Download the pyramid presentation. Presentation on the theme "pyramids". What determines the type of pyramid

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Multidisciplinary gymnasium №79 OPEN LESSON "GEOMETRIC PYRAMID AND ITS PROJECTION" Teacher: Volkova Lydia Nikolaevna 2009 Almaty city

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The presentation was prepared

Dasieva Roza, Naboko Mikhail, Ibragimova Karina, Egizbaeva Ainura, Asanova Elvira, Uskenbaeva Madia.

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About the word pyramid.

Pyramid. The word "pyramid" was introduced into geometry by the Greeks, who are believed to have borrowed it from the Egyptians, who created the most famous pyramids in the world. Another theory derives this term from the Greek word "pyros" (rye) - it is believed that the Greeks baked bread that had the shape of a pyramid.

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What is a pyramid?

A pyramid is a polyhedron, whose base is a polygon, the side faces are triangles with a common vertex.

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Pyramids: Full Truncated Incorrect Correct

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What determines the type of pyramid?

The type of pyramid depends on the polygon that lies at the base.

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Pyramid projection

Triangular pyramid

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A pyramid is a polyhedron, one of the faces of which is an arbitrary n-gon A1A2… An, and the other faces are triangles with a common vertex. This n - gon A1A2 ... An is called the base of the pyramid. Triangular faces are called side faces. The common top of all side faces is called the top of the pyramid. The segments connecting the top of the pyramid with the top of the base are called side edges. The union of the side faces of the pyramid is called its side surface. The perpendicular drawn from the top of the pyramid to the plane of the base is called the height of the pyramid. O S C D B A ABCD - base S - top SO - height

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A pyramid is called regular if its base is a regular polygon, and the segment connecting the top of the pyramid with the center of the base is its height. The height of the side face of a regular pyramid, drawn from its top, is called the apothem of this pyramid. All apothems are equal to each other. If there is an n-gon at the base of the pyramid, then the pyramid is called n-gonal. A triangular pyramid is called a tetrahedron. The tetrahedron is defined by four vertices; the tetrahedron faces are four triangles. A tetrahedron is called regular if all of its edges are equal.

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Pyramid properties

· All lateral ribs are equal. · All side faces are equal isosceles triangles. · All dihedral angles at the base are equal. · All planar apex angles are equal. · All planar angles at the base are equal · The apothems of the lateral faces are the same in length. · A sphere can be inscribed in any regular pyramid.

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Pyramid area

The total surface area of \u200b\u200ba pyramid is the sum of the areas of all its faces. S total \u003d S side + S main The area of \u200b\u200bthe lateral surface of the pyramid is the sum of the areas of its lateral faces. The area of \u200b\u200bthe lateral surface of a regular pyramid: S side.pov. \u003d 1/2 * (Psn * m), where m is the apothem, P is the perimeter of the base

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The volume of the pyramid

The volume of the pyramid is V \u003d (1/3) * Sbase * h, where S is the area of \u200b\u200bthe base, h is the height of the pyramid.

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Truncated pyramid

A truncated pyramid is a part of a pyramid lying between the base and a section parallel to the base. The truncated pyramid is a special case of the pyramid. Definition.

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The base of the truncated pyramid is the base of the original pyramid and the polygon obtained by intersecting it by the plane (A1A2… An and B1B2… Bn). Segments A1B1, A2B2,…, AnBn are called lateral edges of a truncated pyramid. The perpendicular drawn from some point of one base to the plane of the other base is called the height of the truncated pyramid. The side faces of the truncated pyramid are trapeziums. A truncated pyramid with bases A1A2… An and B1B2… Bn is designated as A1A2… AnB1B2… Bn. A truncated pyramid is called regular if it is obtained by sectioning a regular pyramid with a plane parallel to the base. The bases of a regular truncated pyramid are regular polygons, and the side faces are isosceles trapezoids. The heights of these trapezoids are called apothems. A1 A2 A3 An B1 B2 Bn O

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Truncated pyramid properties.

1. The lateral edges and the height of the pyramid are divided by the cutting plane into proportional segments. 2. In the section, a polygon is obtained, similar to the polygon lying at the base. 3. The cross-sectional and base areas will be related to each other as the squares of their distances from the top of the pyramid.

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Surface area of \u200b\u200ba regular truncated pyramid: S \u003d (1/2) * m * (P + P1), where m is the apothem, P is the perimeter of the bases, P1 is the perimeter of the lateral surface. The area of \u200b\u200bthe lateral surface of a regular truncated pyramid is equal to the product of the half-sum of the perimeters of the bases by the apothem: Sside \u003d 1/2 * (Рв + Рн) * m, where m is the apothem, Рв, Рн - the perimeter of the upper and lower bases of the volumetric pyramid: V \u003d (1 / 3) * h * (S1 + √S1S2 + S2), where S1, S2 are the areas of the bases. Side face area: S side.gr. \u003d 1/2 * m * (g + g1), where m - apothem, g, g1 - base of the side face.

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Flat sections of the pyramid

The sections of the pyramid by planes passing through its apex are triangles. In particular, diagonal sections are triangles. These are sections by planes passing through two non-adjacent side edges of the pyramid. A C D S B E F A C D S B ∆SDB is the diagonal section of the SABCD pyramid.

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Construct a section of a quadrangular pyramid with a plane passing through line g and point E є square (SCD). K G H L M N F S B A C D E g Solution: 1. Draw the line CD, CD ∩ g ≡ F, F Є (SCD). 2. Draw the line FE, we obtain the points of intersection with the edges of the pyramid: SD ∩ FE ≡ H, SC ∩ FE ≡ G. 3. Construct the line AD. AD ∩ g ≡ K, K Є (SAD). 4. Through points K and H draw a straight line KH. KH∩SA≡L. 5. Construct the line AB, AB ∩ g ≡ M, M Є (SAB). 6. Through points M and L we construct ML ∩ SB ≡ N. 7. We connect points G, H, L, N. The section GHLM is constructed. Building a section.

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Pyramid presentation

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historical information about the pyramid
The Egyptian pyramids are one of the seven wonders of the world. What are pyramids? The tombs of the Egyptian pharaohs. The largest of them - the pyramids of Cheops, Khafre and Mikerin in El-Giza in ancient times were considered one of the Seven Wonders of the World. The largest of the three is the pyramid of Cheops (the architect Khemiun, 27th century BC). Its height was originally 147 m, and the length of the side of the base was 232 m.

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PYRAMID
A polyhedron made up of an n-gon AB… E and n-triangles is called a pyramid. The area of \u200b\u200bthe total surface of a pyramid is the sum of the areas of all its faces, and the area of \u200b\u200bthe lateral surface of the pyramid is the sum of the areas of its lateral faces.
S full \u003d S side + S main

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The polygon AB ... E is called the base, and the triangles are called the side faces of the pyramid. Point M is called the top of the pyramid, and the segments MA, ME,…, MB are its lateral edges.
pyramid

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Correct pyramid
A pyramid is called regular if its base is a regular polygon, and the segment PO connecting the top of the pyramid with the center of the base * is its height. PE is the apothem of the pyramid.
* The center of a regular polygon is the center of a circle inscribed in it (or circumscribed about it).
The height of the side face of a regular pyramid, drawn from its top, is called apothem.

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All the side edges of a regular pyramid are equal, and the side edges are isosceles triangles. Any lateral edge represents the hypotenuse of the right-angled triangle A₁PO, one leg of which is the height PO of the pyramid, and the other - the radius of the circle described near the base.
Correct pyramid

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THEOREM:
the lateral surface area of \u200b\u200ba regular pyramid is half the product of the base perimeter times the apothem. S full \u003d ⅟₂ Pbasis * d

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Truncated pyramid
A polyhedron whose faces are n-gons A 1 A 2 ... A n and B 1 B 2 ... B n (lower and upper base), located in parallel planes, and n quadrangles A 1 A 2 B 2 B 1, A 2 A 3 B 3 B 2,…, A n A 1 B 1 B n (side faces), is called a truncated pyramid. The segments A 1 B 1, A 2 B 2,…, A n B n are called the lateral edges of the truncated pyramid. The perpendicular CO drawn from some point of one base to the plane of the other base is called the height of the truncated pyramid.
P
A 2
A 3
A 1
A n
B n
B1
B 2
B 3
C
Sside \u003d ⅟₂ (P₁ + P₂) * d

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Slide captions:

Pyramid

Pyramid from the tomb

Great Pyramid of Cheops

Man made pyramid

Pyramids created by nature

Modern buildings

Again the pyramid

A C D E H B S Top Ribs Base O Height of pyramid Pyramid Height of side edge Side edge

S C B A Types of pyramids A M D B C Triangular pyramid Quadrangular pyramid Side surface

C B A S O M N K AB \u003d BC \u003d AC, ∆ABC -equilateral. Regular pyramid r R Apothem

PO (leg) - common; All side edges of a regular pyramid are equal. P A 2 A n A 1 PA 1 A 2… A n - regular pyramid O h R R OPA 1 \u003d OPA 2 \u003d… 2. OA 1 \u003d OA 2 \u003d… R (legs) So PA 1 \u003d PA 2 \u003d…

PA 2 A 3 \u003d… \u003d PA 1 A 2 \u003d All lateral faces of a regular pyramid are equal isosceles triangles. A 1 A 2 A 3 A 4 A 5 A n P PA 1 A 2 A 3… A n - regular pyramid PA 1 A n (on three sides) A \u200b\u200b1 A 2 \u003d A 2 A 3 \u003d A 3 A 4 \u003d. .; PA 1 \u003d PA 2 \u003d PA 3 \u003d…

The lateral surface area of \u200b\u200ba regular pyramid is equal to half the product of the base perimeter and the apothem A 1 A 2 A 3 A 4 A n P H S bp. \u003d SA 1 A 2 P + SA 2 A 3 P + SA 3 A 4 P \u003d… \u003d ½ A 1 A 2 · PH + ½A 2 A 3 · PH + + ½A 3 A 4 · PH… \u003d \u003d ½ PH · (A 1 A 2 + A 2 A 3 + A 3 A 4 +…) \u003d ½ P BASIC PH or S side. \u003d ½P stems h, where h is apothem

A pyramid is a polyhedron that consists of a flat polygon - the base of the pyramid, a point that does not lie in the plane of the base - the top of the pyramid and all segments connecting the top of the pyramid with points.

vertex

• vertex

side ribs

side faces

base

The pyramid is called correct , if its base is a regular polygon, and the vertex is projected to the center of the base.

In a regular pyramid, all side faces are equal isosceles triangles .

Apothem - the height of the side face of the regular pyramid.

S p \u003d S main + S b.p.

ABCD - base

SO - height

• The surface of the pyramid consists of a base and side faces. Each side face is a triangle. One of its peaks is the top of the pyramid, and the opposite side is the side of the base of the pyramid.

• The height of the pyramid is called the perpendicular dropped from the top of the pyramid to the base plane.

∆ SDB - diagonal section

pyramids SABCD.

The theorem on the lateral surface area of \u200b\u200ba regular pyramid

The lateral surface area of \u200b\u200ba regular pyramid is equal to half the product of the base perimeter and the apothem

S side \u003d ½ P main  SH

Dock - in:

S side = (½al + ½al + ½al +…) \u003d

= ½ l (a + a + a +…) \u003d ½Pl

Building regular pyramids

Truncated quadrangular pyramid

C 1

D 1

Upper base

ABOUT 1

Apothem

A 1

B 1

Side faces

(trapezoid)

Bottom base

Lateral surface area of \u200b\u200ba regular truncated pyramid

The lateral surface area of \u200b\u200ba regular truncated pyramid is equal to the product of the half-sum of the base perimeters and the apothem .

S side =½ ( P 1base. + P 2 main )  l

D 1

FROM 1

Dock - in:

S side = (½ (a + b) l + ½ (a + b) l + + ½ (a + b) l +…) \u003d

= ½ l ( (a + a +…) + (b + b +…) ) =

=½ ( P 1base. + P 2 main )  l

ABOUT 1

A 1

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