A right parallelepiped whose base is square. Rectangular parallelepiped

It will be useful for high school students to learn how to solve Unified State Examination problems to find the volume and other unknown parameters of a rectangular parallelepiped. The experience of previous years confirms the fact that such tasks are quite difficult for many graduates.

At the same time, high school students with any level of training should understand how to find the volume or area of ​​a rectangular parallelepiped. Only in this case will they be able to count on receiving competitive scores based on the results of passing the unified state exam in mathematics.

Key points to remember

• The parallelograms that make up a parallelepiped are its faces, their sides are its edges. The vertices of these figures are considered the vertices of the polyhedron itself.
• All diagonals of a rectangular parallelepiped are equal. Since this is a straight polyhedron, the side faces are rectangles.
• Since a parallelepiped is a prism with a parallelogram at its base, this figure has all the properties of a prism.
• The lateral edges of a rectangular parallelepiped are perpendicular to the base. Therefore, they are its heights.

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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 in three different ways, since every two opposite faces can be taken as bases (in Fig. 5, faces ABCD and A"B"C"D", or ABA"B" and CDC"D", or BCB "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", is a parallelogram.
Definition 7 . A right parallelepiped is a parallelepiped that is also a straight prism, that is, a parallelepiped whose side edges 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 right prism in only one way.
Definition 9 . The lengths of three edges of a rectangular parallelepiped, of which no two are parallel to each other (for example, three edges emerging from the same 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 . An 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 of great importance have a rhombohedron shape, for example, Iceland spar crystals.) In a rhombohedron you can find a vertex (and even two opposite vertices) such that all the angles adjacent to it are equal to each other.
Theorem 4 . The diagonals of a rectangular parallelepiped are equal to each other. The square of the diagonal is equal to the sum of the squares of the 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.

In geometry, the 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 simpler figures that 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 the basic formulas for calculating the area and volume for each type of parallelepiped.

Definition

A 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. A brick is a good example of a rectangular parallelepiped that even a child can imagine. Other examples include multi-storey panel houses, cabinets, food storage containers of appropriate shape, etc.

Varieties of figure

There are only two types of parallelepipeds:

1. Rectangular, all side faces of which are at an angle of 90° to the base and are rectangles.
2. Sloping, the side edges of which are located at a 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

1. It is always built symmetrically with respect to the middle of the diagonal.
2. The intersection point of all diagonals divides each diagonal into two equal segments.
3. Opposite faces are equal in length and lie on parallel lines.
4. 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 the absolute value of the triple scalar product of the 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 - total surface area;
• 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 total and lateral surfaces.

Translated from Greek, parallelogram means plane. A parallelepiped is a prism with a parallelogram at its base. There are five types of parallelogram: oblique, straight and cuboid. The cube and rhombohedron also belong to the parallelepiped and are its variety.

Before moving on to the basic concepts, let's give some definitions:

• The diagonal of a parallelepiped is a segment that unites the vertices of the parallelepiped that are opposite each other.
• If two faces have a common edge, then we can call them adjacent edges. If there is no common edge, then the faces are called opposite.
• Two vertices that do not lie on the same face are called opposite.

What properties does a parallelepiped have?

1. The faces of a parallelepiped lying on opposite sides are parallel to each other and equal to each other.
2. If you draw diagonals from one vertex to another, then the point of intersection of these diagonals will divide them in half.
3. The sides of the parallelepiped lying at the same angle to the base will be equal. In other words, the angles of the co-directed sides will be equal to each other.

What types of parallelepiped are there?

Now let's figure out what kind of parallelepipeds there are. As mentioned above, there are several types of this figure: straight, rectangular, inclined parallelepiped, as well as cube and rhombohedron. How do they differ from each other? It's all about the planes that form them and the angles they form.

Let's look in more detail at each of the listed types of parallelepiped.

• As is already clear from the name, an inclined parallelepiped has inclined faces, namely those faces that are not at an angle of 90 degrees in relation to the base.
• But for a right parallelepiped, the angle between the base and the edge is exactly ninety degrees. It is for this reason that this type of parallelepiped has such a name.
• If all the faces of the parallelepiped are identical squares, then this figure can be considered a cube.
• A rectangular parallelepiped received this name because of the planes that form it. If they are all rectangles (including the base), then this is a cuboid. This type of parallelepiped is not found very often. Translated from Greek, rhombohedron means face or base. This is the name given to a three-dimensional figure whose faces are rhombuses.

Basic formulas for a parallelepiped

The volume of a parallelepiped is equal to the product of the area of ​​the base and its height perpendicular to the base.

The area of ​​the lateral surface will be equal to the product of the perimeter of the base and the height.
Knowing the basic definitions and formulas, you can calculate the base area and volume. The base can be chosen at your discretion. However, as a rule, a rectangle is used as the base.