fact that mere observation of random events will allow children to discover probabilistic laws; it is necessary to introduce elements of competition, activities must be exciting and excite the child’s natural curiosity, confront him with reality, and also refute false ideas,” say M. Gleman and T. Varga. And one cannot but agree with them.

Tasks play a huge role in a person’s life. The tasks that a person sets for himself, and the tasks that other people and life circumstances set for him, guide all his activities throughout his life. The famous Russian methodologist V.A. Evtushevsky characterized the functions of tasks in teaching as follows: “The tasks offered in the classroom contain living material for exercising the student’s thinking, for deriving mathematical rules and for practicing applying these rules in solving particular practical issues.”

The presented entertaining tasks with cubes are varied, since you can select cubes whose faces depict numbers, letters, drawings, and colors. Such tasks are applicable for children of a wide age group at various stages of mathematics lessons and in extracurricular activities. They all contribute to:

• learning to read graphic information, images of geometric objects;
• development of spatial imagination;
• formation of skills to mentally imagine different positions of an object and changes in its position depending on different points reference and ability to fix this representation in the image;
• teaching logical justifications of geometric facts;
• development of design abilities, modeling;
• development cognitive processes: perception, attention, memory, thinking;
• development of research skills.

Entertaining tasks with playing cubes will attract the attention of children and make their interest in mathematics quite persistent; they will help master mathematical skills not only for strong students, but also for those for whom this school subject is the most difficult.

Tasks.

Task No. 1. Number the 8 vertices of the cube with serial numbers (1, 2, 3, 4, 5, 6, 7, 8) so that the sum of the numbers on each of its six faces is the same (Fig. 1a).

Answer. Each vertex of the cube belongs to three faces, so the sum 1 + 2 + : + 8 should be multiplied by 3, then divided by 6 (by the number of faces), you get 18 - the sum of the numbers on each face (Fig. 1b).

Picture 1.

Problem No. 2. Is it possible to “number” all the edges with integers so that the sums of the numbers of the edges converging at each vertex are the same if these are the numbers: a) 1; 2; :; 12; b) -6; -5; :; -1; 1; 2; :; 6?

a) No. Let's assume that this is possible and the sum of the numbers of edges converging at each vertex is equal to x. Then the sum of the numbers on all eight edges of the cube is 8x. On the other hand, since each number was included in this sum twice, then this same sum is equal to: (1 + 2 + : + 11 + 12) 2 = (1 + 12) 12 = 156. Equation 8x = 156 in integer solutions does not, so our assumption is incorrect.

b) Yes. The sum of the numbers of edges converging at each vertex is equal to 0 (Fig. 2).

Figure 2.

Task No. 3. In Fig. Figure 3 shows a figure that is a development of a cube. Thin lines are fold lines. Mentally fold the cube out of the unfolded pattern. Determine which face is the top if the shaded face is the bottom.

Figure 3.

Answer. "V".

Problem No. 4. Letters are written on the faces of an opaque cube as shown in Fig. 4a. The cube was thrown, and it fell so that one of the letters began to be located as shown in Fig. 4b. Draw the corresponding letters on the remaining faces of the cube (they may be rotated). Check your answer using the cube model.

Figure 4.

Answer. Rice. 4c.

Task No. 5. Mentally roll a cube from each development given in Fig. 5 and determine which face is the top if the bottom face is shaded.

Answer. a) D, b) B, c) D, d) C.

Figure 5.

Problem No. 6. All the cubes in Fig. 6a are the same. Redraw the pattern of one of the cubes (Fig. 6b) and put the missing letters on it.

Figure 6.

Answer. Rice. 6th century

Problem No. 7. We threw the cube (Fig. 7a) so that it fell, as shown in Fig. 7b fill in the blanks visible edges Cuba.

Answer. Rice. 7th century

Figure 7.

Problem No. 8. B in the right place On the front side of the cube scan, write down the letters G and P in the correct position (Fig. 8).

Figure 8.

Answer. Rice. 9.

Figure 9.

Task No. 9. Looking at the frame of the cube first from the front (view A), then from the left (view B) and finally from the top (view C), read the word formed by bold lines (Fig. 10).

Answer. 1) BOR, 2) SPRUCE, 3) BES.

Figure 10.

Problem No. 10. In Fig. 11 shows a figure that is a development of a cube ( fine lines- these are fold lines). Which points will coincide with point A when gluing the development shown in the figure?

Figure 11.

Answer. M, H.

Task No. 11. Having correctly depicted three rectangular projections of cubes with letters shifted between each other, read the Russian folk wisdom(Fig. 12a).

Answer. Laziness is the mother of vices (Fig. 12b).

Figure 12.

Problem No. 12. A cube is glued from cardboard, with letters written on its edges. In Fig. 13a shows one version of the development of this cube with the image of letters on its faces.

Figure 13.

Draw letters on the empty faces of another version of the development of this cube (Fig. 13b-d).

Answer. Rice. 14.

Figure 14.

Task No. 13. If you guess how to arrange the letters on the cubes (on the front faces), then the letters on the top faces will form a new word (Fig. 15).

Answer. KITTEN - MONKEY.

Figure 15.

Problem No. 14. From the figures shown in Fig. 16, select those that are the dimensions of the cube. Highlight them with color. Redraw the image data, cut it out and check your choice.

Figure 16.

Answer. “a”, “b”, “d”, “d”, “f”, “g”.

Problem No. 15. Which of the cubes shown in Figures 17b-h can be glued from the layout (Figure 17a)?

Figure 17.

Answer. "e".

Problem No. 16. In Fig. 18 you see three children's blocks. All of them are turned towards us with the same pattern - a herringbone pattern. Indicate what pictures we will see on each of the cubes, looking at them from above, taking into account the development of the cube.

Figure 18.

Answer. a) ball, b) leaf, c) cloud.

Problem No. 17. Indicate the coloring of the faces of the cube on the development shown in Fig. 19a-b, if in Fig. The 19v-d cube is presented in three different positions.

Figure 19.

Answer. Rice. 20.

Figure 20.

Problem No. 18. The faces of the cube are colored as shown in Fig. 21. The cube was tossed. It fell so that the front edge became a transparent edge. Color the remaining faces of the cube in the appropriate colors (Fig. 21). Consider all possible options. Make the necessary sweep. Cut it out and check your answer.

Figure 21.

Answer. Rice. 22.

Figure 22.

Task No. 19. A toy was made from multi-colored cubes (Fig. 23a). Color the cubes if red is between blue and yellow, and yellow is below green.

Figure 23.

Answer. Rice. 23b.

Problem No. 20. Color the maximum number of vertices of the cube red so that among the red vertices it is impossible to choose three that form an equilateral triangle.

Answer. The maximum possible number of red vertices is four. Let's prove it.

It is possible to paint four vertices. For example, you can paint four vertices of one face. In this case, the red vertices form a square and there are not three among them that form an equilateral triangle.

Let us prove that it is impossible to color five vertices of a cube that satisfy the condition. Let's color the four vertices of the cube Blue colour, and the remaining ones - in green (Fig. 24). Note that there is the same distance between any two vertices of the same color. Let us be able to recolor the five vertices red. Then some three of them were painted the same color. Therefore, they form an equilateral triangle.

Figure 24.

Problem No. 21. On the faces of the cube there are figures such as in Fig. 25a. The cube is successively rolled from face to face, as shown in Fig. 25b. What figures should be located on the top and right side faces of the last image of the cube?

Figure 25.

Answer. On the top face there is a circle, on the right side face there is a square.

Problem No. 22. A white cube, the edge of which is 3 cm, was painted with blue paint, and then sawed into cubes with an edge 1 cm long. How many of them have one colored face, two colored faces, three colored faces? Is there a cube with uncolored faces?

Answer. They have one painted face - 6 cubes, two painted faces - 12 cubes, three painted faces - 8 cubes, a cube with unpainted faces - 1 cube.

Problem No. 23. Two cubes, the opposite faces of which are painted the same color, were connected together in different ways. They forgot to color some of the faces of the cubes. Color them in appropriate colors (Fig. 26).

Figure 26.

Answer. Rice. 27.

Figure 27.

Problem No. 24. After the development is folded into a cube, which of the following cubes will be obtained (Fig. 28)? (Ignore the placement of the pictures.)

Answer. "G".

Figure 28.

Problem No. 25. Which cube is glued together from this development (Fig. 29)?

Answer. "A".

Figure 29.

Problem No. 26. Find the union of the three parts of the cube to the left of the equal signs (Fig. 30a, b), and draw it to the right of the equal signs as shown in the example (Fig. 31).

Figure 30.

Figure 31.

Answer. Rice. 32.

Figure 32.

Problem No. 27. Each of the figures depicted to the left of the equal signs (Fig. 33) is a union of two parts of the cube obtained by cutting it with a plane passing through the center. Restore these parts, depicting the answer in a form similar to the previous task.

Figure 33.

Answer. Rice. 34.

Figure 34.

Problem No. 28. All faces of the cube are painted in different colors, and each face is painted the same color. If you look at this cube from one side, you can see blue, yellow and white faces. On the other side, black, blue and red edges are visible. On the third side, green, black and white edges are visible. Which edge is opposite the white one?

Answer. Opposite the white edge is the red edge.

Problem No. 29. How many cubes were used to build the tower (Fig. 35)?

Figure 35.

Answer. a) 28; b) 44.

Problem No. 30. How many cubes are needed to fold such a figure (Fig. 36)?

Answer. 106 cubes.

Figure 36.

Problem No. 31. In Fig. 37a shows four cubes. They are colored differently, but each of them has opposite edges of the same color. From these cubes they built the figures “pedestal” and then a parallelepiped. They were built so that the touching faces of the cubes were the same color. Finish coloring the figures in Fig. 37b,c and indicate the numbers of the cubes.

Figure 37.

Answer. Rice. 38.

Figure 38.

Problem No. 32. Journey of a fly. A fly, starting from point A, can go around the four sides of the base of the cube in 4 minutes. How long will it take her to get from A to the opposite vertex B (Fig. 39a).

Figure 39.

Answer. A smart fly would choose the path marked in Fig. 39b with a solid line, it will take 2.236 minutes to overcome it. Path marked dotted line, is longer and will take more time.

Problem No. 33. A large cube is glued together from small wooden cubes. 6 through holes were drilled in it, parallel to the ribs (Fig. 40). How many small cubes are left intact?

Figure 40.

Answer. 44 cubes.

Problem No. 34. I have a piece of cheese in the shape of a cube. How should I make one straight knife cut so that the two new edges are regular hexagons? Of course, if we cut the cheese in the direction of the dotted line in Fig. 41a, then we get two squares. Try to get hexagons.

Answer. Mark the midpoints of the ribs BC, CH, HE, EF, FG and GB. Then, starting from the top, make a cut along the plane indicated by the dotted line (Fig. 41b). Then each of the two new surfaces will be a regular hexagon, and the right piece will look something like shown in Fig. 41c.

Figure 41.

Problem No. 35. An advertising agency sent these drawings to the customer - a packaging manufacturer. He was asked to decide what color should be on the side of the package that is opposite the yellow side in the figure. 42 Q. The next day the customer called. What question did he ask?

Figure 42.

Answer. He asked: “Is there a mistake here or did you deliberately repeat yellow? " The complete diagram is shown in Fig. 43.

Figure 43.

Problem No. 36. On these architectural models, each cube is a separate apartment (Fig. 44). The construction contract will go to the architect whose model has the most apartments. Which layout meets this requirement?

Figure 44.

Answer. The layout of building A meets this requirement; this building has 80 apartments, while building B has only 79.

Problem No. 37. The legend associated with the problem of doubling the surface of a cube has become a classic. Philoponus tells how the Athenians, frightened by the plague epidemic of 432 BC. e., turned to Plato for advice. But before coming to the great philosopher, they appealed to Apollo, who, through the mouth of the Delphic oracle, commanded them to double the size of the golden altar in their temple. However, the Athenians proved unable to do this. Plato said that misfortune befell them because of their malicious neglect of the sublime science of geometry, and lamented that among them there was not a single person wise enough to solve this problem.

The problem of the Delphic Oracle, where it is simply a matter of doubling the cube, is so closely related to the problem of Plato's cubes that authors who are not very experienced in mathematics often confuse them. The last problem is also called the problem of Plato's geometric numbers, usually claiming that almost nothing is known about its true conditions. Some even believe that its terms have been lost.

Exists ancient description a massive cube erected in the center of a tiled platform, and it does not require much imagination to connect this monument with Plato's problem. In Figure 45 you see Plato contemplating such a massive marble cube, which is composed of a number of smaller cubes. The monument rises in the center of a square platform lined with the same small marble cubes.

Figure 45.

The number of cubes in the site and in the monument is the same. Tell me how many cubes it takes to build a monument and a square platform, and you will have solved the great problem of Plato's geometric numbers.

Answer. The problem requires finding a number that, when cubed, will give an exact square. This happens, it turns out, with any number that is itself a square. The smallest square (not counting 1) is 4, so the monument could contain 64 small cubes (4 * 4 * 4) and stand in the center of an 8 * 8 square. Of course, this does not agree with the proportions shown in the figure. We will therefore try the next square, 9, which results in a monument of 729 cubes standing on a square 27 * 27. This is the correct answer, for it is the only one that agrees with the figure.

Problem No. 38. In the East, the art of mixing different types of tea does not neglect millionths of an ounce! They say that the secrets of some mixtures were kept in deep secrecy and could not be repeated for centuries.

To illustrate how difficult it is to penetrate the art of tea blending, we present to your attention one simple task where only two varieties are mixed.

The blender received two boxes of tea. They were both cubic in shape, but had different sizes. The larger box contained black tea, and the smaller one contained green tea. Having mixed the contents of these boxes, the man discovered that the resulting mixture managed to fill exactly 22 boxes of cubic shape and the same size. Let us assume that the internal dimensions of the boxes are expressed as a finite decimal fraction. Can you determine the proportion of black and green tea in this mixture? In other words, find two different rational numbers such that, when adding their cubes, you get a result that, when divided by 22 and then taking the cube root, would also lead to a rational number.

Answer. A cube with an edge of 17.299 inches and a cube with an edge of 25.469 inches have a total volume (21697.794418608 cubic inches) exactly equal to the total volume of 22 cubes with an edge each of 9.954 inches. Therefore, green and black tea were mixed in the ratio of (17299) 3 to (25469) 3.

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Appendix No. 2

Exercises to develop the imagination

№1. There are two ways to lay out a parallelepiped from four cubes. Will the surface area of ​​the parallelepiped be the same in the first and second cases?

№2. The volume of the parallelepiped is 64 cm 3 , width - 4 cm, height - 2 cm. The length of this parallelepiped was reduced by 3 cm. Determine the volume of the resulting parallelepiped. (With the help of the teacher, students must imagine that the parallelepiped specified in the condition is cut into two parallelepipeds, and the length of the “cut” parallelepiped is 3 cm. Hence, in order to solve the problem, it is necessary to reduce the volume of the original parallelepiped by the volume of the “cut off” part.)

№ 3 . Draw a straight line and label it the letter a. Construct several points located from line a at a distance of 2 cm. Where are all such points located?

№ 4. The figure shows a wireframe model of a cube. Name the edges coming out of vertex M.

№ 5. Shade the visible faces of the cube, using a different color for each face.

№ 6 . The figure shows a rectangular parallelepiped facing the viewer with edge LN. Outline visible edges with solid lines, invisible edges with dashed lines..

№ 7. Figure 11 shows how to construct a rectangle. Describe the proposed method in words and complete the construction.

№ 8. Think about which of the figures shown in the figure could be a net of a cube?

№ 9. Which points will align when gluing the development shown in drawing?

The completion of this task must be preceded by the students making this scan from a sheet of paper.

№ 10. On the surface of the glass cube passes broken line made from wire. Draw this polyline on the image of the cube from the front, top and left.

№ 11. How many cubes are needed to build the tower shown in the picture?

№ 13. Arrange the letters on the cube's unfoldings in accordance with those already outlined. B – side edge, B – top, H – bottom.

№ 14 . The cube was looked at from the top right. Draw solid lines so that the cube is visible from the bottom left, top right, bottom right.

№ 15. How many different squares with vertices at given points can be drawn in the picture?

№ 16. What is the smallest number of cubes needed to build a tower? The illustration shows the front view and the left view.

№ 17. Construct a segment AC if it is known that point B is the midpoint of this segment and the segment BC is equal to 4 cm 2 mm.

№ 18 . Draw the bisector of the unfolded angle and the bisector of each of the resulting angles. How many angles are there in total? Find the magnitudes of these angles.

№ 19. How many faces will the polyhedron that results from cutting off all the vertices of a cube have?

№ 20. Draw solid lines around the edges of the cube so that it is visible from the top right (bottom left; top left; bottom right).

No. 21. A sheet of paper shaped like a rectangle was folded in half, as shown in the figure. Then cut along the dotted line

and the smaller cut part was unrolled. What is the development shape of the smaller cut piece?

No. 22. See how you can make a drawing by connecting the dots. Try drawing something yourself by connecting the dots.