With the help of this video tutorial, everyone will be able to independently get acquainted with the topic “The concept of a polyhedron. Prism. Prism surface area. During the lesson, the teacher will explain what these geometric figures, as a polyhedron and prisms, will give the appropriate definitions and explain their essence on concrete examples.
With the help of this lesson, everyone will be able to independently get acquainted with the topic “The concept of a polyhedron. Prism. Prism surface area.
Definition. A surface composed of polygons and bounding a certain geometric body will be called a polyhedral surface or a polyhedron.
Consider the following examples of polyhedra:
1. Tetrahedron ABCD is a surface made up of four triangles: ABC, adb, bdc and ADC(Fig. 1).
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2. Parallelepiped ABCDA 1 B 1 C 1 D 1 is a surface composed of six parallelograms (Fig. 2).
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The main elements of a polyhedron are faces, edges, vertices.
The faces are the polygons that make up the polyhedron.
Edges are sides of faces.
The vertices are the ends of the edges.
Consider a tetrahedron ABCD(Fig. 1). Let us indicate its main elements.
Facets: triangles ABC, ADB, BDC, ADC.
Ribs: AB, AC, BC, DC, AD, BD.
Peaks: A, B, C, D.
Consider a box ABCDA 1 B 1 C 1 D 1(Fig. 2).
Facets: parallelograms AA 1 D 1 D, D 1 DCC 1, BB 1 C 1 C, AA 1 B 1 B, ABCD, A 1 B 1 C 1 D 1 .
Ribs: AA 1 , BB 1 , SS 1 , DD 1 , AD, A 1 D 1 , B 1 C 1 , BC, AB, A 1 B 1 , D 1 C 1 , DC.
Peaks: A, B, C, D, A 1 ,B 1 ,C 1 ,D 1 .
An important special case of a polyhedron is a prism.
ABSA 1 IN 1 WITH 1(Fig. 3).
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Equal Triangles ABC and A 1 B 1 C 1 are located in parallel planes α and β so that the edges AA 1 , BB 1 , SS 1 are parallel.
That is ABSA 1 IN 1 WITH 1- triangular prism, if:
1) Triangles ABC and A 1 B 1 C 1 are equal.
2) Triangles ABC and A 1 B 1 C 1 located in parallel planes α and β: ABC║A 1 B 1 C (α ║ β).
3) Ribs AA 1 , BB 1 , SS 1 are parallel.
ABC and A 1 B 1 C 1- the base of the prism.
AA 1 , BB 1 , SS 1- side ribs of the prism.
If from an arbitrary point H 1 one plane (for example, β) drop the perpendicular HH 1 onto the plane α, then this perpendicular is called the height of the prism.
Definition. If the lateral edges are perpendicular to the bases, then the prism is called straight, otherwise it is called oblique.
Consider a triangular prism ABSA 1 IN 1 WITH 1(Fig. 4). This prism is straight. That is, its side edges are perpendicular to the bases.
For example, rib AA 1 perpendicular to the plane ABC. Edge AA 1 is the height of this prism.
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Note that the side face AA 1 V 1 V perpendicular to the bases ABC and A 1 B 1 C 1, since it passes through the perpendicular AA 1 to the foundations.
Now consider an inclined prism ABSA 1 IN 1 WITH 1(Fig. 5). Here the lateral edge is not perpendicular to the plane of the base. If we drop from the point A 1 perpendicular A 1 H on the ABC, then this perpendicular will be the height of the prism. Note that the segment AN is the projection of the segment AA 1 to the plane ABC.
Then the angle between the line AA 1 and plane ABC is the angle between the line AA 1 and her AN projection onto a plane, that is, the angle A 1 AH.
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Consider a quadrangular prism ABCDA 1 B 1 C 1 D 1(Fig. 6). Let's see how it turns out.
1) Quadrilateral ABCD equal to a quadrilateral A 1 B 1 C 1 D 1: ABCD = A 1 B 1 C 1 D 1.
2) Quadrangles ABCD and A 1 B 1 C 1 D 1 ABC║A 1 B 1 C (α ║ β).
3) Quadrangles ABCD and A 1 B 1 C 1 D 1 arranged so that the lateral ribs are parallel, that is: AA 1 ║BB 1 ║SS 1 ║DD 1.
Definition. The diagonal of a prism is a segment that connects two vertices of a prism that do not belong to the same face.
For instance, AC 1- diagonal of a quadrangular prism ABCDA 1 B 1 C 1 D 1.
Definition. If the side edge AA 1 perpendicular to the plane of the base, then such a prism is called a straight line.
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A special case of a quadrangular prism is the known parallelepiped. Parallelepiped ABCDA 1 B 1 C 1 D 1 shown in fig. 7.
Let's see how it works:
1) Equal figures lie in the bases. In this case - equal parallelograms ABCD and A 1 B 1 C 1 D 1: ABCD = A 1 B 1 C 1 D 1.
2) Parallelograms ABCD and A 1 B 1 C 1 D 1 lie in parallel planes α and β: ABC║A 1 B 1 C 1 (α ║ β).
3) Parallelograms ABCD and A 1 B 1 C 1 D 1 arranged in such a way that the side ribs are parallel to each other: AA 1 ║BB 1 ║SS 1 ║DD 1.
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From a point A 1 drop the perpendicular AN to the plane ABC. Section A 1 H is the height.
Let's take a look at how it works hexagonal prism(Fig. 8).
1) Equal hexagons lie at the base ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1: ABCDEF= A 1 B 1 C 1 D 1 E 1 F 1.
2) Planes of hexagons ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1 parallel, that is, the bases lie in parallel planes: ABC║A 1 B 1 C (α ║ β).
3) Hexagons ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1 arranged so that all side edges are parallel to each other: AA 1 ║BB 1 …║FF 1.
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Definition. If any side edge is perpendicular to the plane of the base, then such a hexagonal prism is called a straight line.
Definition. A right prism is called regular if its bases are regular polygons.
Consider a regular triangular prism ABSA 1 IN 1 WITH 1.
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triangular prism ABSA 1 IN 1 WITH 1- correct, this means that regular triangles lie at the bases, that is, all sides of these triangles are equal. Also, this prism is straight. This means that the side edge is perpendicular to the plane of the base. And this means that everything side faces are equal rectangles.
So if a triangular prism ABSA 1 IN 1 WITH 1 is correct, then:
1) The side edge is perpendicular to the plane of the base, that is, it is the height: AA 1 ⊥ ABC.
2) The base is a regular triangle: ∆ ABC- right.
Definition. The total surface area of a prism is the sum of the areas of all its faces. Denoted S full.
Definition. The area of the lateral surface is the sum of the areas of all lateral faces. Denoted S side.
The prism has two bases. Then the total surface area of the prism is:
S full \u003d S side + 2S main.
The area of the lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism.
The proof will be carried out on the example of a triangular prism.
Given: ABSA 1 IN 1 WITH 1- direct prism, i.e. AA 1 ⊥ ABC.
AA 1 = h.
Prove: S side \u003d R main ∙ h.
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Proof.
triangular prism ABSA 1 IN 1 WITH 1- straight, so AA 1 B 1 B, AA 1 C 1 C, BB 1 C 1 C - rectangles.
Find the area of the lateral surface as the sum of the areas of the rectangles AA 1 B 1 B, AA 1 C 1 C, BB 1 C 1 C:
S side \u003d AB ∙ h + BC ∙ h + CA ∙ h \u003d (AB + BC + CA) ∙ h \u003d P main ∙ h.
We get S side \u003d R main ∙ h, Q.E.D.
We got acquainted with polyhedrons, a prism, its varieties. We proved the theorem on the lateral surface of a prism. In the next lesson, we will solve problems on a prism.
- Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and supplemented - M .: Mnemosyne, 2008. - 288 p. : ill.
- Geometry. Grade 10-11: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
- Geometry. Grade 10: Textbook for general educational institutions with in-depth and profile study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M. : Bustard, 008. - 233 p. :ill.
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Definition. Prism- this is a polyhedron, all the vertices of which are located in two parallel planes, and in the same two planes there are two faces of the prism, which are equal polygons with respectively parallel sides, and all edges that do not lie in these planes are parallel.
Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).
All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).
All side faces form side surface prisms .
All side faces of a prism are parallelograms .
Edges that do not lie at the bases are called lateral edges of the prism ( AA 1, B.B. 1, CC 1, DD 1, EE 1).
Prism Diagonal a segment is called, the ends of which are two vertices of the prism that do not lie on one of its faces (AD 1).
The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .
Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in the bypass order, the vertices of one base are indicated, and then, in the same order, the vertices of the other; the ends of each side edge are designated by the same letters, only the vertices lying in one base are indicated by letters without an index, and in the other - with an index)
The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1, the base is a pentagon, so the prism is called pentagonal prism. But since such a prism has 7 faces, then it heptahedron(2 faces are the bases of the prism, 5 faces are parallelograms, are its side faces)
Among the straight prisms stands out private view: regular prisms.
A straight prism is called correct, if its bases are regular polygons.
Parallelepiped
Parallelepiped- This is a quadrangular prism, at the base of which lies a parallelogram (oblique parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.cuboid - a right parallelepiped whose base is a rectangle.
Properties and theorems:
Some properties of a parallelepiped are similar known properties parallelogram. A rectangular parallelepiped having equal measurements, are called cube
.A cube has all faces equal squares. The square of a diagonal is equal to the sum of the squares of its three dimensions 
,
where d is the diagonal of the square;
a - side of the square.
The idea of a prism is given by:
- various architectural structures;
- Kids toys;
- packing boxes;
- designer items etc.
Total and lateral surface area of the prism
Total surface area of the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its side faces. the bases of the prism are equal polygons, then their areas are equal. SoS full \u003d S side + 2S main,
where S full- total surface area, S side- side surface area, S main- base area
The area of the lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism.
S side\u003d P main * h,
where S side is the area of the lateral surface of a straight prism,
P main - the perimeter of the base of a straight prism,
h is the height of a straight prism, equal to side rib.
Prism Volume
The volume of a prism is equal to the product of the area of the base and the height.