A straight parallelepiped whose bases are called rectangular. Geometric figures
In this lesson, everyone will be able to study the topic "Rectangular parallelepiped". At the beginning of the lesson, we will repeat what an arbitrary and straight parallelepiped are, recall the properties of their opposite faces and diagonals of a parallelepiped. Then we will consider what a rectangular parallelepiped is and discuss its main properties.
Topic: Perpendicularity of lines and planes
Lesson: Rectangular Parallelepiped
A surface made up 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 (base), 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 parallelograms that make up the parallelepiped.
1. Opposite faces of the box are parallel and equal.
(the shapes are equal, that is, they can be combined by overlay)
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 the parallelepiped intersect at one point and are halved 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 by this point in half (Fig. 2).
Rice. 2 The diagonals of the parallelepiped intersect and are halved 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 lateral 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 rectangles lie in the side faces. Arbitrary parallelograms lie at the bases. Denote, ∠BAD = φ, the angle φ can be any.
Rice. 3 Straight parallelepiped
So, a straight 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 ribs are perpendicular to the base. The bases are rectangles.
Parallelepiped ABCDA 1 B 1 C 1 D 1 - 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 °, that is, there is a rectangle at the base.
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 rectangular parallelepiped.
So, rectangular parallelepiped is a parallelepiped with side edges perpendicular to the base. The base of the rectangular parallelepiped is a rectangle.
1. In a rectangular parallelepiped, all six faces are rectangles.
ABCD and A 1 B 1 C 1 D 1 - rectangles by definition.
2. Side ribs are perpendicular to the base... This means that all the side faces of a rectangular parallelepiped are rectangles.
3. All dihedral corners of a rectangular parallelepiped are straight.
Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, that is, the dihedral angle between the planes ABB 1 and ABC.
AB is an edge, point A 1 lies in one plane - in plane ABB 1, and point D in another - in plane A 1 B 1 C 1 D 1. Then the considered dihedral angle can also be denoted as follows: ∠A 1 ABD.
Take point A on edge AB. AA 1 - perpendicular to the edge AB in the plane ABB-1, AD perpendicular to the edge AB in the plane ABC. Hence, ∠А 1 АD is the linear angle of the given dihedral angle. ∠А 1 АD = 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 in a similar way that any dihedral angles of a rectangular parallelepiped are straight.
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 outgoing from one vertex of the rectangle are the dimensions of the rectangular parallelepiped. 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 CC 1 is perpendicular to the plane ABC, and hence to the straight line AC. This means that triangle CC 1 A is rectangular. By the Pythagorean theorem:
Consider a right-angled triangle ABC. By the Pythagorean theorem:
But BC and AD are opposite sides of the rectangle. Hence, BC = AD. Then:
As , but , then. Since CC 1 = AA 1, then what was required to prove.
The diagonals of a rectangular parallelepiped are equal.
Let's designate the measurements of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =
In this lesson, everyone will be able to study the topic "Rectangular parallelepiped". At the beginning of the lesson, we will repeat what an arbitrary and straight parallelepiped are, recall the properties of their opposite faces and diagonals of a parallelepiped. Then we will consider what a rectangular parallelepiped is and discuss its main properties.
Topic: Perpendicularity of lines and planes
Lesson: Rectangular Parallelepiped
A surface made up 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 (base), 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 parallelograms that make up the parallelepiped.
1. Opposite faces of the box are parallel and equal.
(the shapes are equal, that is, they can be combined by overlay)
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 the parallelepiped intersect at one point and are halved 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 by this point in half (Fig. 2).
Rice. 2 The diagonals of the parallelepiped intersect and are halved 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 lateral 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 rectangles lie in the side faces. Arbitrary parallelograms lie at the bases. Denote, ∠BAD = φ, the angle φ can be any.
Rice. 3 Straight parallelepiped
So, a straight 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 ribs are perpendicular to the base. The bases are rectangles.
Parallelepiped ABCDA 1 B 1 C 1 D 1 - 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 °, that is, there is a rectangle at the base.
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 rectangular parallelepiped.
So, rectangular parallelepiped is a parallelepiped with side edges perpendicular to the base. The base of the rectangular parallelepiped is a rectangle.
1. In a rectangular parallelepiped, all six faces are rectangles.
ABCD and A 1 B 1 C 1 D 1 - rectangles by definition.
2. Side ribs are perpendicular to the base... This means that all the side faces of a rectangular parallelepiped are rectangles.
3. All dihedral corners of a rectangular parallelepiped are straight.
Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, that is, the dihedral angle between the planes ABB 1 and ABC.
AB is an edge, point A 1 lies in one plane - in plane ABB 1, and point D in another - in plane A 1 B 1 C 1 D 1. Then the considered dihedral angle can also be denoted as follows: ∠A 1 ABD.
Take point A on edge AB. AA 1 - perpendicular to the edge AB in the plane ABB-1, AD perpendicular to the edge AB in the plane ABC. Hence, ∠А 1 АD is the linear angle of the given dihedral angle. ∠А 1 АD = 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 in a similar way that any dihedral angles of a rectangular parallelepiped are straight.
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 outgoing from one vertex of the rectangle are the dimensions of the rectangular parallelepiped. 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 CC 1 is perpendicular to the plane ABC, and hence to the straight line AC. This means that triangle CC 1 A is rectangular. By the Pythagorean theorem:
Consider a right-angled triangle ABC. By the Pythagorean theorem:
But BC and AD are opposite sides of the rectangle. Hence, BC = AD. Then:
As , but , then. Since CC 1 = AA 1, then what was required to prove.
The diagonals of a rectangular parallelepiped are equal.
Let's designate the measurements of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =
TEXT CODE OF THE LESSON:
Consider these items:
Building bricks, dice, microwave oven. These objects are united by the form.
Surface consisting of two equal parallelograms ABCD and A1B1C1D1
and four parallelograms АА1В1В and ВВ1С1С, СС1D1D, АА1D1D is called a parallelepiped.
The parallelograms that make up the parallelepiped are called faces. Face A1B1C1D1. VV1S1S edge. Edge ABCD.
In this case, the faces ABCD and A1B1C1D1 are often called bases, and the rest of the faces are lateral.
The sides of parallelograms are called the edges of the parallelepiped. Rib A1B1. Rib CC1. Rib AD.
The CC1 edge does not belong to the bases, it is called the lateral edge.
The vertices of the parallelograms are called the vertices of the parallelepiped.
Vertex D1. Vershina V. Vershina S.
Vertices D1 and B
do not belong to the same face and are called opposite.
The box can be drawn in different ways.
A parallelepiped at the base of which a rhombus lies. In this case, the images of the faces are parallelograms.
A parallelepiped at the base of which a square lies. Invisible edges AA1, AB, AD are depicted by dashed lines.
A parallelepiped at the base of which a square lies
Box at the base, which is a rectangle or parallelogram
A box with all its faces as squares. More often it is called a cube.
All considered parallelepipeds have properties. Let us formulate and prove them.
Property 1. Opposite faces of a parallelepiped are parallel and equal.
Consider the parallelepiped ABCDA1B1C1D1 and prove, for example, the parallelism and equality of the faces BB1C1C and AA1D1D.
By the definition of a parallelepiped, the face ABCD is a parallelogram, so by the property of a parallelogram the edge BC is parallel to the edge AD.
The face ABB1A1 is also a parallelogram, which means the edges BB1 and AA1 are parallel.
This means that two intersecting straight lines BC and BB1 of one plane, respectively, are parallel to two straight lines AD and AA1, respectively, of another plane, which means that the planes ABB1A1 and BCC1D1 are parallel.
All faces of the parallelepiped are parallelogram and therefore BC = AD, BB1 = AA1.
In this case, the sides of the angles В1ВС and А1АD are correspondingly codirectional, which means they are equal.
Thus, the two adjacent sides and the angle between them of the parallelogram ABB1A1 are respectively equal to the two adjacent sides and the angle between them of the parallelogram BCC1D1, which means that these parallelograms are equal.
The parallelepiped also has the property of diagonals. The diagonal of a parallelepiped is a segment connecting non-adjacent vertices. In the drawing, the dashed line shows the diagonals B1D, BD1, A1C.
So, property 2. The diagonals of the parallelepiped intersect at one point and the intersection point is divided in half.
To prove the property, consider the quadrilateral BB1D1D. Its diagonals B1D, BD1 are the diagonals of the parallelepiped ABCDA1B1C1D1.
In the first property, we have already found out that the edge BB1 is parallel and equal to the edge AA1, but the edge AA1 is parallel and equal to the edge DD1. Consequently, the edges BB1 and DD1 are parallel and equal, which proves the quadrilateral BB1D1D-parallelogram. And in a parallelogram, by the property of the diagonal B1D, BD1 intersect at some point O and this point is divided in half.
The quadrilateral BC1D1A is also a parallelogram and its diagonals C1A intersect at one point and are divided by this point in half. The diagonals of the parallelogram C1A, BD1 are the diagonals of the parallelepiped, which means that the formulated property is proved.
To consolidate theoretical knowledge about a parallelepiped, consider a proof problem.
Points L, M, N, P are marked on the edges of the parallelepiped so that BL = CM = A1N = D1P. Prove that ALMDNB1C1P is a parallelepiped.
The face BB1A1A is a parallelogram, so the edge BB1 is equal to and parallel to the edge AA1, but by the condition of the segments BL and A1N, it means that the segments LB1 and NA are equal and parallel.
3) Consequently, the LB1NA quadrilateral is based on the parallelogram feature.
4) Since CC1D1D is a parallelogram, it means the edge CC1 is equal to and parallel to the edge D1D, and CM is equal to D1P by condition, it means that the segments MC1 and DP are equal and parallel
Therefore, the quadrilateral MC1PD is also a parallelogram.
5) The angles LB1N and MC1P are equal as angles with respectively parallel and equally directed sides.
6) We got that for parallelograms and MC1PD the corresponding sides are equal and the angles between them are equal, so the parallelograms are equal.
7) The segments are equal by condition, which means that BLMC is a parallelogram and the BC side is parallel to the LM side and is parallel to the B1C1 side.
8) Similarly, it follows from the parallelogram NA1D1P that side A1D1 is parallel to side NP and parallel to side AD.
9) Opposite faces ABB1A1 and DCC1D1 of the parallelepiped are parallel by property, and the segments of parallel straight lines between the parallel planes are equal, so the segments B1C1, LM, AD, NP are equal.
It was found that in the quadrangles ANPD, NB1C1P, LB1C1M, ALMD two sides are parallel and equal, so they are parallelograms. Then our surface ALMDNB1C1P consists of six parallelograms, two of which are equal, and by definition it is a parallelepiped.
In this lesson, we will give a definition of a parallelepiped, discuss its structure and its elements (parallelepiped diagonals, parallelepiped sides and their properties). And also consider the properties of the faces and diagonals of the parallelogram. Next, we will solve a typical problem of constructing a section in a parallelepiped.
Topic: Parallelism of lines and planes
Lesson: Parallelepiped. Box face and diagonal properties
In this lesson we will give a definition of a parallelepiped, discuss its structure, properties and its elements (sides, diagonals).
The parallelepiped is formed by two equal parallelograms ABCD and A 1 B 1 C 1 D 1, which are in parallel planes. Designation: ABCDA 1 B 1 C 1 D 1 or AD 1 (Fig. 1.).
2. Festival of pedagogical ideas "Open Lesson" ()
1. Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, revised and supplemented - M .: Mnemozina, 2008. - 288 p .: ill.
Tasks 10, 11, 12 p. 50
2. Construct a section of a rectangular parallelepiped ABCDA1B1C1D1 plane passing through the points:
a) A, C, B1
b) B1, D1 and the middle of the rib AA1.
3. The edge of the cube is equal to a. Construct a section of the cube with a plane passing through the midpoints of three edges extending from one vertex, and calculate its perimeter and area.
4. What shapes can be obtained as a result of intersection of a parallelepiped plane?
Definition
Polyhedron we will call a closed surface made up of polygons and bounding some part of the space.
The line segments that are the sides of these polygons are called ribs polyhedron, and the polygons themselves are facets... The vertices of the polygons are called the vertices of the polyhedron.
We will consider only convex polyhedra (this is a polyhedron that is located on one side of each plane containing its face).
The polygons of which the polyhedron is composed form its surface. The part of the space that a given polyhedron limits 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. Polytope 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 \) -gonal) prism.
Polygons \ (A_1A_2A_3 ... A_n \) and \ (B_1B_2B_3 ... B_n \) are called the bases of the prism, 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 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 \) with a convex pentagon at its base.
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 oblique(fig. 1), otherwise - straight... For a straight prism, the side edges are heights, and the side edges are equal rectangles.
If a regular polygon lies at the base of a straight prism, then the prism is called correct.
Definition: the concept of volume
The unit of measurement of volume is a unit cube (a cube of \ (1 \ times1 \ times1 \) units \ (^ 3 \), where unit is some unit of measurement).
We can say that the volume of a polyhedron is the amount of space that this polyhedron limits. Otherwise: it 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 disjoint 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 by the height of the prism: \
Definition: parallelepiped
Parallelepiped Is a prism with a parallelogram at its base.
All faces of a parallelepiped (their \ (6 \): \ (4 \) side faces and \ (2 \) bases) are parallelograms, and 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 (their \ (8 \): \ (AC_1, \ A_1C, \ BD_1, \ B_1D \) etc.).
Rectangular parallelepiped is a straight parallelepiped with a rectangle at its base.
Because it is a straight parallelepiped, the side faces are rectangles. Hence, 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 \
The total surface area of a rectangular parallelepiped is \
Theorem
The volume of a rectangular parallelepiped is equal to the product of its three edges extending from one vertex (three dimensions of a rectangular parallelepiped): \
Proof
Because for a rectangular parallelepiped, the side edges are perpendicular to the base, then they are also its heights, that is, \ (h = AA_1 = c \) Because there is a rectangle at the base, then \ (S _ (\ text (base)) = AB \ cdot AD = ab \)... Hence the given formula follows.
Theorem
The diagonal \ (d \) of a rectangular parallelepiped is found by the formula (where \ (a, b, c \) are the dimensions of the parallelepiped) \
Proof
Consider fig. 3. Because there is a rectangle at the base, 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 \). So, \ (\ triangle BB_1D \) is rectangular. Then by the Pythagorean theorem \ (B_1D = BB_1 ^ 2 + BD ^ 2 = a ^ 2 + b ^ 2 + c ^ 2 \), etc.
Definition: cube
Cube is a rectangular parallelepiped, all faces of which are equal squares.
Thus, the three dimensions are equal: \ (a = b = c \). Hence, the following are true.
Theorems
1. The volume of a cube with an edge \ (a \) is \ (V _ (\ text (cube)) = a ^ 3 \).
2. The diagonal of the cube is found by the formula \ (d = a \ sqrt3 \).
3. Total surface area of a cube \ (S _ (\ text (full cube)) = 6a ^ 2 \).
Loading...