The Unified State Examination in mathematics includes whole line problems to determine the surface area and volume of composite polyhedra. This is probably one of the most simple tasks by stereometry. BUT! There is a nuance. Despite the fact that the calculations themselves are simple, it is very easy to make a mistake when solving such a problem.
What's the matter? Not everyone has good spatial thinking to immediately see all the faces and parallelepipeds that make up polyhedra. Even if you know how to do this very well, you can mentally make such a breakdown, you should still take your time and use the recommendations from this article.
By the way, while I was working on this material, I found an error in one of the tasks on the site. You need attentiveness and attentiveness again, like this.
So, if the question is about surface area, then on a sheet of paper in a checkerboard, draw all the faces of the polyhedron and indicate the dimensions. Next, carefully calculate the sum of the areas of all the resulting faces. If you are extremely careful when constructing and calculating, the error will be eliminated.
We use the specified method. It's visual. On a checkered sheet we build all the elements (edges) to scale. If the lengths of the ribs are large, then simply label them.
Answer: 72
Decide for yourself:
Find the surface area of the polyhedron shown in the figure (all dihedral angles straight).
Find the surface area of the polyhedron shown in the figure (all dihedral angles are right angles).
Find the surface area of the polyhedron shown in the figure (all dihedral angles are right angles).
More tasks... They provide solutions in a different way (without construction), try to figure out what came from where. Also solve using the method already presented.
* * *
If you need to find the volume of a composite polyhedron. We divide the polyhedron into its constituent parallelepipeds, carefully record the lengths of their edges and calculate.
Volume of the polyhedron shown in the figure equal to the sum volumes of two polyhedra with edges 6,2,4 and 4,2,2
Answer: 64
Decide for yourself:
Find the volume of the polyhedron shown in the figure (all dihedral angles of the polyhedron are right angles).
Find the volume of the spatial cross shown in the figure and made up of unit cubes.
Find the volume of the polyhedron shown in the figure (all dihedral angles are right angles).
Latest solutions
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u84236168 ✎ 1) The table shows that if there are more than 5 chicks in the nest, then the proportion of surviving chicks decreases sharply, therefore, we agree with this statement. 2) The death of the chicks is not explained in any way in the table, therefore, we cannot say anything about this statement. 3) Yes, the table shows that the fewer eggs in the clutch, the higher the care for the offspring, so, the most high percent surviving chicks (100%) correlates with their smallest number (1), so we agree with this statement. 4) Regarding the fourth statement, we do not have any accurate information + the proportion of surviving chicks is decreasing, which means we do not agree with this statement. 5) The table does not contain information on what the number of eggs in a clutch is related to, therefore, we ignore this statement. Answer: 1, 3. to the problem
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u84236168 ✎ 1) The speed cannot be directly proportional, otherwise, as the temperature decreases, the speed would strictly increase, which we do not observe on the graph. 2) The graph does not say anything about environmental resources, so we cannot say anything about this statement. 3) Pro genetic program There is no information on the graph either, therefore, we cannot say anything. 4) The graph shows that the reproduction rate increases in the interval from 20 to 36 degrees, then we agree with this statement. 5) The graph shows that after 36 degrees the speed drops, which means we agree with this statement. Answer: 4, 5. to the problem
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SURFACE AREA OF A POLYHEDON The surface area of a polyhedron, by definition, is the sum of the areas included in this surface of the polygons. The surface area of a prism consists of the area of the lateral surface and the areas of the bases. The surface area of a pyramid consists of the lateral surface area and the base area.
Find the surface area of the polyhedron shown in the figure, all of whose dihedral angles are right angles. Answer. 22. Solution. The surface of a polyhedron consists of two squares of area 4, four rectangles of area 2 and two non-convex hexagons of area 3. Therefore, the surface area of the polyhedron is 22. Exercise 6
Find the surface area of the polyhedron shown in the figure, all of whose dihedral angles are right angles. Answer. 22. Solution. The surface of a polyhedron consists of two squares of area 4, four rectangles of area 2, and two non-convex hexagons of area 3. Therefore, the surface area of the polyhedron is 22. Exercise 7
Find the surface area of the polyhedron shown in the figure, all of whose dihedral angles are right angles. Answer. 22. Solution. The surface of a polyhedron consists of two squares of area 4, four rectangles of area 2 and two non-convex hexagons of area 3. Therefore, the surface area of the polyhedron is 22. Exercise 8
Answer. 38. Solution. The surface of a polyhedron consists of a square with area 9, seven rectangles with area 3, and two non-convex octagons with area 4. Therefore, the surface area of the polyhedron is 38. Exercise 9
Find the surface area of the polyhedron shown in the figure, all of whose dihedral angles are right angles. Answer. 24. Solution. The surface of a polyhedron consists of three squares of area 4, three squares of area 1, and three non-convex hexagons of area 3. Therefore, the surface area of the polyhedron is 24. Exercise 10
Find the surface area of the polyhedron shown in the figure, all of whose dihedral angles are right angles. Answer. 92. Solution. The surface of a polyhedron consists of two squares of area 16, a rectangle of area 12, three rectangles of area 4, two rectangles of area 8, and two non-convex octagons of area 10. Therefore, the surface area of the polyhedron is 92. Exercise 11
29
Exercise 26 Axial section cylinder - square. The area of the base is 1. Find the surface area of the cylinder. Answer: 6.
The radii of the two balls are 6 and 8. Find the radius of a ball whose surface area is equal to the sum of their surface areas. Answer. 10. Solution. The surface areas of these balls are equal to and. Their sum is equal. Therefore, the radius of a ball whose surface area is equal to this sum is 10. Exercise 30
"We have already considered the theoretical points that are necessary for solving. The Unified State Examination in mathematics contains a number of problems on determining the surface area and volume of composite polyhedra. These are probably one of the simplest problems in stereometry. BUT! There is a nuance. Despite the fact that the calculations themselves are simple, it is very easy to make a mistake when solving such a problem.
What's the matter? Not everyone has good spatial thinking to immediately see all the faces and parallelepipeds that make up polyhedra. Even if you know how to do this very well, you can mentally make such a breakdown, you should still take your time and use the recommendations from this article.
By the way, while I was working on this material, I found an error in one of the tasks on the site. You need attentiveness and attentiveness again, like this.
So, if the question is about surface area, then on a sheet of paper in a checkerboard, draw all the faces of the polyhedron and indicate the dimensions. Next, carefully calculate the sum of the areas of all the resulting faces. If you are extremely careful when constructing and calculating, the error will be eliminated.
We use the specified method. It's visual. On a checkered sheet we build all the elements (edges) to scale. If the lengths of the ribs are large, then simply label them.
Decide for yourself:
Find the surface area of the polyhedron shown in the figure (all dihedral angles are right angles).
Find the surface area of the polyhedron shown in the figure (all dihedral angles are right angles).
More tasks... They provide solutions in a different way (without construction), try to figure out what came from where. Also solve using the method already presented.
If you need to find the volume of a composite polyhedron. We divide the polyhedron into its constituent parallelepipeds, carefully record the lengths of their edges and calculate.
The volume of the polyhedron shown in the figure is equal to the sum of the volumes of two polyhedra with edges 6,2,4 and 4,2,2
Decide for yourself:
Find the volume of the polyhedron shown in the figure (all dihedral angles of the polyhedron are right angles).
First of all, let's define what a polyhedron is. This is a three-dimensional geometric figure, the edges of which are presented in the form of flat polygons. There is no single formula for finding the volume of a polyhedron, since polyhedra can be different shapes. In order to find the volume of a complex polyhedron, it is conditionally divided into several simple ones, such as a parallelepiped, a prism, a pyramid, and then the volumes of simple polyhedra are added up and the desired volume of the figure is obtained.
How to find the volume of a polyhedron - parallelepiped
First, let's find the area of a rectangular parallelepiped. This one has geometric figure all faces are presented in the form of flat rectangular figures.
- The simplest rectangular parallelepiped is a cube. All edges of the cube are equal to each other. In total, such a parallelepiped has 6 faces, that is, 6 identical squares. The volume of such a figure is calculated as follows:
where a is the length of any edge of the cube.
- Volume rectangular parallelepiped, the sides of which have different dimensions, is calculated using the following formula:
where a, b and c are the lengths of the ribs.
How to find the volume of a polyhedron - an inclined parallelepiped
An inclined parallelepiped also has 6 faces, 2 of them are the bases of the figure, 4 more are side faces. Inclined parallelepiped differs from direct topics, that its side edges in relation to the base are not located at right angles. The volume of such a figure is calculated as the product between the area of the base and the height:
where S is the area of the quadrilateral lying at the base, h is the height of the desired figure.
How to find the volume of a polyhedron - prism
A three-dimensional geometric figure, the base of which is represented by a polygon of any shape, and the side faces are parallelograms having common aspects with a base - called a prism. A prism has two bases, and there are as many side faces as there are sides to the figure that is the base.
To find the volume of any prism, both straight and inclined, multiply base area to height:
where S is the area of the polygon at the base of the figure, and h is the height of the prism.
How to find the volume of a polyhedron - a pyramid
If there is a polygon at the base of the figure, and the side faces are presented in the form of triangles meeting at a common vertex, then such a figure is called a pyramid. It differs from the above figures in that it has only one base, in addition to this, it has a top. To find the volume of a pyramid, multiply its base by its height and divide the result by 3:
here S is the base area of the desired geometric figure, and h is the height.
It is quite easy to find the area of a simple polyhedron; it is much more difficult to find the area of a figure consisting of many polyhedra. Special attention you will have to pay attention to correctly dividing a complex polyhedron into simple ones.
We continue to decide tasks from open bank Unified State Exam assignments in mathematics category “No. 8” . Today we are looking at problems that involve compound polyhedra. (We have already encountered problems on composite polyhedra).
Task 1.
Find the surface area of the polyhedron shown in the figure (all dihedral angles are right angles).
Solution:
The surface area of a polyhedron is equal to the difference between the surface area of a rectangular parallelepiped with dimensions 3, 3 and 2 and two areas of 1x1 squares.
Task 2.
The correct one is cut from a unit cube quadrangular prism with a base side of 0.4 and a side edge of 1. Find the surface area of the remaining part of the cube.
Solution:
The surface area of the remaining part of the cube is the sum of the surface area of the cube (edge 1) and the area of the lateral surface of the prism, reduced by double area square (with side 0.4).
Answer: 7.28.
Task 3.
How many times will the surface area of the octahedron increase if all its edges are increased by 6 times?
Solution:
If all edges are increased by 6 times, the area of each face will change by 36 times, therefore the sum of the areas of all faces (surface area) of the enlarged octahedron will be 36 times more area surface of the original octahedron.
Task 4.
The surface area of a tetrahedron is 1. Find the surface area of a polyhedron whose vertices are the midpoints of the sides of the given tetrahedron.
Solution:
The surface of the required polyhedron consists of 8 faces - triangles.
The area of each such triangle from a pair (highlighted in the same color in the figure)
4 times less area the corresponding face of the tetrahedron.
Then the sum of the areas of the faces of the polyhedron is half the surface of the tetrahedron. That is
Answer: 0.5.
You can also watch the video for task 4:
Task 5.
Find the volume of the spatial cross shown in the figure and made up of unit cubes.
Solution:
The volume of this spatial cross is 7 volumes of unit cubes. That's why
Task 6.
Find the volume of the polyhedron shown in the figure (all dihedral angles are right angles).
Solution:
The volume of a given polyhedron is the volume of a cuboid with dimensions 3, 6 and 2 without the volume of a cuboid with dimensions 1, 2, 2.
Task 7.
The volume of a tetrahedron is 1.5. Find the volume of a polyhedron whose vertices are the midpoints of the sides of the given tetrahedron.
Lesson objectives:
learning objectives:
Summarize knowledge about polyhedra and their elements;
Repeat the formulas for calculating the surface areas of a rectangular parallelepiped, prism, pyramid;
Strengthen practical skills in calculating the surface areas of polyhedra;
Expand knowledge in the field of mathematics, show its applied nature;
development goals:
Development of cognitive interest, logical thinking, spatial imagination and research abilities;
Development of computing skills, key competencies, abilities to compare, analyze, and justify the choice made;
educational goals:
Developing responsibility, the ability to work in a team, and make independent decisions;
Show the significance of knowledge and the possibility of its application in practice.
Lesson type: lesson on the integrated application of knowledge.
Equipment: models of polyhedrons, task cards, presentation, office equipment, reflection forms, dummy boxes with gifts.
During the classes
- Organizational moment (2 min.).
- Updating knowledge (4 min.).
- Statement of the problem, definition of lesson goals (8 min.).
- Conducting research work (27 min.).
- Reflection (4 min.).
Lesson epigraph: What makes sense is the knowledge that is used in practice. (Tom Bradesford)
Lesson summary.
Good afternoon
Good afternoon to smiling faces!
May the good day last until the evening!
I invite you to a math lesson.
We begin the countdown of our 45-minute life together. It depends on me and you how interesting and useful this period of life will be.
Here are case folders containing materials for our lesson.
Find “MG Score Sheet”. Select a person from the group who will fill it out. Fill out 1 column, write the names of the students in your MG (microgroup). In the next column, give a plus sign to whoever from your group answers the question correctly next questions and so on throughout our lesson.
Please look at the table, what do you see? (Student answers: prism, cuboid, pyramid).
What do we call them? (Student answers: polyhedra).
And, indeed, the topic “Polyhedra” is the main one in the stereometry course. We see multifaceted shapes every day. This is a matchbox, a juice box, a milk box, a book, a room, multi-story buildings, Kremlin towers, Egyptian pyramids.
Let's see what surfaces this prism consists of? (Student answers: from the bases and the lateral surface).
What surfaces does this pyramid consist of? (Student answers: from the base and side surface).
How can you find the area full surface this parallelepiped? (Students' answers).
A this pyramid? (Students' answers).
So, what will the topic of today's lesson be? (Students' answers).“Surface areas of polyhedra.” (Slide 1).
Why this particular topic? You, future employees of trade enterprises, will more than once encounter polyhedrons in your professional field. For example, when placing boxes of goods in a warehouse, on the sales floor; when wrapping gifts.
So imagine this situation: You work as a sales consultant in the Gifts store. You need to collect and pack a gift for your buyer, which consists of 4 items. The amount of the gift should not exceed 15,000 rubles. You do not charge money for packaging materials. Therefore, the less you spend, the greater your profit.
Hence arises contradiction: on the one hand, gift wrapping, and on the other hand, saving on packaging material. Tell me, what problem arises from this? Work in MG. (Students' answers).
To summarize your statements, let us formulate problem: to save packaging material, in the form of which polyhedron should you pack a gift - a rectangular parallelepiped or a pyramid?
Suggest solutions to this problem. (Work in MG)
To resolve this problem, we will work with you research mode, which has a certain structure.
It's there hypothesis: Let's assume that if we pack a gift in the form of a pyramid, then the consumption of packaging material will be less.
What will you research? (Work in MG)
Object of research: Gift set - a kind of polyhedron (parallelepiped, pyramid - from the point of view of mathematics).
What subject does the research relate to? (Work in MG)
Subject of study: mathematics.
Why are we doing research? (Work in MG)
Purpose of the study: to calculate the total surface area of a rectangular parallelepiped and a pyramid.
What should you do for this? (Work in MG)
For work, I propose the following algorithm.
Research objectives:
Choose up to 4 gifts worth no more than 15,000 rubles;
- “put” the gifts together (in the form of a certain polygon);
Determine the basic formulas for work;
Take the necessary measurements;
Calculate the areas of polyhedra;
Make a comparative analysis;
Formulate conclusions.
Each group will work on their own gift. Find in the case folders a sign with the name of the gift (for new arrivals, for anniversaries, for newlyweds).
Choose gifts. The list of gifts is in the case-folders, mark the selected products in the table, and immediately select from the display case. (Work in MG)
Find the “Work Algorithm” table in the case folders and work according to it. For work we set the time to 18 minutes. When you're done, prepare your speakers.
Conclusion. Returning to our goals and problem, tell me, in what form is it more economical to pack a gift, in the form of a rectangular parallelepiped or a pyramid? (Students' answers). - Who will answer what this is connected with? (Students' answers).
This means that it follows that the put forward the hypothesis was refuted.
Refer back to score sheets. Note what contribution each of you made to the common cause? How successful were you in phases 1 and 2 of the study? Rate everyone, put a rating in the “Group Rating” column. Give the evaluation sheets to the teacher. (The teacher comments on the grades of individual students)
Now I’ll ask you to fill out the reflection sheet.
Attention to the screen. The period of our life together is ending. I would like to end the lesson with the words of Tom Braidsford: “It is the knowledge that is used in practice that makes sense.” Thank you for the lesson. Thank you for attention. Good afternoon until the evening.
Handout.
MG score sheet
| F.I. students | Answers on questions | Research | Performance | Grade | ||
| Stage 1 | Stage 2 | groups | final | |||
| 1 | ||||||
| 2 | ||||||
| 3 | ||||||
| 4 | ||||||
| 5 | ||||||
Situation: You work as a sales consultant in the Gifts store. You need to collect and pack a gift for your buyer, which consists of 4 names. The gift amount should not exceed 15000 rubles
Name of gifts for MG
- Gift for new residents.
- Gift for newlyweds.
- Gift for anniversaries.
The price of the product
| № | Gift name | Price (in rubles) |
| 1 | Iron | 800-00 |
| 2 | Bedding set | 1000-00 |
| 3 | Microwave | 3500-00 |
| 4 | Landline phone | 500-00 |
| 5 | Chocolate | 200-00 |
| 6 | Food processor | 2500-00 |
| 7 | Cellular telephone | 7000-00 |
| 8 | Satellite dish “Tricolor TV” | 9000-00 |
| 9 | DVD player | 2000-00 |
| 10 | Camera | 1500-00 |
| 11 | Hair clipper | 1700-00 |
| 12 | Vacuum cleaner | 4500-00 |
| 13 | Tea-set | 1000-00 |
Work algorithm
| Select gifts | ||||
| Name | 1 | 2 | 3 | 4 |
| “Stack” gifts | ||||
| Determine the basic formulas for finding the area of surfaces | ||||
| Name of the polyhedron | Formula | Designation | ||
| Rectangular parallelepiped | S= | S - total surface area, | ||
| Quadrangular pyramid | S= | S - total surface area, S base - base area, S b - lateral surface area, |
||
| Regular quadrangular pyramid | S b= | |||
| Measure the linear dimensions of the gift | ||||
a) A gift in the form of a rectangular parallelepiped |
width | length | height | Calculate area |
b) Gift in the form of a pyramid |
||||
| The shape of the gift was chosen in the form of ..., because the total surface area of ... turned out to be ... than that of ... | ||||
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