In this lesson, everyone will be able to study the topic " cuboid". 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 , then. 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 =
A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will parallelograms.
Each parallelepiped can be considered as a prism with three different ways, since every two opposite faces can be taken as bases (in Fig. 5, faces ABCD and A "B" C "D", or ABA "B" and CDC "D", or BC "C" and ADA "D") .
The body under consideration has twelve edges, four equal and parallel to each other.
Theorem 3
. The diagonals of the parallelepiped intersect at one point, coinciding with the midpoint of each of them.
The parallelepiped ABCDA"B"C"D" (Fig. 5) has four diagonals AC", BD", CA", DB". We must prove that the midpoints of any two of them, for example, AC and BD, coincide. This follows from the fact that the figure ABC "D", which has equal and parallel sides AB and C "D", is a parallelogram.
Definition 7
. A right parallelepiped is a parallelepiped that is also a straight prism, that is, a parallelepiped whose side edges are perpendicular to the base plane.
Definition 8
. A rectangular parallelepiped is a right parallelepiped whose base is a rectangle. In this case, all its faces will be rectangles.
A rectangular parallelepiped is a right prism, no matter which of its faces we take as the base, since each of its edges is perpendicular to the edges coming out of the same vertex with it, and will therefore be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, box can be viewed as a right prism in only one way.
Definition 9
. The lengths of three edges of a cuboid, of which no two are parallel to each other (for example, three edges coming out of the same vertex), are called its dimensions. Two |rectangular parallelepipeds having correspondingly equal dimensions are obviously equal to each other.
Definition 10
A cube is a rectangular parallelepiped, all three dimensions of which are equal to each other, so that all its faces are squares. Two cubes whose edges are equal are equal. 
Definition 11
. An inclined parallelepiped in which all edges are equal and the angles of all faces are equal or complementary is called a rhombohedron.
All faces of a rhombohedron are equal rhombuses. (The shape of a rhombohedron has some crystals of great importance, for example, crystals of Iceland spar.) In a rhombohedron, one can find such a vertex (and even two opposite vertices) that all angles adjacent to it are equal to each other.
Theorem 4
. The diagonals of a rectangular parallelepiped are equal to each other. The square of the diagonal is equal to the sum of the squares of three dimensions.
In a rectangular parallelepiped ABCDA "B" C "D" (Fig. 6), the diagonals AC "and BD" are equal, since the quadrilateral ABC "D" is a rectangle (line AB is perpendicular to the plane BC "C", in which lies BC ") .
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the hypotenuse square theorem. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC "2 \u003d AB 2 + AA" 2 + A "D" 2 \u003d AB 2 + AA "2 + AD 2.
Definition
polyhedron we will call a closed surface composed of polygons and bounding some part of the space.
The segments that are the sides of these polygons are called ribs polyhedron, and the polygons themselves - faces. The vertices of the polygons are called the vertices of the polyhedron.
We will consider only convex polyhedra (this is a polyhedron that is on one side of each plane containing its face).
The polygons that make up a polyhedron form its surface. The part of space 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 \(A_1B_1, \A_2B_2, ..., A_nB_n\) are parallel. Polyhedron formed by 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 (\(n\)-coal) prism.
The polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) are called the bases of the prism, parallelogram \(A_1B_1B_2A_2, \A_2B_2B_3A_3, ...\)– side faces, segments \(A_1B_1, \A_2B_2, \ ..., A_nB_n\)- side ribs.
Thus, the side edges of the prism are parallel and equal to each other.
Consider an example - a prism \(A_1A_2A_3A_4A_5B_1B_2B_3B_4B_5\), whose base is a convex pentagon.
Height A prism is a perpendicular from any point on one base to the plane of another base.
If the side edges are not perpendicular to the base, then such a prism is called oblique(Fig. 1), otherwise - straight. For 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 right prism, then the prism is called correct.
Definition: concept of volume
The volume unit is a unit cube (cube with dimensions \(1\times1\times1\) units\(^3\) , where unit is some unit of measure).
We can say that the volume of a polyhedron is the amount of space that this polyhedron limits. Otherwise: it is a value whose numerical value indicates 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 value.
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: box
Parallelepiped It is a prism whose base is a parallelogram.
All faces of the parallelepiped (their \(6\) : \(4\) side faces and \(2\) bases) are parallelograms, and the opposite faces (parallel to each other) are equal parallelograms (Fig. 2).
Diagonal of the box is a segment connecting two vertices of a parallelepiped that do not lie in the same face (their \(8\) : \(AC_1, \A_1C, \BD_1, \B_1D\) etc.).
cuboid is a right parallelepiped with a rectangle at its base.
Because is a right parallelepiped, then the side faces are rectangles. So, in general, all the faces of a rectangular parallelepiped are rectangles.
All diagonals of a cuboid are equal (this follows from the equality of triangles \(\triangle ACC_1=\triangle AA_1C=\triangle BDD_1=\triangle BB_1D\) etc.).
Comment
Thus, the parallelepiped has all the properties of a prism.
Theorem
The area of the lateral surface of a rectangular parallelepiped is equal to \
Square full surface rectangular parallelepiped is equal to \
Theorem
The volume of a cuboid is equal to the product of three of its edges coming out of one vertex (three dimensions of a cuboid): \
Proof
Because for a rectangular parallelepiped, the lateral edges are perpendicular to the base, then they are also its heights, that is, \(h=AA_1=c\) the base is a rectangle \(S_(\text(main))=AB\cdot AD=ab\). This is where the formula comes from.
Theorem
The diagonal \(d\) of a cuboid is searched for by the formula (where \(a,b,c\) are the dimensions of the cuboid)\
Proof
Consider 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 line in this plane, i.e. \(BB_1\perp BD\) . So \(\triangle BB_1D\) 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 sides of which 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 \(V_(\text(cube))=a^3\) .
2. The cube diagonal is searched by the formula \(d=a\sqrt3\) .
3. Total surface area of a cube \(S_(\text(full cube iterations))=6a^2\).