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 are additional properties that are derived from the definition of a cuboid.
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. This means that all the lateral faces of a rectangular parallelepiped are rectangles.
3. All dihedral angles of a rectangular parallelepiped are right.
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:
Consider the 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, 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 =
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, 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 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).
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:
- 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. A cube is a type of 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 figure below shows an example of the correct execution of task conditions.
Having examined the drawing made and remembering all the properties of the geometric body, we come to the only correct method of solution. 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, 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:
- Rectangular, all side faces of which are at an angle of 90° to the base and are rectangles.
- 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
- 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 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.
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