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 lateral ribs 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 a-priory),
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 in the following way: ∠A 1 ABD.
Let's take point A on edge AB. AA 1 - perpendicular to edge AB in the plane АВВ-1, AD perpendicular to edge AB in ABC plane. So, ∠A 1 AD - linear angle 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.
Square diagonal of a cuboid equal to the sum 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 - 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 =
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 only consider 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
Let's consider two equal polygon\(A_1A_2A_3...A_n\) 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), in otherwise – straight. In a straight prism, the side edges are heights, and the side faces are equal rectangles.
If the base of a straight prism lies regular polygon, 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 the quantity numeric value which shows how many times a unit cube and its parts fit into a given polyhedron.
Volume has the same properties as area:
1. Volumes 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. Prism volume equal to the product base area per prism height: \
Definition: parallelepiped
Parallelepiped is a prism with a parallelogram at its base.
All faces of a parallelepiped (there are \(6\) of them: \(4\) side faces and \(2\) bases) are parallelograms, and the opposite faces ( parallel friend friend) 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 \(S_(\text(full cube))=6a^2\).
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 given properties, they are easy to understand and are logically derived based on the type and features geometric body. However, simple statements can be incredibly helpful in deciding typical tasks Unified State Exam 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 geometric problem should 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.
Instructions
Method 2. Let's assume that the rectangular parallelepiped is a cube. A cube is a rectangular parallelepiped, each face is represented by a square. Therefore, all its sides are equal. Then to calculate the length of its diagonal it will be expressed as follows:
Sources:
- rectangle diagonal formula
Parallelepiped - special case a prism in which all six faces are parallelograms or rectangles. Parallelepiped with rectangular edges also called rectangular. A parallelepiped has four intersecting diagonals. If three edges a, b, c are given, you can find all the diagonals of a rectangular parallelepiped by performing additional constructions.
Instructions
Find the diagonal of the parallelepiped m. To do this, find the unknown hypotenuse in a, n, m: m² = n² + a². Substitute known values, then calculate the square root. The result obtained will be the first diagonal of the parallelepiped m.
In the same way, draw sequentially all the other three diagonals of the parallelepiped. Also, for each of them, perform additional construction of diagonals of adjacent faces. Considering the formed right triangles and using the Pythagorean theorem, find the values of the remaining diagonals.
Video on the topic
Sources:
- finding a parallelepiped
The hypotenuse is the side opposite right angle. Legs are the sides of a triangle adjacent to a right angle. Applied to triangles ABC and ACD: AB and BC, AD and DC–, AC is the common hypotenuse for both triangles (the desired diagonal). Therefore, AC = square AB + square BC or AC b = square AD + square DC. Substitute the side lengths rectangle into the above formula and calculate the length of the hypotenuse (diagonal rectangle).
For example, the sides rectangle ABCD are equal to the following values: AB = 5 cm and BC = 7 cm. The square of the diagonal AC of a given rectangle according to the Pythagorean theorem: AC squared = square AB + square BC = 52+72 = 25 + 49 = 74 sq.cm. Use a calculator to calculate the value square root 74. You should get 8.6 cm (rounded value). Please note that according to one of the properties rectangle, its diagonals are equal. So the length of the second diagonal BD rectangle ABCD is equal to the length of diagonal AC. For the above example, this value
a, towards the base of the PP;
with its height.
- what do we need to know, what data do we have?
- what properties does a rectangular parallelepiped have?
- does the Pythagorean Theorem apply here? How?
- Is there enough data to apply the Pythagorean theorem, or are some other calculations needed?
Diagonal square, a square parallelepiped (see properties of a square parallelepiped) is equal to the sum of the squares of three times it different sides(width, height, thickness), and accordingly the diagonal of a square parallelepiped is equal to the root of this sum.
I remember the school curriculum in geometry, we can say this: the diagonal of a parallelepiped is equal to the square root obtained from the sum of its three sides (they are designated by small letters a, b, c).
The length of the diagonal of a rectangular parallelepiped is equal to the square root of the sum of the squares of its sides.
As far as I know since school curriculum, class 9 if I’m not mistaken, and if memory serves, then the diagonal of a rectangular parallelepiped is equal to the square root of the sum of the squares of all three sides.
the square of the diagonal is equal to the sum of the squares of the width, height and length, based on this formula we get the answer, the diagonal is equal to the square root of the sum of its three different dimensions, they designate letters nсz abc
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:
Another definition can be given by considering the Cartesian rectangular system coordinates:
The PP diagonal is the radius vector of any point in space, given by coordinates x, y and z in Cartesian system coordinates 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 coordinate axes. The projections coincide with the vertices of this parallelepiped.
A rectangular parallelepiped is a type of polyhedron consisting of 6 faces, at the base of which is a rectangle. A diagonal is a line segment that connects opposite vertices parallelogram.
The formula for finding the length of a diagonal is that the square of the diagonal is equal to the sum of the squares of the three dimensions of the parallelogram.
I found a good diagram-table on the Internet with a complete listing of everything that is in the parallelepiped. There is a formula to find the diagonal, which is denoted by d.
There is an image of the edge, vertex and other important things for the parallelepiped.
If the length, height and width (a,b,c) of a rectangular parallelepiped are known, then the formula for calculating the diagonal will look like this:
Typically, teachers do not offer their students a bare formula, but make efforts so that they can derive it on their own by asking leading questions:
Usually, after answering the questions posed, students can easily derive this formula on their own.
The diagonals of a rectangular parallelepiped are equal. As well as the diagonals of its opposite faces. The length of the diagonal can be calculated by knowing the length of the edges of the parallelogram emanating from one vertex. This length is equal to the square root of the sum of the squares of the lengths of its edges.
A cuboid is one of the so-called polyhedra, which consists of 6 faces, each of which is a rectangle. A diagonal is a segment that connects opposite vertices of a parallelogram. If the length, width and height of a rectangular parallelepiped are taken to be a, b, c, respectively, then the formula for its diagonal (D) will look like this: D^2=a^2+b^2+c^2.
Diagonal of a rectangular parallelepiped is a segment connecting its opposite vertices. So we have cuboid with diagonal d and sides a, b, c. One of the properties of a parallelepiped is that the square diagonal length d is equal to the sum of the squares of its three dimensions a, b, c. Hence the conclusion is that diagonal length can be easily calculated using the following formula:
