The sum of the squares of three dimensions is equal. Definitions of a parallelepiped. Basic properties and formulas

In this lesson, everyone will be able to study the topic “Rectangular parallelepiped”. At the beginning of the lesson, we will repeat what arbitrary and straight parallelepipeds are, remember the properties of their opposite faces and diagonals of the parallelepiped. Then we'll look at what a cuboid is and discuss its basic properties.

Topic: Perpendicularity of lines and planes

Lesson: Cuboid

A surface composed of two equal parallelograms ABCD and A 1 B 1 C 1 D 1 and four parallelograms ABV 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1 is called parallelepiped(Fig. 1).

Rice. 1 Parallelepiped

That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (bases), they lie in parallel planes so that the side edges AA 1, BB 1, DD 1, CC 1 are parallel. Thus, a surface composed of parallelograms is called parallelepiped.

Thus, the surface of a parallelepiped is the sum of all the parallelograms that make up the parallelepiped.

1. The opposite faces of a parallelepiped are parallel and equal.

(the shapes are equal, that is, they can be combined by overlapping)

For example:

ABCD = A 1 B 1 C 1 D 1 (equal parallelograms by definition),

AA 1 B 1 B = DD 1 C 1 C (since AA 1 B 1 B and DD 1 C 1 C are opposite faces of the parallelepiped),

AA 1 D 1 D = BB 1 C 1 C (since AA 1 D 1 D and BB 1 C 1 C are opposite faces of the parallelepiped).

2. The diagonals of a parallelepiped intersect at one point and are bisected by this point.

The diagonals of the parallelepiped AC 1, B 1 D, A 1 C, D 1 B intersect at one point O, and each diagonal is divided in half by this point (Fig. 2).

Rice. 2 The diagonals of a parallelepiped intersect and are divided in half by the intersection point.

3. There are three quadruples of equal and parallel edges of a parallelepiped: 1 - AB, A 1 B 1, D 1 C 1, DC, 2 - AD, A 1 D 1, B 1 C 1, BC, 3 - AA 1, BB 1, CC 1, DD 1.

Definition. A parallelepiped is called straight if its lateral edges are perpendicular to the bases.

Let the side edge AA 1 be perpendicular to the base (Fig. 3). This means that straight line AA 1 is perpendicular to straight lines AD and AB, which lie in the plane of the base. This means that the side faces contain rectangles. And the bases contain arbitrary parallelograms. Let us denote ∠BAD = φ, the angle φ can be any.

Rice. 3 Right parallelepiped

So, a right parallelepiped is a parallelepiped in which the side edges are perpendicular to the bases of the parallelepiped.

Definition. The parallelepiped is called rectangular, if its lateral edges are perpendicular to the base. The bases are rectangles.

The parallelepiped ABCDA 1 B 1 C 1 D 1 is rectangular (Fig. 4), if:

1. AA 1 ⊥ ABCD (lateral edge perpendicular to the plane of the base, that is, a straight parallelepiped).

2. ∠BAD = 90°, i.e. the base is a rectangle.

Rice. 4 Rectangular parallelepiped

A rectangular parallelepiped has all the properties of an arbitrary parallelepiped. But there is additional properties, which are derived from the definition rectangular parallelepiped.

So, cuboid is a parallelepiped whose side edges are perpendicular to the base. The base of a cuboid is a rectangle.

1. In a rectangular parallelepiped, all six faces are rectangles.

ABCD and A 1 B 1 C 1 D 1 are rectangles by definition.

2. Lateral ribs are perpendicular to the base. So that's it side faces rectangular parallelepiped - rectangles.

3. All dihedral angles rectangular parallelepiped straight lines.

Let us consider, for example, the dihedral angle of a rectangular parallelepiped with edge AB, i.e., the dihedral angle between planes ABC 1 and ABC.

AB is an edge, point A 1 lies in one plane - in the plane ABB 1, and point D in the other - in the plane A 1 B 1 C 1 D 1. Then the dihedral angle under consideration can also be denoted as follows: ∠A 1 ABD.

Let's take point A on edge AB. AA 1 is perpendicular to edge AB in the plane АВВ-1, AD is perpendicular to edge AB in the plane ABC. This means that ∠A 1 AD is the linear angle of a given dihedral angle. ∠A 1 AD = 90°, which means that the dihedral angle at edge AB is 90°.

∠(ABB 1, ABC) = ∠(AB) = ∠A 1 ABD= ∠A 1 AD = 90°.

Similarly, it is proved that any dihedral angles of a rectangular parallelepiped are right.

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Note. The lengths of the three edges emanating from one vertex of a cuboid are the measurements of the cuboid. They are sometimes called length, width, height.

Given: ABCDA 1 B 1 C 1 D 1 - rectangular parallelepiped (Fig. 5).

Prove: .

Rice. 5 Rectangular parallelepiped

Proof:

Straight line CC 1 is perpendicular to plane ABC, and therefore to straight line AC. This means that the triangle CC 1 A is right-angled. According to the Pythagorean theorem:

Let's consider right triangle ABC. According to the Pythagorean theorem:

But BC and AD - opposite sides rectangle. So BC = AD. Then:

Because , A , That. Since CC 1 = AA 1, this is what needed to be proven.

The diagonals of a rectangular parallelepiped are equal.

Let us denote the dimensions of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =

In geometry, the following types of parallelepipeds are distinguished: rectangular parallelepiped (the faces of the parallelepiped are rectangles); a right parallelepiped (its side faces act as rectangles); inclined parallelepiped (its side faces act as perpendiculars); a cube is a parallelepiped with absolutely identical dimensions, and the faces of the cube are squares. Parallelepipeds can be either inclined or straight.

The main elements of a parallelepiped are that two faces of the presented geometric figure that do not have a common edge are opposite, and those that do are adjacent. The vertices of the parallelepiped, which do not belong to the same face, act opposite to each other. A parallelepiped has a dimension - these are three edges that have a common vertex.

The line segment that connects opposite vertices is called a diagonal. The four diagonals of a parallelepiped, intersecting at one point, are simultaneously divided in half.

In order to determine the diagonal of a parallelepiped, you need to determine the sides and edges, which are known from the conditions of the problem. With three known ribs A , IN , WITH draw a diagonal in the parallelepiped. According to the property of a parallelepiped, which says that all its angles are right, the diagonal is determined. Construct a diagonal from one of the faces of the parallelepiped. The diagonals must be drawn in such a way that the diagonal of the face, the desired diagonal of the parallelepiped and the known edge create a triangle. After a triangle is formed, find the length of this diagonal. The diagonal in the other resulting triangle acts as the hypotenuse, so it can be found using the Pythagorean theorem, which must be taken under the square root. This way we know the value of the second diagonal. In order to find the first diagonal of the parallelepiped in the formed right triangle, it is also necessary to find the unknown hypotenuse (using the Pythagorean theorem). Using the same example, sequentially find the remaining three diagonals existing in the parallelepiped, performing additional constructions of diagonals that form right triangles and solve using the Pythagorean theorem.

A rectangular parallelepiped (PP) is nothing more than a prism, the base of which is a rectangle. For a PP, all diagonals are equal, which means that any of its diagonals is calculated using the formula:

a, c - sides of the base of the PP;

c is its height.

Another definition can be given by considering the Cartesian rectangular coordinate system:

The PP diagonal is the radius vector of any point in space specified by x, y and z coordinates in the Cartesian coordinate system. This radius vector to the point is drawn from the origin. And the coordinates of the point will be the projections of the radius vector (diagonals of the PP) onto the coordinate axes. The projections coincide with the vertices of this parallelepiped.

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 case 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 we need to use the relation opposite side(base sides) and hypotenuse (diagonals). 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 already contains known parties“c”, “x” and the one that needs to be counted, “v”:

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, which 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 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).

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).

A parallelepiped is geometric figure, all 6 faces of which are parallelograms.

Depending on the type of these parallelograms there are the following types parallelepiped:

straight;
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 above properties, they are easy to understand and are derived logically based on the type and characteristics of the geometric body. However, simple statements can be incredibly useful when solving typical USE tasks 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

Exercise 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 to a geometric problem must 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.

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 \

Square full surface rectangular parallelepiped is equal to \

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 \(S_(\text(full cube))=6a^2\).