The formula for the volume of an oblique triangular prism. Lesson "Volume of an inclined prism

"Geometric body prism" - Rectangular parallelepiped. Rectangle. Diagonal sections. Pythagorean theorem. The amount of areas. Vertices. base of the prism. What is the name of the prism shown in the figure. Math fight. Solution. Prism. What is a straight prism. Received knowledge. Diagonal of a regular triangular prism.

"Figure prism" - Definition of a prism. Inclined and straight prism. Let us first prove the theorem for a triangular prism. Prism types. The volume of an inclined prism. Prism. The area of ​​the lateral surface of the prism. The total surface area of ​​the prism. Let us now prove the theorem for an arbitrary prism. correct prism.

"Volume of the prism" - Area S of the base of the original prism. The solution of the problem. Lesson goals. The volume of the original prism is equal to the product S · h. The volume of a straight prism. The prism can be divided into straight triangular prisms with height h. The concept of a prism. Draw the altitude of triangle ABC. Questions. Study of the prism volume theorem. Basic steps in proving the direct prism theorem?

"The concept of a prism" - The area of ​​\u200b\u200bthe total surface of a prism. direct prism. The area of ​​the lateral surface of the prism. Polygon. Prism sections. correct prism. Prisms encountered in life. triangular prisms. Proof. The volume of an inclined prism. Definition of a prism. Inclined and straight prism. Prism types. Prism.

"Properties of a prism" - Are there inclined prisms in which a sphere can be inscribed. prism properties. The condition formulated for a straight prism. Cylinder. Prism. Cross section of a cylinder. Formula of three cosines. Base. triangular prism. The sine theorem for a trihedral angle. The edge of a triangular prism. Around which of the varieties of prisms can you always describe a sphere.

“The concept of a prism polyhedron” - A parallelogram is formed in the section. Consequence. prism properties. The term “prism” is of Greek origin and literally means “sawn off” (body). The surface area of ​​the prism and the lateral surface area of ​​the prism. Such a section is called the diagonal section of the prism. Given: The side of the base of a regular triangular prism is 8 cm, the side edge is 6 cm.

Prism definition:

  • А1А2…AnВ1В2Вn– prism

  • Polygons А1А2…An and В1В2…Вn – prism bases

  • Parallelograms A1A2B2B1, A1A2B2B1, ... AnA1B1Bn - side faces

  • Segments А1В1, А2В2…AnBn – side edges of the prism


Prism types

  • Hexagonal Triangular Quadrangular Prism Prism Prism


Inclined and straight prism

  • If the side edges of the prism are perpendicular to the bases, then the prism is called straight , otherwise - oblique .


Correct prism

  • The prism is called correct if it is a straight line and its bases are regular polygons.


Total surface area of ​​the prism


Prism side surface area

  • Theorem

  • The area of ​​the lateral surface of a straight prism is equal to half the product of the perimeter of the base and the height of the prism.


Volume of an inclined prism

  • Theorem

  • The volume of an inclined prism is equal to the product of the area of ​​the base and the height.


Proof

  • Proof

  • Let us first prove the theorem for a triangular prism, and then for an arbitrary prism.

    • 1. Consider a triangular prism with volume V, base area S and height h. Mark a point O on one of the bases of the prism and direct the Ox axis perpendicular to the bases. Consider a section of a prism by a plane perpendicular to the Ox axis and, therefore, parallel to the plane of the base. We denote by the letter x the abscissa of the point of intersection of this plane with the axis Ox, and through S (x) - the area of ​​the resulting section.

    Let us prove that the area S (x) is equal to the area S of the base of the prism. To do this, note that the triangles ABC (the base of the prism) and A1B1C1 (the section of the prism by the considered plane) are equal. Indeed, the quadrilateral AA1BB1 is a parallelogram (segments AA1 and BB1 are equal and parallel), so A1B1=AB. Similarly, it is proved that B1C1=BC and A1C1=AC. So, triangles A1B1C1 and ABC are equal on three sides. Therefore, S(x)=S. Applying now the basic formula for calculating the volumes of bodies at a=0 and b=h, we obtain



2. h h h, S S*h. The theorem has been proven.

    2. Let us now prove the theorem for an arbitrary prism with height h and base area S. Such a prism can be divided into triangular prisms with a total height h. We express the volume of each triangular prism according to the formula we have proved and add these volumes. Bracketing the common factor h, we get in parentheses the sum of the areas of the bases of triangular prisms, i.e. the area S the base of the original prism. Thus, the volume of the original prism is S*h. The theorem has been proven.


Volume is a characteristic of any figure that has non-zero dimensions in all three dimensions of space. In this article, from the point of view of stereometry (the geometry of spatial figures), we will consider a prism and show how to find the volumes of prisms of various types.

Stereometry has an exact answer to this question. A prism in it is understood as a figure formed by two identical polygonal faces and several parallelograms. The figure below shows four different prisms.

Each of them can be obtained as follows: you need to take a polygon (triangle, quadrilateral, and so on) and a segment of a certain length. Then each vertex of the polygon should be transferred using parallel segments to another plane. In the new plane, which will be parallel to the original one, a new polygon will be obtained, similar to the one chosen initially.

Prisms can be of different types. So, they can be straight, oblique and correct. If the lateral edge of the prism (the segment connecting the tops of the bases) is perpendicular to the bases of the figure, then the latter is a straight line. Accordingly, if this condition is not met, then we are talking about an inclined prism. A regular figure is a right prism with an equiangular and equilateral base.

Volume of regular prisms

Let's start with the simplest case. We give the formula for the volume of a regular prism with an n-gonal base. The volume formula V for any figure of the class under consideration has the following form:

That is, to determine the volume, it is enough to calculate the area of ​​\u200b\u200bone of the bases S o and multiply it by the height h of the figure.

In the case of a regular prism, we denote the length of the side of its base by the letter a, and the height, which is equal to the length of the side edge, by the letter h. If the base of the n-gon is correct, then the easiest way to calculate its area is to use the following universal formula:

S n \u003d n / 4 * a2 * ctg (pi / n).

Substituting in equality the value of the number of sides n and the length of one side a, you can calculate the area of ​​the n-coal base. Note that the cotangent function here is calculated for the angle pi/n, which is expressed in radians.

Taking into account the equality written for S n, we obtain the final formula for the volume of a regular prism:

Vn = n/4*a2*h*ctg(pi/n).

For each specific case, one can write down the corresponding formulas for V, but they all follow unambiguously from the general expression written down. For example, for a regular quadrangular prism, which in the general case is a rectangular parallelepiped, we get:

V 4 \u003d 4/4 * a2 * h * ctg (pi / 4) \u003d a2 * h.

If we take h=a in this expression, then we get a formula for the volume of a cube.

Volume of straight prisms

We note right away that for straight figures there is no general formula for calculating volume, which was given above for regular prisms. When finding the quantity under consideration, the original expression should be used:

Here h is the length of the side edge, as in the previous case. As for the base area S o , it can take on a variety of values. The task of calculating a straight prism of volume is reduced to finding the area of ​​its base.

The calculation of the value of S o should be carried out based on the characteristics of the base itself. For example, if it is a triangle, then the area can be calculated as follows:

Here h a is the apothem of the triangle, that is, its height lowered to the base a.

If the base is a quadrilateral, then it can be a trapezoid, a parallelogram, a rectangle, or a completely arbitrary type. For all these cases, you should use the appropriate planimetry formula to determine the area. For example, for a trapezoid, this formula looks like:

S o4 \u003d 1/2 * (a 1 + a 2) * h a .

Where h a is the height of the trapezoid, a 1 and a 2 are the lengths of its parallel sides.

To determine the area for polygons of a higher order, one should break them into simple figures (triangles, quadrangles) and calculate the sum of the areas of the latter.

Volume of inclined prisms

This is the most difficult case of calculating the volume of a prism. The general formula for such figures also applies:

However, to the complexity of finding the area of ​​the base, representing an arbitrary type of polygon, is added the problem of determining the height of the figure. In an inclined prism, it is always less than the length of the side edge.

The easiest way to find this height is if you know any angle of the figure (flat or dihedral). If such an angle is given, then one should use it to construct a right-angled triangle inside the prism, which would contain the height h as one of the sides and, using trigonometric functions and the Pythagorean theorem, find the value h.

Geometric volume problem

Given a regular prism with a triangular base, having a height of 14 cm and a side length of 5 cm. What is the volume of a triangular prism?

Since we are talking about the correct figure, we have the right to use the well-known formula. We have:

V 3 = 3/4*a2*h*ctg(pi/3) = 3/4*52*14*1/√3 = √3/4*25*14 = 151.55 cm3.

A triangular prism is a fairly symmetrical figure, in the form of which various architectural structures are often performed. This glass prism is used in optics.

The concept of a prism. Volume formulas for prisms of different types: regular, straight and oblique. Problem solving - all about traveling to the site

Volume of an inclined prism





All prisms are divided into straight and oblique .


Straight prism, base

which serves the right

polygon is called

correct prism.

Properties of the correct prism:

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


Section of a PRISM.

An 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 rib.

S b \u003d P ortho.sec C




1. Distances between the ribs of the inclined

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

Lateral surface of the prism - 45cm 2 .Find its side edge.

Solution:

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

So the side edge is 45:9=5(cm)


Find unknown elements

regular triangular

Prisms

by elements specified in the table.


ANSWERS.


Thank you for the lesson.

Homework.

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