How to find the volume of a straight prism using its base. Prism base area: from triangular to polygonal
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What is the volume of a prism and how to find it
The volume of a prism is the product of the area of its base and its height.
However, we know that at the base of the prism there can be a triangle, a square or some other polyhedron.
Therefore, to find the volume of a prism, you simply need to calculate the area of the base of the prism, and then multiply this area by its height.
That is, if there is a triangle at the base of the prism, then first you need to find the area of the triangle. If the base of the prism is a square or other polygon, then first you need to look for the area of the square or other polygon.
It should be remembered that the height of the prism is the perpendicular drawn to the bases of the prism.
What is a prism
Now let's remember the definition of a prism.
A prism is a polygon, two faces (bases) of which are in parallel planes, and all edges located outside these faces are parallel.
To put it simply:
A prism is any geometric figure that has two equal bases and flat faces.
The name of a prism depends on the shape of its base. When the base of a prism is a triangle, then such a prism is called triangular. A polyhedral prism is a geometric figure whose base is a polyhedron. Also, a prism is a type of cylinder.
What types of prisms are there?
If we look at the picture above, we will see that prisms are straight, regular and oblique.
Exercise
1. Which prism is called correct?
2. Why is it called that?
3. What is the name of a prism whose bases are regular polygons?
4. What is the height of this figure?
5. What is the name of a prism whose edges are not perpendicular?
6. Define a triangular prism.
7. Can a prism be a parallelepiped?
8. What geometric figure is called a semiregular polygon?
What elements does a prism consist of?
A prism consists of elements such as a lower and upper base, side faces, edges and vertices.
Both bases of the prism lie in planes and are parallel each other.
The side faces of the pyramid are parallelograms.
The lateral surface of a pyramid is the sum of its lateral faces.
The common sides of the side faces are nothing more than the side edges of a given figure.
The height of the pyramid is the segment connecting the planes of the bases and perpendicular to them.
Prism properties
A geometric figure, like a prism, has a number of properties. Let's take a closer look at these properties:
Firstly, the bases of a prism are equal polygons;
Secondly, the side faces of a prism are presented in the form of a parallelogram;
Thirdly, this geometric figure the edges are parallel and equal;
Fourthly, the total surface area of the prism is:
Now let's look at the theorem that provides the formula used to calculate the lateral surface area and proof.
Have you ever thought about this interesting fact that a prism can be not only geometric body, but also other objects around us. Even an ordinary snowflake, depending on temperature regime can turn into an ice prism, taking the shape of a hexagonal figure.
But calcite crystals have such a unique phenomenon as breaking up into fragments and taking on the shape of a parallelepiped. And what’s most amazing is that no matter how small the calcite crystals are crushed into, the result is always the same: they turn into tiny parallelepipeds.
It turns out that the prism has gained popularity not only in mathematics, demonstrating its geometric body, but also in the field of art, since it is the basis of paintings created by such great artists as P. Picasso, Braque, Griss and others.
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).
- An inclined prism (Figure 5).
- The correct prism (Fig. 2, Fig. 3).
- Diagonal section of the prism (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 between teacher and class (2-3 min.).
- PHYSICAL MINUTE (3 min)
- PROBLEM SOLVING (10 min)
- While solving problems, students compare their answers with those shown by the teacher. This is a sample solution to a problem with detailed comments... Individual work of a teacher with “strong” students (10 min.). Independent work
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 described around triangle ABC is 2.5 cm and the height of the prism is 10 cm. (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 grade each other), learning how to solve problems on the 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 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, pages 66-67
Problem 14 The base of a straight prism is a rhombus, one of the diagonals of which is equal to its side.
Chapter 2, §3, pages 66-67
Calculate the perimeter of the section with a plane passing through the major diagonal of the lower base, if the volume of the prism is equal and all side faces are squares (Fig. 13). Problem 30
Chapter 2, §3, pages 66-67
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). 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).
students working on a test at the computer
1) 152) 45 3) 104) 125) 18
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.
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) The volume of a straight prism is equal to half the 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) The volume of a regular hexagonal prism is calculated by the formula V = 1.5a 2 h, where a is the side of the base, h is the height of the prism.
1) 92) 9 3) 4,54) 2,255) 1,125
3. The side of the base of a regular triangular prism is equal to .
In the school curriculum for a stereometry course, the study of three-dimensional figures usually begins with a simple geometric body - the polyhedron of a prism. 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 quadrangles, 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, the bases of which are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure is a straight parallelepiped.
A drawing showing a quadrangular prism is shown below.
You can also see in the picture essential elements, of which the geometric body consists. These include:
Sometimes in geometry problems you can come across the concept of a section. The definition will sound like this: a section is all the points of a volumetric body belonging to a cutting plane. The section can be perpendicular (intersects 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 constructed 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.
To find the reduced prismatic elements, various relations and formulas are used. Some of them are known from the planimetry course (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 its base and height:
V = Sbas h
Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in 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 development.
From the drawing it can be seen that the side surface is 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 = Posn h
Taking into account that the perimeter of the square is equal to P = 4a, the formula takes the form:
Sside = 4a h
For cube:
Sside = 4a²
To calculate the total surface area of the prism, you need to add 2 base areas to the lateral area:
Sfull = Sside + 2Smain
In relation to a quadrangular regular prism, the formula looks like:
Stotal = 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, the formulas can be derived:
- base side length: a = Sside / 4h = √(V / h);
- height or side rib length: h = Sside / 4a = V / a²;
- base area: Sbas = V / h;
- side face area: Side gr = Sside / 4.
To determine how much area the 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 a prism, use the formula:
dprize = √(2a² + h²)
To understand how to apply the given relationships, you can practice and solve several simple tasks.
Examples of problems with solutions
Here are some tasks found on 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 sand level be if you move it into a container of the same shape, but with a base twice as long?
It should be reasoned 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 denote the length of the base by 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, we can equate the expressions:
10a² = 4ha²
After reducing both sides of the equation by a², we get:
As a result new level sand will be h = 10 / 4 = 2.5 cm.
Task 2.
ABCDA₁B₁C₁D₁ — correct 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 at the base there is a square with a diagonal of 6√2. The diagonal of the side face has the same size, therefore, the 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 a known diagonal:
a = d / √2 = 6√2 / √2 = 6
The total surface area is found using the formula for a 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, i.e. 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 area 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 involving a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and rectangle, as well as to know the formulas for finding the volume and surface area.
How to find the area of a cube
Schoolchildren who are preparing to take the Unified State Exam in mathematics 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 many students consider such geometry tasks to be quite difficult.
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 regular prism is one whose side edges are perpendicular to the base in which the regular polygon is located. The side faces of this figure are equal rectangles. A correct prism is always straight.
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Specialists of the Shkolkovo educational project propose 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 "Catalog" section presents large selection exercises of varying degrees of difficulty.
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.
