The base of the prism is a triangle. Prism volume

In physics, a triangular prism made of glass is often used to study the spectrum of white light because it can resolve it into its individual components. In this article we will consider the volume formula

What is a triangular prism?

Before giving the volume formula, let's consider the properties of this figure.

To get this, you need to take a triangle of any shape and move it parallel to itself to some distance. The vertices of the triangle in the initial and final positions should be connected by straight segments. Received volumetric figure called a triangular prism. It consists of five sides. Two of them are called bases: they are parallel and equal each other. The bases of the prism in question are triangles. The three remaining sides are parallelograms.

In addition to the sides, the prism in question is characterized by six vertices (three for each base) and nine edges (6 edges lie in the planes of the bases and 3 edges are formed by the intersection of the sides). If lateral ribs perpendicular to the bases, then such a prism is called rectangular.

Difference triangular prism from all other figures of this class is that it is always convex (four-, five-, ..., n-gonal prisms may also be concave).

This rectangular figure, at the base of which lies an equilateral triangle.

Volume of a general triangular prism

How to find the volume of a triangular prism? Formula in general view similar to that for any type of prism. It has the following mathematical notation:

Here h is the height of the figure, that is, the distance between its bases, S o is the area of the triangle.

The value of S o can be found if some parameters for the triangle are known, for example, one side and two angles or two sides and one angle. The area of a triangle is equal to half the product of its height and the length of the side by which this height is lowered.

As for the height h of the figure, it is easiest to find for rectangular prism. IN the latter case h coincides with the length of the side edge.

Volume of a regular triangular prism

General formula the volume of a triangular prism, which is given in the previous section of the article, can be used to calculate the corresponding value for a regular triangular prism. Since its base is an equilateral triangle, its area is equal to:

Anyone can get this formula if they remember that in equilateral triangle all angles are equal to each other and amount to 60 o. Here the symbol a is the length of the side of the triangle.

The height h is the length of the edge. It is in no way connected with the base of a regular prism and can take arbitrary values. As a result, the formula for the volume of a triangular prism is the right kind looks like that:

Having calculated the root, you can rewrite this formula as follows:

Thus, to find the volume of a regular prism with triangular base, it is necessary to square the side of the base, multiply this value by the height and multiply the resulting value by 0.433.

Prism volume. Problem solving

Geometry is the most powerful means for sharpening our mental faculties and enabling us to think and reason correctly.

G. Galileo

The purpose of the lesson:

teach solving problems on calculating the volume of prisms, summarize and systematize the information students have about a prism and its elements, develop the ability to solve problems of increased complexity;
develop logical thinking, ability to work independently, skills of mutual control and self-control, ability to speak and listen;
develop the habit of constant employment in some useful activity, fostering responsiveness, hard work, and accuracy.

Lesson type: lesson on applying knowledge, skills and abilities.

Equipment: control cards, media projector, presentation “Lesson. Prism Volume”, computers.

During the classes

Lateral ribs of the prism (Fig. 2).
The lateral surface of the prism (Figure 2, Figure 5).
The height of the prism (Fig. 3, Fig. 4).
Straight prism (Figure 2,3,4).
Inclined prism(Figure 5).
The correct prism (Fig. 2, Fig. 3).
Diagonal section prisms (Figure 2).
Diagonal of the prism (Figure 2).
Perpendicular section of the prism (Fig. 3, Fig. 4).
The lateral surface area of the prism.
Square full surface prisms.
Prism volume.

HOMEWORK CHECK (8 min)

Exchange notebooks, check the solution on the slides and mark it (mark 10 if the problem has been compiled)

Make up a problem based on the picture and solve it. The student defends the problem he has compiled at the board. Figure 6 and Figure 7.

Chapter 2,§3
Problem.2. The lengths of all edges of a regular triangular prism are equal to each other. Calculate the volume of the prism if its surface area is cm 2 (Fig. 8)

Chapter 2,§3
Problem 5. The base of the straight prism ABCA 1B 1C1 is right triangle ABC (angle ABC=90°), AB=4cm. Calculate the volume of the prism if the radius of the circle circumscribed about triangle ABC, is 2.5 cm, and the height of the prism is 10 cm. (Figure 9).

Chapter2,§3
Problem 29. The length of the side of the base is regular quadrangular prism equal to 3cm. The diagonal of the prism forms an angle of 30° with the plane of the side face. Calculate the volume of the prism (Figure 10).

Collaboration teachers with the class (2-3 min.).

Purpose: summing up the results of the theoretical warm-up (students grade each other), learning how to solve problems on the topic.

PHYSICAL MINUTE (3 min)
PROBLEM SOLVING (10 min)

On at this stage The teacher organizes frontal work on repeating methods for solving planimetric problems and planimetric formulas. The class is divided into two groups, some solve problems, others work at the computer. Then they change. Students are asked to solve all No. 8 (orally), No. 9 (orally). Then they divide into groups and proceed to solve problems No. 14, No. 30, No. 32.

Chapter 2, §3, pages 66-67

Problem 8. All edges of a regular triangular prism are equal to each other. Find the volume of the prism if the cross-sectional area of the plane passing through the edge of the lower base and the middle of the side of the upper base is equal to cm (Fig. 11).

Chapter 2,§3, page 66-67
Problem 9. The base of a straight prism is a square, and its side edges are twice the size of the side of the base. Calculate the volume of the prism if the radius of the circle described near the cross section of the prism by a plane passing through the side of the base and the middle of the opposite side edge is equal to cm (Fig. 12)

Chapter 2,§3, page 66-67
Problem 14 The base of a straight prism is a rhombus, one of the diagonals of which is equal to its side. Calculate the perimeter of the section by a plane passing through large diagonal lower base, if the volume of the prism is equal and all side faces are squares (Fig. 13).

Chapter 2,§3, page 66-67
Problem 30 ABCA 1 B 1 C 1 is a regular triangular prism, all edges of which are equal to each other, the point is the middle of edge BB 1. Calculate the radius of the circle inscribed in the section of the prism by the AOS plane, if the volume of the prism is equal to (Fig. 14).

Chapter 2,§3, page 66-67
Problem 32.In a regular quadrangular prism, the sum of the areas of the bases is equal to the area of the lateral surface. Calculate the volume of the prism if the diameter of the circle described near the cross section of the prism by a plane passing through the two vertices of the lower base and the opposite vertex of the upper base is 6 cm (Fig. 15).

While solving problems, students compare their answers with those shown by the teacher. This is a sample solution to the problem with detailed comments... Individual work teachers with “strong” students (10 min.).

Independent work students working on a test at the computer

1. The side of the base of a regular triangular prism is equal to , and the height is 5. Find the volume of the prism.

1) 152) 45 3) 104) 125) 18

2. Choose the correct statement.

1) The volume of a right prism whose base is a right triangle is equal to the product of the area of the base and the height.

2) The volume of a regular triangular prism is calculated by the formula V = 0.25a 2 h - where a is the side of the base, h is the height of the prism.

3) Volume of a straight prism equal to half product of the area of the base and the height.

4) The volume of a regular quadrangular prism is calculated by the formula V = a 2 h-where a is the side of the base, h is the height of the prism.

5)Volume correct hexagonal prism calculated by the formula V = 1.5a 2 h, where a is the side of the base, h is the height of the prism.

3. The side of the base of a regular triangular prism is equal to . Through the side of the lower base and the opposite vertex A plane is drawn from the upper base, which passes at an angle of 45° to the base. Find the volume of the prism.

1) 92) 9 3) 4,54) 2,255) 1,125

4. The base of a right prism is a rhombus, the side of which is 13, and one of the diagonals is 24. Find the volume of the prism if the diagonal of the side face is 14.

Schoolchildren who are preparing for passing the Unified State Exam in mathematics, you should definitely learn how to solve problems on finding the area of a straight and regular prism. Many years of practice confirm the fact that similar tasks Geometry is considered quite difficult by many students.

At the same time, high school students with any level of training should be able to find the area and volume of a regular and straight prism. Only in this case will they be able to count on receiving competitive scores based on the results of passing the Unified State Exam.

Key Points to Remember

If the lateral edges of a prism are perpendicular to the base, it is called a straight line. All side faces of this figure are rectangles. The height of a straight prism coincides with its edge.
A correct prism is one whose side edges are perpendicular to the base in which it is located. regular polygon. The lateral edges of this figure are equal rectangles. A correct prism is always straight.

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Specialists educational project“Shkolkovo” proposes to go from simple to complex: first we give theory, basic formulas, theorems and elementary problems with solutions, and then gradually move on to expert-level tasks.

Basic information is systematized and clearly presented in the “Theoretical Information” section. If you have already managed to repeat the necessary material, we recommend that you practice solving problems on finding the area and volume of a right prism. The “Catalogue” section presents a large selection of exercises varying degrees difficulties.

Try to calculate the area of a straight and regular prism or right now. Analyze any task. If it does not cause any difficulties, you can safely move on to expert-level exercises. And if certain difficulties do arise, we recommend that you regularly prepare for the Unified State Exam online together with the Shkolkovo mathematical portal, and tasks on the topic “Straight and Regular Prism” will be easy for you.

Definition.

This is a hexagon whose bases are two equal square, and the side faces are equal rectangles

Side rib- This common side two adjacent side faces

Prism height- this is a segment perpendicular to the bases of the prism

Prism diagonal- a segment connecting two vertices of the bases that do not belong to the same face

Diagonal plane- a plane that passes through the diagonal of the prism and its lateral edges

Diagonal section- the boundaries of the intersection of the prism and the diagonal plane. The diagonal cross section of a regular quadrangular prism is a rectangle

Perpendicular section (orthogonal section)- this is the intersection of a prism and a plane drawn perpendicular to its lateral edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are indicated by the corresponding letters:

The bases ABCD and A 1 B 1 C 1 D 1 are equal and parallel to each other
Side faces AA 1 D 1 D, AA 1 B 1 B, BB 1 C 1 C and CC 1 D 1 D, each of which is a rectangle
Side surface- the sum of the areas of all lateral faces of the prism
Total surface - the sum of the areas of all bases and side faces (sum of the area of the side surface and bases)
Side ribs AA 1, BB 1, CC 1 and DD 1.
Diagonal B 1 D
Base diagonal BD
Diagonal section BB 1 D 1 D
Perpendicular section A 2 B 2 C 2 D 2.

Properties of a regular quadrangular prism

The bases are two equal squares
The bases are parallel to each other
The side faces are rectangles
The side edges are equal to each other
Side faces are perpendicular to the bases
The lateral ribs are parallel to each other and equal
Perpendicular section perpendicular to all side ribs and parallel to the bases
Angles perpendicular section- straight
The diagonal cross section of a regular quadrangular prism is a rectangle
Perpendicular (orthogonal section) parallel to the bases

Formulas for a regular quadrangular prism

Instructions for solving problems

When solving problems on the topic " regular quadrangular prism" means that:

Correct prism- a prism at the base of which lies a regular polygon, and the side edges are perpendicular to the planes of the base. That is, a regular quadrangular prism contains at its base square. (see properties of a regular quadrangular prism above) Note. This is part of a lesson with geometry problems (section stereometry - prism). Here are problems that are difficult to solve. If you need to solve a geometry problem that is not here, write about it in the forum. To indicate the action of retrieving square root the symbol is used in solving problems√ .

Task.

In the correct four carbon prism the base area is 144 cm 2 and the height is 14 cm. Find the diagonal of the prism and the total surface area.

Solution.
A regular quadrilateral is a square.
Accordingly, the side of the base will be equal

√144 = 12 cm.
From where the diagonal of the base of a regular rectangular prism will be equal to
√(12 2 + 12 2 ) = √288 = 12√2

The diagonal of a regular prism forms a right triangle with the diagonal of the base and the height of the prism. Accordingly, according to the Pythagorean theorem, the diagonal of a given regular quadrangular prism will be equal to:
√((12√2) 2 + 14 2 ) = 22 cm

Answer: 22 cm

Task

Determine the total surface of a regular quadrangular prism if its diagonal is 5 cm and the diagonal of its side face is 4 cm.

Solution.
Since the base of a regular quadrangular prism is a square, we find the side of the base (denoted as a) using the Pythagorean theorem:

A 2 + a 2 = 5 2
2a 2 = 25
a = √12.5

The height of the side face (denoted as h) will then be equal to:

H 2 + 12.5 = 4 2
h 2 + 12.5 = 16
h 2 = 3.5
h = √3.5

The total surface area will be equal to the sum of the lateral surface area and twice the base area

S = 2a 2 + 4ah
S = 25 + 4√12.5 * √3.5
S = 25 + 4√43.75
S = 25 + 4√(175/4)
S = 25 + 4√(7*25/4)
S = 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.