Similar polyhedra for exam tests. Area of a polyhedron where all angles are right angles
The Unified State Examination in mathematics includes a number of problems on determining the surface area and volume of composite polyhedra. This is 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.
Answer: 72
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).
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
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
u84236168 ✎ Biotic factor - the impact of living organisms on each other. Abiotic factor is the effect of the inorganic environment on living organisms (chemical and physical). A) An increase in pressure is a physical factor, therefore, we classify it as abiotic. B) Earthquake is a physical abiotic factor. C) The epidemic is caused by microorganisms, therefore there is a biotic factor here. D) The interaction of wolves in a pack is a biotic factor. D) Competition between pines is a biotic factor, because Pines are living organisms. Answer: 11222 to the problem
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 percentage 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
u84236168 ✎ A) Cactus spines and barberry spines are plant organs, an example is used in the comparative anatomical method of studying evolution. B) Remains are fossilized parts of ancient living beings, whose study is the science of paleontology, therefore, this is a paleontological method. C) Phylogenesis is the process of historical development of nature and individual organisms. In the phylogenetic series of a horse there may be its ancient ancestors, therefore, this is a paleontological method. D) Human multi-nipple refers to the comparative anatomical method, because the norm (two nipples) and atavism are compared. D) The appendix in humans is a rudiment, therefore, the norm and the rudiment are also compared here. Answer: 21122 to the problem
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) There is also no information about the genetic program on the graph, 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
u84236168 ✎ In this picture, the external auditory canal, eardrum and cochlea (as can be seen from the shape) are correctly labeled. The remaining elements: 3 - chamber of the inner ear, 4 - hammer, 5 - incus. Answer: 1, 2, 6. to the problem
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 The axial section of a cylinder is a 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 come in 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. In such a geometric figure, all faces are presented in the form of flat rectangular shapes.
- 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.
- The volume of a rectangular parallelepiped whose sides have different measurements, 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 base of the figure, another 4 are the side faces. An inclined parallelepiped differs from a straight parallelepiped in that its side faces are not at right angles to the base. 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 that have common sides with the base, is 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 the area of the base by the 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. Particular attention will have to be paid to the correct division of a complex polyhedron into simple ones.
We continue to decide problems from the open bank of Unified State Examination tasks 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.
A regular quadrangular prism with a base side of 0.4 and a side edge of 1 is cut from a unit cube. 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 twice the area of the square (with a side of 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:
When 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 greater than the surface area 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 than the area of the corresponding tetrahedron face.
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.
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