Volume of an inclined prism. Presentation "Volume of an Inclined Prism" of Applied Mathematics

Lesson plan Calculating the volumes of bodies using a definite integral Calculating the volumes of bodies using a definite integral Calculating the volumes of bodies using a definite integral Calculating the volumes of bodies using a definite integral Volume of an inclined prism Volume of an inclined prism Volume of an inclined prism Volume of an inclined prism Volume of a pyramid Volume of a pyramid Volume of a pyramid Volume of a pyramid Volume of a truncated pyramid Volume of a truncated pyramid Volume of a truncated pyramid Volume of a truncated pyramid Volume of a cone Volume of a cone Volume of a cone Volume of a cone Volume of a truncated cone Volume of a truncated cone Volume of a truncated cone Volume of a truncated cone Questions for consolidation Questions for consolidation Questions for consolidation Questions for consolidation

Calculation of volumes of bodies The approximate value of the volume of a body is equal to the sum of the volumes of straight prisms, the bases of which are equal to the cross-sectional areas of a body of height equal to i = x i – x i – 1 The approximate value of the volume of a body is equal to the sum of the volumes of straight prisms, the bases of which are equal to the cross-sectional areas of the body, and the heights are equal to i = x i – x i – 1 a x i-1 x i b α β S(x i) The segment is divided into n parts

Volume of a pyramid The volume of a triangular pyramid is equal to one-third of the product of the base area and the height. Theorem: The volume of a triangular pyramid is equal to one-third of the product of the base area and the height or a certain integral of the base area in the interval from 0 to h B C O A M h

Presentation on the topic PRISMA This presentation is designed for visual use in a lesson in the academic discipline “mathematics” for 2nd year students within the framework of the topic: “Polyhedra”. The presentation includes slides of a training and control nature. The purpose of this project: 1. Instilling interest in mathematics as an element of universal human culture. Creating motivation among students for the academic discipline “mathematics”, saving time for the purpose of deeper assimilation of the material for quick analysis of problems in the lesson, and for a better perception of spatial figures in space in the lesson. 2. Development of cognitive interest, spatial imagination, intelligence, logical thinking, intuition, attention. 3.Formation of communication skills, the ability to work in a team. This presentation is used to accompany several stages of the lesson. Using the “Living Geometry” program, a visual demonstration of various types of prisms is carried out from various angles: rotation of the prism, tilt, change in the height of the prism, demonstration of the faces of the prism, its visible and invisible edges. During the lesson, various forms and methods of work and the use of ICT were thought through. The developed project will assist teachers of educational institutions in preparing and conducting a lesson on the topic: “Prism, its elements and properties

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"Presentation on PRISMA"

LESSON TOPIC:

"PRISM,

its elements

and properties »

1.) Definition of a prism.

2.) types of prisms:

- straight prism;

- inclined prism;

- correct prism;

3.) The total surface area of ​​the prism.

4.) The area of ​​the lateral surface of the prism.

5.) Volume of the prism.

6.) Let's prove the theorem for a triangular prism.

7.) Let us prove the theorem for an arbitrary prism.

8.) Prism sections:

- perpendicular section of the prism;

Definition of a prism

Prism -

This polyhedron, consisting from two flat polygons , lying in different planes and combined by parallel transfer,

and all segments , connecting the corresponding points these polygons.

HEIGHT

EDGE

LATERAL

Prism elements

EDGE

BASE

EDGE

Prism elements

Base rib

Upper base

vertex

Side rib

Side edge

diagonal

Bottom base

height

Prism elements

• Grounds –

These are faces that are combined by parallel translation.

• Side edge –

this is an edge that is not a base.

• Side ribs –

these are segments connecting the corresponding vertices of the bases.

• Peaks –

these are the points that are the tops of the bases.

• Height –

it is a perpendicular dropped from one base to another.

• Diagonal –

This is a segment connecting two vertices that do not lie on the same face.

If the lateral edges of a prism are perpendicular to the bases, then the prism is called straight ,

otherwise - inclined .

types of prisms

inclined

correct

Straight a prism is called correct, if in her basis lies regular polygon

If in basis prism lies - n- square , then the prism is called n- coal

Quadrangular

Hexagonal Triangular

prism prism prism

Diagonal section - a section of a prism by a plane passing through two side edges that do not belong to the same face.

In the cross section it is formed

parallelogram.

In some

cases may

it turns out to be a rhombus, rectangle or square.

Diagonal sections parallelepiped

Prism properties

1. The bases of the prism are equal polygons.

2. The lateral faces of the prism are parallelograms, if the prism is straight, then they are rectangles

3. The lateral edges of the prism and the base are parallel and equal.

4. Opposite edges are parallel and equal.

5. Opposite side faces are parallel and equal.

6. The height is perpendicular to each base.

7. Diagonals intersect at one point and bisect at it.

Prism lateral surface area

Theorem on the lateral surface area of ​​a straight prism

Square lateral surface the direct prism is equal to the product base perimeter on height prisms

P- perimeter

h– prism height

Total surface area of ​​the prism

The total surface area of ​​a prism is the sum of the areas of all its faces.

Prism volume

THEOREM:

Volume

prism is equal

product of area

base to height

V= S basic ∙h

Volume of an inclined prism

THEOREM:

Inclined volume

prism is equal

product of area

base to height.

V= S basic ∙h

Problem No. 229 (b), p. 68

In a regular n-gonal prism, the side of the base is equal to A and the height is h. Calculate the areas of the lateral and total surfaces of the prism if: n = 4, A= 12 dm, h = 8 dm.

A= 12 dm

mutual verification

SOLUTION:

T.K. n = 4, then the prism is quadrangular.

Sside = = 4 A h

Sside = 4 8 12 = 384 (dm 2)

Spol = 2Smain + Sside

Sbas = A 2 = 12 2 = 144 (dm 2)

Spol = 2 144 + 384 = 672 (dm 2)

Answer: 384 dm 2, 672 dm 2

Checking the answer

SOLUTION:

T.K. n = 6, then the prism is hexagonal.

Sside = 6 50 23 = 6900 (cm2) = 69 (dm 2)

Spol = 3 A· (2h + √3 · A)

Spol = 69 · (100 + 23√3) = 69 · 140 = 9660 (cm 2) = 97 (dm 2)

Answer: 69 dm 2, 97 dm 2

Heron of Alexandria

Heron's formula

Ancient Greek scientist, mathematician,

physicist, mechanic, inventor.

allows you to calculate

Heron's mathematical works

area of ​​a triangle ( S )

are an encyclopedia of ancient

on its sides a, b, c :

applied mathematics. In the best of

them - "Metrica" ​​- given the rules and

formulas for exact and approximate

calculating areas of correct

Where R - semi-perimeter of a triangle:

polygons, truncated volumes

cones and pyramids, given

Heron's formula for determining

area of ​​the triangle on three sides,

rules for numerical solution are given

quadratic equations and approximate

extracting square and cubic

roots .

unknown

probably

Solve a problem

• In a right triangular prism, the sides of the base are 10 cm, 17 cm and 21 cm, and the height of the prism is 18 cm. Find the total surface area and volume of the prism.

Checking the answer

SOLUTION:

P = 10+17 +21 = 48(cm)

Sside = 48 18 = 864 (cm 2)

Spol = 864 + 168 = 1032 (cm 2 )

V= S basic ∙h = 84 ·18 = 1512(cm 3)

1032 (cm 2 )

, 1512 (cm 3)

The lesson is over!

Continue the sentence:

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Volume of an inclined prism

All prisms are divided into straight And inclined .

Straight prism, base

which serves the correct

a polygon is called

correct prism.

Properties of a regular prism:

1. The bases of a regular prism are regular polygons. 2. The lateral faces of a regular prism are equal rectangles. 3. The lateral edges of a regular prism are equal .

PRISM cross section.

The orthogonal section of a prism is a section formed by a plane perpendicular to the side edge.

The lateral surface of the prism is equal to the product of the perimeter of the orthogonal section and the length of the lateral edge.

S b =P orth.section C

1. Distances between inclined ribs

triangular prism are equal to: 2cm, 3cm and 4cm

The lateral surface of the prism is 45cm 2 .Find its side edge.

Solution:

In the perpendicular section of the prism there is a triangle whose perimeter is 2+3+4=9

This means the side edge is equal to 45:9 = 5 (cm)

Find unknown elements

regular triangular

Prisms

by elements specified in the table.

ANSWERS.

Thank you for the lesson.

Homework.

“Volumes” - Exercise 9*. B. Cavalieri. Volume of an inclined prism 3. Find the volume of a parallelepiped. Answer: Yes. Volume of an inclined prism 1. Exercise 8*. Three parallelepipeds are given in space. Cavalieri principle. Answer: 1:3. The face of a parallelepiped is a rhombus with a side of 1 and an acute angle of 60°.

“Scope of concept” - MAIN PURPOSE of the lesson. The presented lesson is the first lesson-lecture on the topic “Volumes”. During the lesson, differentiated testing work is carried out using tests. Control questions. S=smain+Sside. Let's fill in the second half of the table. What is the volume of a rectangular parallelepiped?

“Volume of bodies” - When a = x and b = x, a point can degenerate into a section, for example, when x = a. Ф(х1). F(x2). F(xi). a x b x. Volume of an inclined prism, pyramid and cone. Ф(x).

“Volumes of bodies” - Volumes of bodies. V=a*b*c. V=S*h. Completed by Alesya Krivodusheva, grade 11-A. Consequence. The ratio of the volumes of similar bodies is equal to the cube of the similarity coefficient, i.e. 2010. Volume of the pyramid. h. Volumes of similar bodies. The volume of the pyramid is equal to one third of the product of the base and the height. The volume of a cylinder is equal to the product of the area of ​​the base and the height.