When you were little and played with cubes, you may have made the shapes shown in Figure 154. These figures give an idea of rectangular parallelepiped. The shape of a rectangular parallelepiped is, for example, a box of chocolates, a brick, matchbox, packing box, juice pack.
Figure 155 shows a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1.
Rectangular parallelepiped limited to six edges. Each face is a rectangle, i.e. The surface of a rectangular parallelepiped consists of six rectangles.
The sides of the faces are called edges of a rectangular parallelepiped, vertices of faces − vertices of a rectangular parallelepiped. For example, segments AB, BC, A 1 B 1 are edges, and points B, A 1, C 1 are vertices of the parallelepiped ABCDA 1 B 1 C 1 D 1 (Fig. 155).
A rectangular parallelepiped has 8 vertices and 12 edges.
The faces AA 1 B 1 B and DD 1 C 1 C do not have common vertices. Such edges are called opposite. In the parallelepiped ABCDA 1 B 1 C 1 D 1 there are two more pairs of opposite faces: rectangles ABCD and A 1 B 1 C 1 D 1, as well as rectangles AA 1 D 1 D and BB 1 C 1 C.
Opposite faces rectangular parallelepiped are equal.
In Figure 155, the face ABCD is called basis rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 .
The surface area of a parallelepiped is the sum of the areas of all its faces.
To have an idea of the dimensions of a rectangular parallelepiped, it is enough to consider any three edges having common top. The lengths of these edges are called measurements rectangular parallelepiped. To distinguish them, they use names: length, width, height(Fig. 156).
A rectangular parallelepiped in which all dimensions are equal is called cube(Fig. 157). The surface of the cube consists of six equal squares.
If a box in the shape of a rectangular parallelepiped is opened (Fig. 158) and cut along four vertical edges (Fig. 159), and then unfolded, we get a figure consisting of six rectangles (Fig. 160). This figure is called development of a rectangular parallelepiped.
Figure 161 shows a figure consisting of six equal squares. It is the development of a cube.
Using a development, you can make a model of a rectangular parallelepiped.
This can be done, for example, like this. Draw its outline on paper. Cut it out, bend it along the segments corresponding to the edges of the rectangular parallelepiped (see Fig. 159), and glue it together.
A rectangular parallelepiped is a type of polyhedron - a figure whose surface consists of polygons. Figure 162 shows polyhedra.
One type of polyhedron is pyramid.
This figure is not new to you. Studying the course Ancient world, you got acquainted with one of the seven wonders of the world - the Egyptian pyramids.
Figure 163 shows the pyramids MABC, MABCD, MABCDE. The surface of the pyramid consists of side faces− triangles having a common vertex, and grounds(Fig. 164). The common vertex of the lateral faces is called edges of the base of the pyramid, and the sides of the side faces that do not belong to the base are lateral edges of the pyramid.
Pyramids can be classified according to the number of sides of the base: triangular, quadrangular, pentagonal (see Fig. 163), etc.
Surface triangular pyramid consists of four triangles. Any of these triangles can serve as the base of a pyramid. This base is a type of pyramid, any face of which can serve as its base.
Figure 165 shows a figure that can serve sweep quadrangular pyramid . It consists of a square and four equal isosceles triangles.
Figure 166 shows a figure consisting of four equal equilateral triangles. Using this figure, you can make a model of a triangular pyramid, all of whose faces are equilateral triangles.
Polyhedra are examples geometric bodies.
Figure 167 shows familiar geometric bodies that are not polyhedra. You will learn more about these bodies in 6th grade.
A parallelepiped is a geometric figure, all 6 faces of which are parallelograms.
Depending on the type of these parallelograms, the following types of parallelepiped are distinguished:
- direct;
- inclined;
- rectangular.
A right parallelepiped is a quadrangular prism whose edges make an angle of 90° with the plane of the base.
A rectangular parallelepiped is a quadrangular prism, all of whose faces are rectangles. Cube is a variety quadrangular prism, in which all faces and edges are equal to each other.
The features of a figure predetermine its properties. These include the following 4 statements:
It is simple to remember all the given properties, they are easy to understand and are logically derived based on the type and features geometric body. However, simple statements can be incredibly helpful in deciding typical tasks Unified State Exam and will save the time needed to pass the test.
Parallelepiped formulas
To find answers to the problem, it is not enough to know only the properties of the figure. You may also need some formulas for finding the area and volume of a geometric body.
The area of the bases is found in the same way as the corresponding indicator of a parallelogram or rectangle. You can choose the base of the parallelogram yourself. As a rule, when solving problems it is easier to work with a prism, the base of which is a rectangle.
The formula for finding the lateral surface of a parallelepiped may also be needed in test tasks.
Examples of solving typical Unified State Exam tasks
Task 1.
Given: a rectangular parallelepiped with dimensions of 3, 4 and 12 cm.
Necessary find the length of one of the main diagonals of the figure.
Solution: Any solution geometric problem should begin with the construction of a correct and clear drawing, on which “given” and the desired value will be indicated. The picture below shows an example correct design task conditions.
Having examined the drawing made and remembering all the properties of the geometric body, we come to the only the right way solutions. Applying the 4th property of a parallelepiped, we obtain the following expression:
After simple calculations we get the expression b2=169, therefore b=13. The answer to the task has been found; you need to spend no more than 5 minutes searching for it and drawing it.
In geometry key concepts are plane, point, straight line and angle. Using these terms, you can describe any geometric figure. Polyhedra are usually described in terms of more simple figures, which lie in the same plane, such as a circle, triangle, square, rectangle, etc. In this article we will look at what a parallelepiped is, describe the types of parallelepipeds, its properties, what elements it consists of, and also give basic formulas to calculate the area and volume for each type of parallelepiped.
Definition
Parallelepiped in three-dimensional space is a prism, all sides of which are parallelograms. Accordingly, it can only have three pairs of parallel parallelograms or six faces.
To visualize a parallelepiped, imagine an ordinary standard brick. Brick - good example a rectangular parallelepiped that even a child can imagine. Other examples include multi-storey panel houses, cabinets, storage containers food products appropriate form, etc.
Varieties of figure
There are only two types of parallelepipeds:
- Rectangular, all side faces which are at an angle of 90° to the base and are rectangles.
- Sloping, the side edges of which are located under certain angle to the base.
What elements can this figure be divided into?
- As in any other geometric figure, in a parallelepiped any 2 faces with a common edge are called adjacent, and those that do not have it are parallel (based on the property of a parallelogram, which has pairs of parallel opposite sides).
- The vertices of a parallelepiped that do not lie on the same face are called opposite.
- The segment connecting such vertices is a diagonal.
- The lengths of the three edges of a cuboid that meet at one vertex are its dimensions (namely, its length, width and height).
Shape Properties
- It is always built symmetrically with respect to the middle of the diagonal.
- The intersection point of all diagonals divides each diagonal into two equal segments.
- Opposite faces are equal in length and lie on parallel lines.
- If you add the squares of all dimensions of a parallelepiped, the resulting value will be equal to the square of the length of the diagonal.
Calculation formulas
The formulas for each particular case of a parallelepiped will be different.
For an arbitrary parallelepiped it is true that its volume is equal to absolute value triple dot product vectors of three sides emanating from one vertex. However, there is no formula for calculating the volume of an arbitrary parallelepiped.
For a rectangular parallelepiped the following formulas apply:
- V=a*b*c;
- Sb=2*c*(a+b);
- Sp=2*(a*b+b*c+a*c).
- V - volume of the figure;
- Sb - lateral surface area;
- Sp - area full surface;
- a - length;
- b - width;
- c - height.
Another special case of a parallelepiped in which all sides are squares is a cube. If any of the sides of the square is designated by the letter a, then the following formulas can be used for the surface area and volume of this figure:
- S=6*a*2;
- V=3*a.
- S- area of the figure,
- V is the volume of the figure,
- a is the length of the figure's face.
The last type of parallelepiped we are considering is a straight parallelepiped. What is the difference between a right parallelepiped and a cuboid, you ask. The fact is that the base of a rectangular parallelepiped can be any parallelogram, but the base of a straight parallelepiped can only be a rectangle. If we denote the perimeter of the base, equal to the sum of the lengths of all sides, as Po, and denote the height by the letter h, we have the right to use the following formulas to calculate the volume and areas of the full and lateral surfaces.
A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will be parallelograms.
Each parallelepiped can be considered as a prism with three in various ways, since every two opposite faces(in Fig. 5, faces ABCD and A"B"C"D", or ABA"B" and CDC"D", or VSV"C" and ADA"D").
The body in question has twelve edges, four equal and parallel to each other.
Theorem 3
. The diagonals of a parallelepiped intersect at one point, coinciding with the middle of each of them.
The parallelepiped ABCDA"B"C"D" (Fig. 5) has four diagonals AC", BD", CA", DB". We must prove that the midpoints of any two of them, for example AC and BD", coincide. This follows from the fact that the figure ABC"D", having equal and parallel sides AB and C"D" are a parallelogram.
Definition 7
. A right parallelepiped is a parallelepiped that is also a straight prism, i.e. a parallelepiped side ribs which are perpendicular to the plane of the base.
Definition 8
. A rectangular parallelepiped is a right parallelepiped whose base is a rectangle. In this case, all its faces will be rectangles.
A rectangular parallelepiped is a right prism, no matter which of its faces we take as the base, since each of its edges is perpendicular to the edges emerging from the same vertex, and will, therefore, be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, parallelepiped can be viewed as a straight prism in only one way.
Definition 9
. Lengths of three The edges of a rectangular parallelepiped, of which no two are parallel to each other (for example, three edges emerging from one vertex), are called its dimensions. Two rectangular parallelepipeds having correspondingly equal dimensions are obviously equal to each other.
Definition 10
.A cube is a rectangular parallelepiped, all three dimensions of which are equal to each other, so that all its faces are squares. Two cubes whose edges are equal are equal. 
Definition 11
. Inclined parallelepiped, in which all edges are equal to each other and the angles of all faces are equal or complementary, is called a rhombohedron.
All faces of a rhombohedron are equal rhombuses. (Some crystals have a rhombohedron shape, having great value, for example, Iceland spar crystals.) In a rhombohedron you can find a vertex (and even two opposite vertices) such that all angles adjacent to it are equal to each other.
Theorem 4
. The diagonals of a rectangular parallelepiped are equal to each other. Diagonal square equal to the sum squares of three dimensions.
In the rectangular parallelepiped ABCDA"B"C"D" (Fig. 6), the diagonals AC" and BD" are equal, since the quadrilateral ABC"D" is a rectangle (the straight line AB is perpendicular to the plane ECB"C", in which BC lies") .
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the theorem about the square of the hypotenuse. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC" 2 = AB 2 + AA" 2 + A" D" 2 = AB 2 + AA" 2 + AD 2.
