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.
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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 DABC. All of its faces 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 side 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 surface areas. 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?
or (equivalently) a polyhedron with six faces that are parallelograms. Hexagon.
The parallelograms that make up a parallelepiped are edges of this parallelepiped, the sides of these parallelograms are edges of a parallelepiped, and the vertices of parallelograms are peaks parallelepiped. In a parallelepiped, each face is parallelogram.
As a rule, any 2 opposite faces are identified and called bases of the parallelepiped, and the remaining faces - lateral faces of the parallelepiped. The edges of the parallelepiped that do not belong to the bases are lateral ribs.
2 faces of a parallelepiped that have a common edge are adjacent, and those that do not have common edges - opposite.
A segment that connects 2 vertices that do not belong to the 1st face is parallelepiped diagonal.
The lengths of the edges of a rectangular parallelepiped that are not parallel are linear dimensions (measurements) parallelepiped. A rectangular parallelepiped has 3 linear dimensions.
Types of parallelepiped.
There are several types of parallelepipeds:
Direct is a parallelepiped with an edge perpendicular to the plane of the base.
A rectangular parallelepiped in which all 3 dimensions are equal is cube. Each of the faces of the cube is equal squares .
Any parallelepiped. The volume and ratios in an inclined parallelepiped are mainly determined using vector algebra. The volume of a parallelepiped is equal to the absolute value of the mixed product of 3 vectors, which are determined by the 3 sides of the parallelepiped (which originate from the same vertex). The relationship between the lengths of the sides of the parallelepiped and the angles between them shows the statement that the Gram determinant of the given 3 vectors is equal to the square of their mixed product.
Properties of a parallelepiped.
- The parallelepiped is symmetrical about the middle of its diagonal.
- Any segment with ends that belong to the surface of a parallelepiped and that passes through the middle of its diagonal is divided by it into two equal parts. All diagonals of the parallelepiped intersect at the 1st point and are divided by it into two equal parts.
- The opposite faces of the parallelepiped are parallel and have equal dimensions.
- The square of the length of the diagonal of a rectangular parallelepiped is equal to
