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.
Decision:
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.
Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
The following are some examples of the types of personal information we may collect and how we may use such information.
What personal information we collect:
- When you submit an application on the site, we may collect various information, including your name, phone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send you important notices and messages.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.
Disclosure to third parties
We do not disclose information received from you to third parties.
Exceptions:
- In the event that it is necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from state bodies in the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest reasons.
- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.
Maintaining your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.
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 written down general expression. 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.
