Rectangular parallelepiped - Hypermarket of knowledge. Volume of a parallelepiped: basic formulas and examples of problems

In this lesson, everyone will be able to study the topic "Rectangular box". At the beginning of the lesson, we will repeat what an arbitrary and straight parallelepipeds are, recall the properties of their opposite faces and diagonals of the parallelepiped. Then we will consider what a cuboid is and discuss its main 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 ABB 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 side ribs AA 1, BB 1, DD 1, SS 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. Opposite faces of a parallelepiped are parallel and equal.

(the figures are equal, that is, they can be combined by overlay)

For example:

ABCD \u003d A 1 B 1 C 1 D 1 (equal parallelograms by definition),

AA 1 B 1 B \u003d 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 \u003d 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 the parallelepiped intersect at one point and bisect that 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 the parallelepiped intersect and bisect the intersection point.

3. There are three quadruples of equal and parallel edges of the 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, SS 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 the line AA 1 is perpendicular to the lines AD and AB, which lie in the plane of the base. And, therefore, rectangles lie in the side faces. And the bases are arbitrary parallelograms. Denote, ∠BAD = φ, the angle φ can be any.

Rice. 3 Right box

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

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

The parallelepiped АВСДА 1 В 1 С 1 D 1 is rectangular (Fig. 4) if:

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

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

Rice. 4 Cuboid

A rectangular box has all the properties of an arbitrary box. But there are additional properties that are derived from the definition of a cuboid.

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

1. In a cuboid, 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 everything side faces cuboid - rectangles.

3. All dihedral angles of a cuboid are right angles.

Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, i.e., the dihedral angle between the planes ABB 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 considered dihedral angle can also be denoted as follows: ∠А 1 АВD.

Take point A on edge AB. AA 1 is perpendicular to the edge AB in the plane ABB-1, AD is perpendicular to the edge AB in the plane ABC. Hence, ∠A 1 AD is the linear angle of the given dihedral angle. ∠A 1 AD \u003d 90 °, which means that the dihedral angle at the edge AB is 90 °.

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

It is proved similarly that any dihedral angles of a rectangular parallelepiped are right.

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

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

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

Prove: .

Rice. 5 Cuboid

Proof:

The line CC 1 is perpendicular to the plane ABC, and hence to the line AC. So triangle CC 1 A is a right triangle. According to the Pythagorean theorem:

Consider right triangle ABC. According to the Pythagorean theorem:

But BC and AD are opposite sides of the rectangle. So BC = AD. Then:

Because , A , That. Since CC 1 = AA 1, then what was required to be proved.

The diagonals of a rectangular parallelepiped are equal.

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

Often students indignantly ask: “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 here it is just possible to say: "It will come in handy."

How, for example, to find out if a parcel will fit in a mailbox? Of course, you can choose the right one by trial and error. What if there is no such possibility? 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.

Parallelepiped and its types

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

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

Its types are distinguished depending on which figure lies at its base and how the side ribs are directed. In general, one speaks of oblique 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 already direct. And at 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, the main difference between mathematicians and artists is observed. 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 edges is completely invisible.

About the introduced notation

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

Formulas for an oblique box

The first and second for areas:

The third one is for calculating the volume of the box:

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

Formulas for a cuboid

Similarly to the first paragraph - two formulas for areas:

And one more for volume:

First task

Condition. Given a rectangular parallelepiped whose volume is 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 question of the problem, you need to find out all the sides in three right triangles. They will give the necessary edge values for 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 which the main diagonal of the parallelogram was drawn. The angle between them will be what you need.

The first triangle, which will give one of the sides of the base, will be the following. It contains the desired side and two diagonals drawn. It is rectangular. Now you need to use the ratio of the opposite leg (base side) 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 marked with 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, the side edge to the 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 marked with the letter "x". It is easy to calculate using the Pythagorean theorem:

x \u003d √ (18 2 - (9 √ 2) 2) \u003d 9 √ 2 (cm).

Now we need to consider another right triangle. It already contains famous parties"s", "x" and the one that needs to be counted, "in":

c \u003d √ ((9 √ 2) 2 - 9 2 \u003d 9 (cm).

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

V \u003d 9 * 9 * 9√2 \u003d 729√2 (cm 3).

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

Second task

Condition. Find the volume of the parallelepiped. It knows the sides of the parallelogram that lies at the base, 3 and 6 cm, as well as its acute angle - 45º. The lateral 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 oblique parallelepiped. But both quantities are unknown in it.

The area of \u200b\u200bthe base, that is, the parallelogram, will be determined by the formula in which you need to multiply the known sides and the sine of the acute angle between them.

S o \u003d 3 * 6 sin 45º \u003d 18 * (√2) / 2 \u003d 9 √2 (cm 2).

The second unknown is the 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. So, you can use the ratio of the leg to the hypotenuse.

n \u003d 4 * sin 30º \u003d 4 * 1/2 \u003d 2.

Now all values are known and you can calculate the volume:

V \u003d 9 √2 * 2 \u003d 18 √2 (cm 3).

Answer: the volume is 18 √2 cm 3 .

Third task

Condition. Find the volume of the parallelepiped if it is known to be a straight line. The sides of its base form a parallelogram and are equal to 2 and 3 cm. Sharp corner between them 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 is the same size as the larger base, they can be denoted by the same letter d. The largest angle of a parallelogram is 120º, since it forms 180º with an acute one. Let the second diagonal of the base be denoted by the letter "x". Now, for the two diagonals of the base, cosine theorems can be written:

d 2 \u003d a 2 + in 2 - 2av cos 120º,

x 2 \u003d a 2 + in 2 - 2ab cos 60º.

Finding values without squares does not make sense, since then they will be raised to the second power again. After substituting the data, it turns out:

d 2 \u003d 2 2 + 3 2 - 2 * 2 * 3 cos 120º \u003d 4 + 9 + 12 * ½ \u003d 19,

x 2 \u003d a 2 + in 2 - 2ab cos 60º \u003d 4 + 9 - 12 * ½ \u003d 7.

Now the height, which is also the side edge of the parallelepiped, will be the leg in the triangle. The hypotenuse will be known diagonal body, and the second leg - "x". You can write the Pythagorean Theorem:

n 2 \u003d d 2 - x 2 \u003d 19 - 7 \u003d 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 \u003d 2 * 3 sin 60º \u003d 6 * √3/2 \u003d 3 √3 (cm 2).

Combining everything into a volume formula, we get:

V = 3√3 * 2√3 = 18 (cm 3).

Answer: V \u003d 18 cm 3.

The 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; side faces are rhombuses; one of the vertices 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 paragraph 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. Again, there are no known quantities in it. However, the area of the base is easy to calculate because it is a square.

S o \u003d 5 2 \u003d 25 (cm 2).

A little more difficult is the case with height. It will be such in three figures: a parallelepiped, a quadrangular pyramid and an isosceles triangle. The last circumstance should be used.

Since it is a 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 of the second degree of the edge and the square of half the diagonal and do not forget to extract the square root:

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

V \u003d 25 * 2.5 √2 \u003d 62.5 √2 (cm 3).

Answer: 62.5 √2 (cm 3).

cuboid

A cuboid is a right cuboid in which 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 differ in color, have a lot of additional details, but if these subtleties are discarded, then we can say that, for example, a cabinet, a 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 do you see rectangular boxes? Look at the book, because it is just such a shape! A brick, a matchbox, a wooden block have the same shape, and even right now you are inside a rectangular cuboid, because the classroom is the brightest interpretation of this geometric figure.

Exercise: What examples of a 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;

Second, the 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.

Task: Look at the picture and say which parallelepipeds are shown in it. How is a cuboid different from a cube?

Properties of a cuboid

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 its four diagonals are absolutely identical.

Thirdly, if all three parameters of the 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 faces? If there are, then show them in the picture. 2. From what geometric shapes are the faces of a rectangular parallelepiped? 3. What is the arrangement of equal faces in relation to each other? 4. Name the number of pairs of equal faces of this figure. 5. Find the edges in the cuboid that indicate its length, width, height. How many did you count?

Task

To beautifully arrange 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 package beautiful, Tanya decided to cover it with beautiful paper, the cost of which is 3 hryvnias per 1 dm2. How much money do you need to spend on wrapping paper?

Did you know that the famous illusionist David Blaine, as part of an experiment, spent 44 days in a glass box suspended over the Thames. These 44 days he did not eat, but only drank water. In his voluntary penitentiary, David took only writing instruments, a pillow and mattress, and handkerchiefs.

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 ABB 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.