Find the volume of the polyhedron; all dihedral angles are right. How to find the volume of a polyhedron

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Find the volume of the spatial cross shown in the figure and made up of unit cubes.

Find the surface area of ​​the spatial cross shown in the figure and composed of unit cubes.

The surface area of ​​a cube is 18. Find its diagonal.

The volume of a cube is 8. Find its surface area.

If each edge of a cube is increased by 1, its surface area increases by 54. Find the edge of the cube.

36) No. 27098_ The diagonal of the cube is . Find its volume.

37) No. 27099_ The volume of the cube is . Find its diagonal.

If each edge of a cube is increased by 1, then its volume will increase by 19. Find the edge of the cube.

The diagonal of a cube is 1. Find its surface area.

The surface area of ​​the cube is 24. Find its volume.

The volume of one cube is 8 times the volume of another cube. How many times the surface area of ​​the first cube? more area surface of the second cube?

The two edges of a cuboid coming from the same vertex are 3 and 4. The surface area of ​​this cuboid is 94. Find the third edge coming from the same vertex.

The two edges of a cuboid extending from the same vertex are 1, 2. The surface area of ​​the cuboid is 16. Find its diagonal.

The area of ​​the face of a rectangular parallelepiped is 12. The edge perpendicular to this face is 4. Find the volume of the parallelepiped.

The volume of a rectangular parallelepiped is 24. One of its edges is 3. Find the area of ​​the face of the parallelepiped perpendicular to this edge.

The volume of a rectangular parallelepiped is 60. The area of ​​one of its faces is 12. Find the edge of the parallelepiped perpendicular to this face.

The two edges of a cuboid coming from the same vertex are 2 and 6. The volume of the cuboid is 48. Find the third edge of the cuboid coming from the same vertex.

Three edges of a rectangular parallelepiped coming from one vertex are equal to 4, 6, 9. Find an edge of a cube of equal size.

The two edges of a rectangular parallelepiped extending from the same vertex are 2, 4. The diagonal of the parallelepiped is 6. Find the volume of the parallelepiped.

Two edges of a rectangular parallelepiped coming from one vertex are equal to 2, 3. The volume of the parallelepiped is 36. Find its diagonal.

51) No. 27103_ The diagonal of a rectangular parallelepiped is equal to and forms angles of 30, 30 and 45 with the planes of the faces of the parallelepiped. Find the volume of the parallelepiped.

The edges of a cuboid extending from one vertex are 1, 2, 3. Find its surface area.

The two edges of a cuboid extending from the same vertex are 2, 4. The diagonal of the cuboid is 6. Find the surface area of ​​the cuboid.

The two edges of a rectangular parallelepiped extending from the same vertex are 1, 2. The volume of the parallelepiped is 6. Find its surface area.

55) No. 27104_ The face of a parallelepiped is a rhombus with a side of 1 and an acute angle of 60. One of the edges of the parallelepiped makes an angle of 60 with this face and is equal to 2. Find the volume of the parallelepiped.

Find the surface area of ​​a straight prism whose base is a rhombus with diagonals equal to 6 and 8 and a lateral edge equal to 10.

Find the lateral edge of a regular quadrangular prism if the side of its base is 20 and its surface area is 1760.

At the base of a straight prism lies a rhombus with diagonals equal to 6 and 8.

Its surface area is 248. Find the side edge of this prism.

Straight base triangular prism serves as a right triangle with legs 6 and 8, the side edge is equal to 5. Find the volume of the prism.

The base of a right triangular prism is a right triangle with legs 3 and 5. The volume of the prism is 30. Find its side edge.

The base of a right triangular prism is a right triangle with legs 6 and 8, the height of the prism is 10. Find its surface area.

The base of a right triangular prism is a right triangle with legs 6 and 8. Its surface area is 288. Find the height of the prism.

In a triangular prism, the two side faces are perpendicular. Their common edge is 10 and is separated from the other lateral edges by 6 and 8. Find the lateral surface area of ​​this prism.

Through the midline of the base of a triangular prism, the lateral surface area of ​​which is 24, a plane parallel to lateral rib. Find the lateral surface area of ​​the trimmed triangular prism.

"We have already considered the theoretical points that are necessary for the solution.

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 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 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.

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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).

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. 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, 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 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 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 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.