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).
- Lateral surface prisms (Fig. 2, Fig. 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.
- The total surface area of the prism.
- Prism volume.
- HOMEWORK CHECK (8 min)
- Collaboration teachers with the class (2-3 min.).
- PHYSICAL MINUTE (3 min)
- PROBLEM SOLVING (10 min)
- Independent work students working on a test at the computer
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 a straight prism ABCA 1B 1C1 is a 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 of a regular quadrangular prism is 3 cm. 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).
Purpose: summing up the theoretical warm-up (students give marks each other), studying ways to solve problems on a topic.
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 lateral rib, 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 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.).
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.
Let us separately draw the base of the prism, i.e. triangle ABC (Fig. 307, a), and build it up to a rectangle, for which we draw a straight line KM through vertex B || AC and from points A and C we lower perpendiculars AF and CE onto this line. We get rectangle ACEF. Drawing the height ВD of triangle ABC, we see that rectangle ACEF is divided into 4 right triangle. Moreover, \(\Delta\)ALL = \(\Delta\)BCD and \(\Delta\)BAF = \(\Delta\)BAD. This means that the area of the rectangle ACEF is doubled more area triangle ABC, i.e. equal to 2S.
To this prism with base ABC we will attach prisms with bases ALL and BAF and height h(Fig. 307, b). We obtain a rectangular parallelepiped with an ACEF base.
If we dissect this parallelepiped with a plane passing through straight lines BD and BB’, we will see that the rectangular parallelepiped consists of 4 prisms with bases BCD, ALL, BAD and BAF.
Prisms with bases BCD and BC can be combined, since their bases are equal (\(\Delta\)BCD = \(\Delta\)BCE) and their side edges, which are perpendicular to the same plane, are also equal. This means that the volumes of these prisms are equal. The volumes of prisms with bases BAD and BAF are also equal.
Thus, it turns out that the volume of a given triangular prism with base ABC is half the volume rectangular parallelepiped with ACEF base.
We know that the volume of a rectangular parallelepiped equal to the product area of its base by height, i.e. in in this case equal to 2S h. Hence the volume of this right triangular prism is equal to S h.
The volume of a right triangular prism is equal to the product of the area of its base and its height.
2. Volume of a right polygonal prism.
To find the volume of a right polygonal prism, for example a pentagonal one, with base area S and height h, let's divide it into triangular prisms (Fig. 308).
Denoting the base areas of triangular prisms by S 1, S 2 and S 3, and the volume of a given polygonal prism by V, we obtain:
V = S 1 h+ S 2 h+ S 3 h, or
V = (S 1 + S 2 + S 3) h.
And finally: V = S h.
In the same way, the formula for the volume of a right prism with any polygon at its base is derived.
Means, The volume of any right prism is equal to the product of the area of its base and its height.
Prism volume
Theorem. The volume of a prism is equal to the product of the area of the base and the height.
First we prove this theorem for a triangular prism, and then for a polygonal one.
1) Let us draw (Fig. 95) through the edge AA 1 of the triangular prism ABCA 1 B 1 C 1 a plane parallel to the face BB 1 C 1 C, and through the edge CC 1 - a plane parallel to the face AA 1 B 1 B; then we will continue the planes of both bases of the prism until they intersect with the drawn planes.
Then we get a parallelepiped BD 1, which is divided by the diagonal plane AA 1 C 1 C into two triangular prisms (one of which is this one). Let us prove that these prisms are equal in size. To do this, we will carry out perpendicular section abcd. The cross-section will produce a parallelogram whose diagonal ac divisible by two equal triangle. This prism is equal in size to a straight prism whose base is \(\Delta\) abc, and the height is edge AA 1. Another triangular prism is equal in area to a straight line whose base is \(\Delta\) adc, and the height is edge AA 1. But two straight prisms with equally And equal heights are equal (because when nested they are combined), which means that the prisms ABCA 1 B 1 C 1 and ADCA 1 D 1 C 1 are equal in size. It follows from this that the volume of this prism is half the volume of the parallelepiped BD 1; therefore, denoting the height of the prism by H, we get:
$$ V_(\Delta ex.) = \frac(S_(ABCD)\cdot H)(2) = \frac(S_(ABCD))(2)\cdot H = S_(ABC)\cdot H $$
2) Let us draw diagonal planes AA 1 C 1 C and AA 1 D 1 D through the edge AA 1 of the polygonal prism (Fig. 96).
Then this prism will be cut into several triangular prisms. The sum of the volumes of these prisms constitutes the required volume. If we denote the areas of their bases by b 1 , b 2 , b 3, and the total height through H, we get:
volume of polygonal prism = b 1H+ b 2H+ b 3 H =( b 1 + b 2 + b 3) H =
= (area ABCDE) H.
Consequence. If V, B and H are numbers expressing in the corresponding units the volume, base area and height of the prism, then, according to what has been proven, we can write:
Other materialsIn 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 to 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 the side edges are perpendicular to the bases, then such a prism is called rectangular.
The difference between a triangular prism and all other figures of this class is that it is always convex (four-, five-, ..., n-gonal prisms may also be concave).
This rectangular figure, which is based on 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 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 an 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.
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Different prisms are different from each other. At the same time, they have a lot in common. To find the area of the base of the prism, you will need to understand what type it has.
General theory
A prism is any polyhedron sides which have the shape of a parallelogram. Moreover, its base can be any polyhedron - from a triangle to an n-gon. Moreover, the bases of the prism are always equal to each other. What does not apply to the side faces is that they can vary significantly in size.
When solving problems, not only the area of the base of the prism is encountered. It may require knowledge of the lateral surface, that is, all the faces that are not bases. Full surface there will already be a union of all the faces that make up the prism.
Sometimes problems involve height. It is perpendicular to the bases. The diagonal of a polyhedron is a segment that connects in pairs any two vertices that do not belong to the same face.
It should be noted that the base area of a straight or inclined prism does not depend on the angle between them and the side faces. If they identical figures in the upper and lower faces, then their areas will be equal.
Triangular prism
It has at its base a figure with three vertices, that is, a triangle. As you know, it can be different. If so, it is enough to remember that its area is determined by half the product of the legs.
The mathematical notation looks like this: S = ½ av.
To find out the area of the base in general, the formulas are useful: Heron and the one in which half of the side is taken by the height drawn to it.
The first formula should be written as follows: S = √(р (р-а) (р-в) (р-с)). This notation contains a semi-perimeter (p), that is, the sum of three sides divided by two.
Second: S = ½ n a * a.
If you want to find out the area of the base of a triangular prism, which is regular, then the triangle turns out to be equilateral. There is a formula for it: S = ¼ a 2 * √3.
Quadrangular prism
Its base is any of the known quadrangles. It can be a rectangle or square, parallelepiped or rhombus. In each case, in order to calculate the area of the base of the prism, you will need your own formula.
If the base is a rectangle, then its area is determined as follows: S = ab, where a, b are the sides of the rectangle.
When we're talking about O four carbon prism, then the area of the base of a regular prism is calculated using the formula for a square. Because it is he who lies at the foundation. S = a 2.
In the case when the base is a parallelepiped, the following equality will be needed: S = a * n a. It happens that the side of a parallelepiped and one of the angles are given. Then to calculate the height you will need to use additional formula: na = b * sin A. Moreover, angle A is adjacent to side “b”, and the height na is opposite to this angle.
If there is a rhombus at the base of the prism, then to determine its area you will need the same formula as for a parallelogram (since it is a special case of it). But you can also use this: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.
Regular pentagonal prism
This case involves dividing the polygon into triangles, the areas of which are easier to find out. Although it happens that figures can have a different number of vertices.
Since the base of the prism is regular pentagon, then it can be divided into five equilateral triangles. Then the area of the base of the prism is equal to the area of one such triangle (the formula can be seen above), multiplied by five.
Regular hexagonal prism
Using the principle described for a pentagonal prism, it is possible to divide the hexagon of the base into 6 equilateral triangles. The formula for the base area of such a prism is similar to the previous one. Only it should be multiplied by six.
The formula will look like this: S = 3/2 a 2 * √3.
Tasks
No. 1. Given a regular straight line, its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of the base of the prism and the entire surface.
Solution. The base of the prism is a square, but its side is unknown. You can find its value from the diagonal of the square (x), which is related to the diagonal of the prism (d) and its height (h). x 2 = d 2 - n 2. On the other hand, this segment “x” is the hypotenuse in a triangle whose legs are equal to the side of the square. That is, x 2 = a 2 + a 2. Thus it turns out that a 2 = (d 2 - n 2)/2.
Substitute the number 22 instead of d, and replace “n” with its value - 14, it turns out that the side of the square is 12 cm. Now just find out the area of the base: 12 * 12 = 144 cm 2.
To find out the area of the entire surface, you need to add twice the base area and quadruple the side area. The latter can be easily found using the formula for a rectangle: multiply the height of the polyhedron and the side of the base. That is, 14 and 12, this number will be equal to 168 cm 2. total area The surface of the prism turns out to be 960 cm 2.
Answer. The area of the base of the prism is 144 cm 2. The entire surface is 960 cm 2.
No. 2. Given At the base there is a triangle with a side of 6 cm. In this case, the diagonal of the side face is 10 cm. Calculate the areas: the base and the side surface.
Solution. Since the prism is regular, its base is an equilateral triangle. Therefore, its area turns out to be 6 squared, multiplied by ¼ and the square root of 3. A simple calculation leads to the result: 9√3 cm 2. This is the area of one base of the prism.
All side faces are the same and are rectangles with sides of 6 and 10 cm. To calculate their areas, just multiply these numbers. Then multiply them by three, because the prism has exactly that many side faces. Then the area of the lateral surface of the wound turns out to be 180 cm 2.
Answer. Areas: base - 9√3 cm 2, lateral surface of the prism - 180 cm 2.