The cube has visible edges. §22
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 visible edges cube, using a different color for each face.
№ 6 . The figure shows a rectangular parallelepiped facing the viewer with edge LN. Circle visible ribs solid, invisible - 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.
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. A rectangular parallelepiped and a 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?
