Different prisms are different from each other. At the same time, they have a lot in common. To find the area of \u200b\u200bthe base of a prism, you need to figure out what kind it looks like.
General theory
A prism is any polyhedron whose sides have the form of a parallelogram. Moreover, any polyhedron can be at its base - from a triangle to an n-gon. Moreover, the bases of the prism are always equal each other. What does not apply to the side faces - they can vary significantly in size.
When solving problems, it is not only the area of \u200b\u200bthe base of the prism that is encountered. It may be necessary to know the lateral surface, that is, all faces that are not bases. The full surface will already be the union of all the faces that make up the prism.
Sometimes heights appear in tasks. 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 area of the base of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures in the upper and lower faces, then their areas will be equal.
triangular prism
It has at the base a figure with three vertices, that is, a triangle. It is known to be different. If then it is enough to recall that its area is determined by half the product of the legs.
Mathematical notation looks like this: S = ½ av.
To find the area of the base in general view, the formulas are useful: Heron and the one in which half of the side is taken to the height drawn to it.
The first formula should be written like this: S \u003d √ (p (p-a) (p-in) (p-s)). This entry 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 know the area of the base triangular prism, which is correct, then the triangle is equilateral. It has its own formula: S = ¼ a 2 * √3.
quadrangular prism
Its base is any of the known quadrilaterals. It can be a rectangle or a square, a parallelepiped or a rhombus. In each case, in order to calculate the area of \u200b\u200bthe base of the prism, you will need your own formula.
If the base is a rectangle, then its area is determined as follows: S = av, where a, b are the sides of the rectangle.
When it comes to a quadrangular prism, the base area of a regular prism is calculated using the formula for a square. Because it is he who lies at the base. S \u003d a 2.
In the case when the base is a parallelepiped, the following equality will be needed: S \u003d a * n a. It happens that a side of a parallelepiped and one of the angles are given. Then, to calculate the height, you need to use additional formula: n a \u003d b * sin A. Moreover, the angle A is adjacent to the side "b", and the height n and opposite to this corner.
If a rhombus lies at the base of the prism, then the same formula will be needed to determine its area as for a parallelogram (since it is a special case of it). But you can also use this one: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.
Regular pentagonal prism
This case involves splitting the polygon into triangles, the areas of which are easier to find out. Although it happens that the figures can be with 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 \u200b\u200bthe 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
According to the principle described for a pentagonal prism, it is possible to divide the base hexagon into 6 equilateral triangles. The formula for the area of the base of such a prism is similar to the previous one. Only in it should be multiplied by six.
The formula will look like this: S = 3/2 and 2 * √3.
Tasks
No. 1. A regular straight line is given. Its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of \u200b\u200bthe base of the prism and the entire surface.
Solution. The base of a prism is a square, but its side is not known. 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 (n). x 2 \u003d 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 \u003d a 2 + a 2. Thus, it turns out that a 2 \u003d (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 it’s easy to find out the base area: 12 * 12 \u003d 144 cm 2.
To find out the area of \u200b\u200bthe entire surface, you need to add twice the value of the base area and quadruple the side. The latter is easy to find by 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. The total surface area of the prism is found to be 960 cm 2 .
Answer. The base area of the prism is 144 cm2. The entire surface - 960 cm 2 .
No. 2. Dana At the base lies 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 equal to 6 squared times 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, it is enough to multiply these numbers. Then multiply them by three, because the prism has exactly so many side faces. Then the area of the side surface is wound 180 cm 2 .
Answer. Areas: base - 9√3 cm 2, side surface of the prism - 180 cm 2.
The volume of the prism. Problem solving
Geometry is the most powerful tool for the refinement of our mental faculties and enables us to think and reason correctly.
G. Galileo
The purpose of the lesson:
- to teach solving problems for calculating the volume of prisms, to generalize and systematize the information that students have about the prism and its elements, to form the ability to solve problems of increased complexity;
- develop logical thinking, the ability to work independently, the skills of mutual control and self-control, the ability to speak and listen;
- develop the habit of constant employment, some useful deed, education of responsiveness, diligence, accuracy.
Type of lesson: a lesson in the application of 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 side surface of the prism (Figure 2, Figure 5).
- The height of the prism (Figure 3, Figure 4).
- Direct prism (Fig. 2,3,4).
- Inclined prism (Figure 5).
- Correct prism (Fig. 2, Fig. 3).
- Diagonal section of a prism (Fig. 2).
- Prism diagonal (Figure 2).
- Perpendicular section of the prism (pi3, fig4).
- The area of the lateral surface of the prism.
- Square full surface prisms.
- The volume of the prism.
- CHECK HOMEWORK (8 min)
- Collaboration teachers with a class (2-3 min.).
- PHYSICAL MINUTE (3 min)
- PROBLEM SOLVING (10 min)
- Independent work students on a test at a computer
Exchange notebooks, check the solution on the slides and mark the mark (mark 10 if the task is composed)
Draw a problem and solve it. The student defends the problem he has compiled at the blackboard. Figure 6 and Figure 7.
Chapter 2, §3
Task.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 direct prism ABCA 1B 1C1 is right triangle ABC (angle ABC=90°), AB=4cm. Calculate the volume of the prism if the radius of the circumscribed triangle ABC is 2.5cm and the height of the prism is 10cm. (Figure 9).
Chapter 2, § 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 results of the theoretical warm-up (students put down marks to each other), learning how to solve problems on the topic.
On this stage the teacher organizes frontal work on repetition of methods for solving planimetric problems, planimetry formulas. The class is divided into two groups, some solve problems, others work at the computer. Then they change. Students are invited to solve all No. 8 (orally), No. 9 (orally). After they are divided into groups and transgress to solve problems No. 14, No. 30, No. 32.
Chapter 2, §3, page 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 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 side of the base. Calculate the volume of the prism if the radius of a circle circumscribed about the section of the prism by a plane passing through the side of the base and the midpoint of the opposite lateral rib, equal to cm. (Fig. 12)
Chapter 2, §3, page 66-67
Task 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 the large diagonal of the lower base, if the volume of the prism is equal and all side faces are square (Fig. 13).
Chapter 2, §3, page 66-67
Problem 30.ABCA 1 B 1 C 1 is a regular triangular prism, all the edges of which are equal to each other, the point about the middle of the 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 (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 circumscribed near the section of the prism by a plane passing through 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 , 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, the base of which is a right triangle, is equal to the product of the base area and the height.
2) The volume of a regular triangular prism is calculated by the formula V \u003d 0.25a 2 h - where a is the side of the base, h is the height of the prism.
3) The volume of a straight prism is equal to half the product of the area of \u200b\u200bthe base and the height.
4) The volume of a regular quadrangular prism is calculated by the formula V \u003d a 2 h-where a is the side of the base, h is the height of the prism.
5) The volume of a regular hexagonal prism is calculated by the formula V \u003d 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. A plane is drawn through the side of the lower base and the opposite top of 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 straight 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.
In physics, a triangular prism made of glass is often used to study the spectrum white light, because it is able to decompose it into separate components. In this article, we will consider the volume formula
What is a triangular prism?
Before giving the volume formula, consider the properties of this figure.
To get this, you need to take a triangle of arbitrary shape and move it parallel to itself for a certain distance. The vertices of the triangle in the initial and final positions should be connected by straight segments. The resulting three-dimensional figure is called a triangular prism. It has five sides. Two of them are called bases: they are parallel and equal to each other. The bases of the considered prism are triangles. The three remaining sides are parallelograms.
In addition to the sides, the prism under consideration 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 can also be concave).
This is a rectangular figure, at the base of which lies an equilateral triangle.
Volume of a triangular prism of a general type
How to find the volume of a triangular prism? The formula in general terms is similar to that for a prism of any kind. 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 a 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 on which this height is lowered.
As for the height h of the figure, it is easiest to find it for a rectangular prism. In the latter case, h coincides with the length of the side edge.
Volume of a regular triangular prism
The general formula for 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:
Everyone can get this formula if they remember that in an equilateral triangle all angles are equal to each other and make up 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 has nothing to do with the base of a regular prism and can take arbitrary values. As a result, the formula for the volume of a triangular prism of the correct form looks like this:
Having calculated the root, we 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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In the school curriculum for the course of solid geometry, the study of three-dimensional figures usually begins with a simple geometric body - a prism polyhedron. The role of its bases is performed by 2 equal polygons lying in parallel planes. A special case is a regular quadrangular prism. Its bases are 2 identical regular quadrilaterals, to which the sides are perpendicular, having the shape of parallelograms (or rectangles if the prism is not inclined).
What does a prism look like
A regular quadrangular prism is a hexagon, at the bases of which there are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure- a straight parallelepiped.
A drawing showing a quadrangular prism is shown below.
You can also see in the picture the most important elements that make up a geometric body. They are commonly referred to as:
Sometimes in problems in geometry you can find the concept of a section. The definition will sound like this: a section is all points of a volumetric body that belong to the cutting plane. The section is perpendicular (crosses the edges of the figure at an angle of 90 degrees). For a rectangular prism, a diagonal section is also considered (the maximum number of sections that can be built is 2), passing through 2 edges and the diagonals of the base.
If the section is drawn in such a way that the cutting plane is not parallel to either the bases or the side faces, the result is a truncated prism.
Various ratios and formulas are used to find the reduced prismatic elements. Some of them are known from the course of planimetry (for example, to find the area of the base of a prism, it is enough to recall the formula for the area of a square).
Surface area and volume
To determine the volume of a prism using the formula, you need to know the area of \u200b\u200bits base and height:
V = Sprim h
Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in a more detailed form:
V = a² h
If we are talking about a cube - a regular prism with equal length, width and height, the volume is calculated as follows:
To understand how to find the lateral surface area of a prism, you need to imagine its sweep.
It can be seen from the drawing that side surface made up of 4 equal rectangles. Its area is calculated as the product of the perimeter of the base and the height of the figure:
Sside = Pos h
Since the perimeter of a square is P = 4a, the formula takes the form:
Sside = 4a h
For cube:
Sside = 4a²
To calculate the total surface area of a prism, add 2 base areas to the side area:
Sfull = Sside + 2Sbase
As applied to a quadrangular regular prism, the formula has the form:
Sfull = 4a h + 2a²
For the surface area of a cube:
Sfull = 6a²
Knowing the volume or surface area, you can calculate the individual elements of a geometric body.
Finding prism elements
Often there are problems in which the volume is given or the value of the lateral surface area is known, where it is necessary to determine the length of the side of the base or the height. In such cases, formulas can be derived:
- base side length: a = Sside / 4h = √(V / h);
- height or side rib length: h = Sside / 4a = V / a²;
- base area: Sprim = V / h;
- side face area: Side gr = Sside / 4.
To determine how much area a diagonal section has, you need to know the length of the diagonal and the height of the figure. For a square d = a√2. Therefore:
Sdiag = ah√2
To calculate the diagonal of the prism, the formula is used:
dprize = √(2a² + h²)
To understand how to apply the above ratios, you can practice and solve a few simple tasks.
Examples of problems with solutions
Here are some of the tasks that appear in the state final exams in mathematics.
Exercise 1.
Sand is poured into a box shaped like a regular quadrangular prism. The height of its level is 10 cm. What will the level of sand be if you move it into a container of the same shape, but with a base length 2 times longer?
It should be argued as follows. The amount of sand in the first and second containers did not change, i.e., its volume in them is the same. You can define the length of the base as a. In this case, for the first box, the volume of the substance will be:
V₁ = ha² = 10a²
For the second box, the length of the base is 2a, but the height of the sand level is unknown:
V₂ = h(2a)² = 4ha²
Because the V₁ = V₂, the expressions can be equated:
10a² = 4ha²
After reducing both sides of the equation by a², we get:
As a result, the new sand level will be h = 10 / 4 = 2.5 cm.
Task 2.
ABCDA₁B₁C₁D₁ is a regular prism. It is known that BD = AB₁ = 6√2. Find the total surface area of the body.
To make it easier to understand which elements are known, you can draw a figure.
Since we are talking about a regular prism, we can conclude that the base is a square with a diagonal of 6√2. The diagonal of the side face has the same value, therefore, side face also has the shape of a square, equal to the base. It turns out that all three dimensions - length, width and height - are equal. We can conclude that ABCDA₁B₁C₁D₁ is a cube.
The length of any edge is determined through the known diagonal:
a = d / √2 = 6√2 / √2 = 6
The total surface area is found by the formula for the cube:
Sfull = 6a² = 6 6² = 216
Task 3.
The room is being renovated. It is known that its floor has the shape of a square with an area of 9 m². The height of the room is 2.5 m. What is the lowest cost of wallpapering a room if 1 m² costs 50 rubles?
Since the floor and ceiling are squares, that is, regular quadrangles, and its walls are perpendicular to horizontal surfaces, we can conclude that it is a regular prism. It is necessary to determine the area of its lateral surface.
The length of the room is a = √9 = 3 m.
The square will be covered with wallpaper Sside = 4 3 2.5 = 30 m².
The lowest cost of wallpaper for this room will be 50 30 = 1500 rubles.
Thus, to solve problems for a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and a rectangle, as well as to know the formulas for finding the volume and surface area.
How to find the area of a cube
