Parallelepiped, cube. Detailed theory with examples

Students often ask indignantly: “How will this be useful to me in life?” On any topic of each subject. The topic about the volume of a parallelepiped is no exception. And this is where you can just say: “It will come in handy.”

How, for example, can you find out whether a package will fit in a postal box? Of course, you can choose the right one by trial and error. What if this is not possible? Then calculations will come to the rescue. Knowing the capacity of the box, you can calculate the volume of the parcel (at least approximately) and answer the question posed.

Parallelepiped and its types

If we literally translate its name from ancient Greek, it turns out that it is a figure consisting of parallel planes. There are the following equivalent definitions of a parallelepiped:

• a prism with a base in the form of a parallelogram;
• a polyhedron, each face of which is a parallelogram.

Its types are distinguished depending on what figure lies at its base and how the lateral ribs are directed. In general, we talk about inclined parallelepiped, whose base and all faces are parallelograms. If the side faces of the previous view become rectangles, then it will need to be called direct. And rectangular and the base also has 90º angles.

Moreover, in geometry they try to depict the latter in such a way that it is noticeable that all the edges are parallel. Here, by the way, is the main difference between mathematicians and artists. It is important for the latter to convey the body in compliance with the law of perspective. And in this case, the parallelism of the ribs is completely invisible.

About the introduced notations

In the formulas below, the notations indicated in the table are valid.

Formulas for an inclined parallelepiped

First and second for areas:

The third is to calculate the volume of a parallelepiped:

Since the base is a parallelogram, to calculate its area you will need to use the appropriate expressions.

Formulas for a rectangular parallelepiped

Similar to the first point - two formulas for areas:

And one more for volume:

First task

Condition. Given a rectangular parallelepiped, the volume of which needs to be found. The diagonal is known - 18 cm - and the fact that it forms angles of 30 and 45 degrees with the plane of the side face and the side edge, respectively.

Solution. To answer the problem question, you will need to know all the sides in three right triangles. They will give the necessary values ​​of the edges by which you need to calculate the volume.

First you need to figure out where the 30º angle is. To do this, you need to draw a diagonal of the side face from the same vertex from where the main diagonal of the parallelogram was drawn. The angle between them will be what is needed.

The first triangle that will give one of the values ​​of the sides of the base will be the following. It contains the required side and two drawn diagonals. It's rectangular. Now you need to use the ratio of the opposite leg (side of the base) and the hypotenuse (diagonal). It is equal to the sine of 30º. That is, the unknown side of the base will be determined as the diagonal multiplied by the sine of 30º or ½. Let it be designated by the letter “a”.

The second will be a triangle containing a known diagonal and an edge with which it forms 45º. It is also rectangular, and you can again use the ratio of the leg to the hypotenuse. In other words, side edge to diagonal. It is equal to the cosine of 45º. That is, “c” is calculated as the product of the diagonal and the cosine of 45º.

c = 18 * 1/√2 = 9 √2 (cm).

In the same triangle you need to find another leg. This is necessary in order to then calculate the third unknown - “in”. Let it be designated by the letter “x”. It can be easily calculated using the Pythagorean theorem:

x = √(18 2 - (9√2) 2) = 9√2 (cm).

Now we need to consider another right triangle. It contains the already known sides “c”, “x” and the one that needs to be counted, “b”:

in = √((9√2) 2 - 9 2 = 9 (cm).

All three quantities are known. You can use the formula for volume and calculate it:

V = 9 * 9 * 9√2 = 729√2 (cm 3).

Answer: the volume of the parallelepiped is 729√2 cm 3.

Second task

Condition. You need to find the volume of a parallelepiped. In it, the sides of the parallelogram that lies at the base are known to be 3 and 6 cm, as well as its acute angle - 45º. The side rib has an inclination to the base of 30º and is equal to 4 cm.

Solution. To answer the question of the problem, you need to take the formula that was written for the volume of an inclined parallelepiped. But both quantities are unknown in it.

The area of ​​the base, that is, of a parallelogram, will be determined by a formula in which you need to multiply the known sides and the sine of the acute angle between them.

S o = 3 * 6 sin 45º = 18 * (√2)/2 = 9 √2 (cm 2).

The second unknown quantity is height. It can be drawn from any of the four vertices above the base. It can be found from a right triangle in which the height is the leg and the side edge is the hypotenuse. In this case, an angle of 30º lies opposite the unknown height. This means that we can use the ratio of the leg to the hypotenuse.

n = 4 * sin 30º = 4 * 1/2 = 2.

Now all the values ​​are known and the volume can be calculated:

V = 9 √2 * 2 = 18 √2 (cm 3).

Answer: the volume is 18 √2 cm 3.

Third task

Condition. Find the volume of a parallelepiped if it is known that it is straight. The sides of its base form a parallelogram and are equal to 2 and 3 cm. The acute angle between them is 60º. The smaller diagonal of the parallelepiped is equal to the larger diagonal of the base.

Solution. In order to find out the volume of a parallelepiped, we use the formula with the base area and height. Both quantities are unknown, but they are easy to calculate. The first one is height.

Since the smaller diagonal of the parallelepiped coincides in size with the larger base, they can be designated by the same letter d. The largest angle of a parallelogram is 120º, since it forms 180º with the acute one. Let the second diagonal of the base be designated by the letter “x”. Now for the two diagonals of the base we can write the cosine theorems:

d 2 = a 2 + b 2 - 2av cos 120º,

x 2 = a 2 + b 2 - 2ab cos 60º.

It makes no sense to find values ​​without squares, since later they will be raised to the second power again. After substituting the data, we get:

d 2 = 2 2 + 3 2 - 2 * 2 * 3 cos 120º = 4 + 9 + 12 * ½ = 19,

x 2 = a 2 + b 2 - 2ab cos 60º = 4 + 9 - 12 * ½ = 7.

Now the height, which is also the side edge of the parallelepiped, will turn out to be a leg in the triangle. The hypotenuse will be the known diagonal of the body, and the second leg will be “x”. We can write the Pythagorean Theorem:

n 2 = d 2 - x 2 = 19 - 7 = 12.

Hence: n = √12 = 2√3 (cm).

Now the second unknown quantity is the area of ​​the base. It can be calculated using the formula mentioned in the second problem.

S o = 2 * 3 sin 60º = 6 * √3/2 = 3√3 (cm 2).

Combining everything into the volume formula, we get:

V = 3√3 * 2√3 = 18 (cm 3).

Answer: V = 18 cm 3.

Fourth task

Condition. It is required to find out the volume of a parallelepiped that meets the following conditions: the base is a square with a side of 5 cm; the side faces are rhombuses; one of the vertices located above the base is equidistant from all the vertices lying at the base.

Solution. First you need to deal with the condition. There are no questions with the first point about the square. The second, about rhombuses, makes it clear that the parallelepiped is inclined. Moreover, all its edges are equal to 5 cm, since the sides of the rhombus are the same. And from the third it becomes clear that the three diagonals drawn from it are equal. These are two that lie on the side faces, and the last one is inside the parallelepiped. And these diagonals are equal to the edge, that is, they also have a length of 5 cm.

To determine the volume, you will need a formula written for an inclined parallelepiped. There are again no known quantities in it. However, the area of ​​the base is easy to calculate because it is a square.

S o = 5 2 = 25 (cm 2).

The situation with height is a little more complicated. It will be like this in three figures: a parallelepiped, a quadrangular pyramid and an isosceles triangle. This last circumstance should be taken advantage of.

Since it is the height, it is a leg in a right triangle. The hypotenuse in it will be a known edge, and the second leg is equal to half the diagonal of the square (the height is also the median). And the diagonal of the base is easy to find:

d = √(2 * 5 2) = 5√2 (cm).

The height will need to be calculated as the difference between the second power of the edge and the square of half the diagonal and then remember to take the square root:

n = √ (5 2 - (5/2 * √2) 2) = √(25 - 25/2) = √(25/2) = 2.5 √2 (cm).

V = 25 * 2.5 √2 = 62.5 √2 (cm 3).

Answer: 62.5 √2 (cm 3).

Definition

Polyhedron we will call a closed surface composed of polygons and bounding a certain part of space.

The segments that are the sides of these polygons are called ribs polyhedron, and the polygons themselves are edges. The vertices of polygons are called polyhedron vertices.

We will consider only convex polyhedra (this is a polyhedron that is located on one side of each plane containing its face).

The polygons that make up a polyhedron form its surface. The part of space that is bounded by a given polyhedron is called its interior.

Definition: prism

Consider two equal polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) located in parallel planes so that the segments \(A_1B_1, \A_2B_2, ..., A_nB_n\) parallel. A polyhedron formed by the polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) , as well as parallelograms \(A_1B_1B_2A_2, \A_2B_2B_3A_3, ...\), is called (\(n\)-gonal) prism.

Polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) are called prism bases, parallelograms \(A_1B_1B_2A_2, \A_2B_2B_3A_3, ...\)– side faces, segments \(A_1B_1, \ A_2B_2, \ ..., A_nB_n\)- lateral ribs.
Thus, the lateral edges of the prism are parallel and equal to each other.

Let's look at an example - a prism \(A_1A_2A_3A_4A_5B_1B_2B_3B_4B_5\), at the base of which lies a convex pentagon.

Height prisms are a perpendicular dropped from any point of one base to the plane of another base.

If the side edges are not perpendicular to the base, then such a prism is called inclined(Fig. 1), otherwise – straight. In a straight prism, the side edges are heights, and the side faces are equal rectangles.

If a regular polygon lies at the base of a straight prism, then the prism is called correct.

Definition: concept of volume

The unit of volume measurement is a unit cube (a cube measuring \(1\times1\times1\) units\(^3\), where unit is a certain unit of measurement).

We can say that the volume of a polyhedron is the amount of space that this polyhedron limits. Otherwise: this is a quantity whose numerical value shows how many times a unit cube and its parts fit into a given polyhedron.

Volume has the same properties as area:

1. The volumes of equal figures are equal.

2. If a polyhedron is composed of several non-intersecting polyhedra, then its volume is equal to the sum of the volumes of these polyhedra.

3. Volume is a non-negative quantity.

4. Volume is measured in cm\(^3\) (cubic centimeters), m\(^3\) (cubic meters), etc.

Theorem

1. The area of ​​the lateral surface of the prism is equal to the product of the perimeter of the base and the height of the prism.
The lateral surface area is the sum of the areas of the lateral faces of the prism.

2. The volume of the prism is equal to the product of the base area and the height of the prism: \

Definition: parallelepiped

Parallelepiped is a prism with a parallelogram at its base.

All faces of the parallelepiped (there are \(6\) : \(4\) side faces and \(2\) bases) are parallelograms, and the opposite faces (parallel to each other) are equal parallelograms (Fig. 2).

Diagonal of a parallelepiped is a segment connecting two vertices of a parallelepiped that do not lie on the same face (there are \(8\) of them: \(AC_1,\A_1C,\BD_1,\B_1D\) etc.).

Rectangular parallelepiped is a right parallelepiped with a rectangle at its base.
Because Since this is a right parallelepiped, the side faces are rectangles. This means that in general all the faces of a rectangular parallelepiped are rectangles.

All diagonals of a rectangular parallelepiped are equal (this follows from the equality of triangles \(\triangle ACC_1=\triangle AA_1C=\triangle BDD_1=\triangle BB_1D\) etc.).

Comment

Thus, a parallelepiped has all the properties of a prism.

Theorem

The lateral surface area of ​​a rectangular parallelepiped is \

The total surface area of ​​a rectangular parallelepiped is \

Theorem

The volume of a cuboid is equal to the product of its three edges emerging from one vertex (three dimensions of the cuboid): \

Proof

Because In a rectangular parallelepiped, the lateral edges are perpendicular to the base, then they are also its heights, that is, \(h=AA_1=c\) Because the base is a rectangle, then \(S_(\text(main))=AB\cdot AD=ab\). This is where this formula comes from.

Theorem

The diagonal \(d\) of a rectangular parallelepiped is found using the formula (where \(a,b,c\) are the dimensions of the parallelepiped) \

Proof

Let's look at Fig. 3. Because the base is a rectangle, then \(\triangle ABD\) is rectangular, therefore, by the Pythagorean theorem \(BD^2=AB^2+AD^2=a^2+b^2\) .

Because all lateral edges are perpendicular to the bases, then \(BB_1\perp (ABC) \Rightarrow BB_1\) perpendicular to any straight line in this plane, i.e. \(BB_1\perp BD\) . This means that \(\triangle BB_1D\) is rectangular. Then, by the Pythagorean theorem \(B_1D=BB_1^2+BD^2=a^2+b^2+c^2\), thd.

Definition: cube

Cube is a rectangular parallelepiped, all of whose faces are equal squares.

Thus, the three dimensions are equal to each other: \(a=b=c\) . So the following are true

Theorems

1. The volume of a cube with edge \(a\) is equal to \(V_(\text(cube))=a^3\) .

2. The diagonal of the cube is found using the formula \(d=a\sqrt3\) .

3. Total surface area of ​​a cube \(S_(\text(full cube))=6a^2\).

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 is the 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.

A parallelepiped is a quadrangular prism with parallelograms at its base. The height of a parallelepiped is the distance between the planes of its bases. In the figure, the height is shown by the segment . There are two types of parallelepipeds: straight and inclined. As a rule, a math tutor first gives the appropriate definitions for a prism and then transfers them to a parallelepiped. We will do the same.

Let me remind you that a prism is called straight if its side edges are perpendicular to the bases; if there is no perpendicularity, the prism is called inclined. This terminology is also inherited by the parallelepiped. A right parallelepiped is nothing more than a type of straight prism, the side edge of which coincides with the height. Definitions of such concepts as face, edge and vertex, which are common to the entire family of polyhedra, are preserved. The concept of opposite faces appears. A parallelepiped has 3 pairs of opposite faces, 8 vertices and 12 edges.

The diagonal of a parallelepiped (the diagonal of a prism) is a segment connecting two vertices of a polyhedron and not lying on any of its faces.

Diagonal section - a section of a parallelepiped passing through its diagonal and the diagonal of its base.

Properties of an inclined parallelepiped:
1) All its faces are parallelograms, and the opposite faces are equal parallelograms.
2)The diagonals of a parallelepiped intersect at one point and bisect at this point.
3)Each parallelepiped consists of six triangular pyramids of equal volume. To show them to the student, the math tutor must cut off half of the paralleleped with its diagonal section and divide it separately into 3 pyramids. Their bases must lie on different faces of the original parallelepiped. A mathematics tutor will find application of this property in analytical geometry. It is used to derive the volume of a pyramid through a mixed product of vectors.

Formulas for the volume of a parallelepiped:
1) , where is the area of ​​the base, h is the height.
2) The volume of a parallelepiped is equal to the product of the cross-sectional area and the lateral edge.
Math tutor: As you know, the formula is common to all prisms and if the tutor has already proven it, there is no point in repeating the same thing for a parallelepiped. However, when working with an average-level student (the formula is not useful to a weak student), it is advisable for the teacher to act exactly the opposite. Leave the prism alone and carry out a careful proof for the parallelepiped.
3) , where is the volume of one of the six triangular pyramids that make up the parallelepiped.
4) If , then

The area of ​​the lateral surface of a parallelepiped is the sum of the areas of all its faces:
The total surface of a parallelepiped is the sum of the areas of all its faces, that is, the area + two areas of the base: .

About the work of a tutor with an inclined parallelepiped:
A math tutor does not often work on problems involving an inclined parallelepiped. The likelihood of them appearing on the Unified State Exam is quite low, and the didactics are indecently poor. A more or less decent problem on the volume of an inclined parallelepiped raises serious problems associated with determining the location of point H - the base of its height. In this case, the math tutor can be advised to cut the parallelepiped to one of its six pyramids (which are discussed in property No. 3), try to find its volume and multiply it by 6.

If the side edge of a parallelepiped has equal angles with the sides of the base, then H lies on the bisector of angle A of the base ABCD. And if, for example, ABCD is a rhombus, then

Math tutor tasks:
1) The faces of a parallelepiped are equal to each other with a side of 2 cm and an acute angle. Find the volume of the parallelepiped.
2) In an inclined parallelepiped, the side edge is 5 cm. The section perpendicular to it is a quadrilateral with mutually perpendicular diagonals having lengths of 6 cm and 8 cm. Calculate the volume of the parallelepiped.
3) In an inclined parallelepiped it is known that , and in ABCD the base is a rhombus with a side of 2 cm and an angle . Determine the volume of the parallelepiped.

Mathematics tutor, Alexander Kolpakov

Theorem. In any parallelepiped, opposite faces are equal and parallel.

Thus, the faces (Fig.) BB 1 C 1 C and AA 1 D 1 D are parallel, because two intersecting lines BB 1 and B 1 C 1 of one face are parallel to two intersecting lines AA 1 and A 1 D 1 of the other. These faces are equal, since B 1 C 1 =A 1 D 1, B 1 B=A 1 A (as opposite sides of parallelograms) and ∠BB 1 C 1 = ∠AA 1 D 1.

Theorem. In any parallelepiped, all four diagonals intersect at one point and are bisected at it.

Let's take (Fig.) some two diagonals in the parallelepiped, for example, AC 1 and DB 1, and draw straight lines AB 1 and DC 1.

Since the edges AD and B 1 C 1 are respectively equal and parallel to the edge BC, then they are equal and parallel to each other.

As a result, the figure ADC 1 B 1 is a parallelogram in which C 1 A and DB 1 are diagonals, and in a parallelogram the diagonals intersect in half.

This proof can be repeated for every two diagonals.

Therefore, diagonal AC 1 intersects BD 1 in half, diagonal BD 1 intersects A 1 C in half.

Thus, all diagonals intersect in half and, therefore, at one point.

Theorem. In a rectangular parallelepiped, the square of any diagonal is equal to the sum of the squares of its three dimensions.

Let (Fig.) AC 1 be some diagonal of a rectangular parallelepiped.

Drawing AC, we get two triangles: AC 1 C and ACB. Both of them are rectangular:

the first because the parallelepiped is straight, and therefore edge CC 1 is perpendicular to the base,

the second because the parallelepiped is rectangular, which means that there is a rectangle at its base.

From these triangles we find:

AC 2 1 = AC 2 + CC 2 1 and AC 2 = AB 2 + BC 2

Therefore, AC 2 1 = AB 2 + BC 2 + CC 2 1 = AB 2 + AD 2 + AA 2 1

Consequence. In a rectangular parallelepiped all diagonals are equal.