The volume of a polyhedron is that all dihedral angles are right angles. How to find the volume of a polyhedron

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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).

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).

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).

Polygons are flat geometric shapes. Three-dimensional (three-dimensional) geometric figures include.

Definition. A polyhedron is a geometric spatial body bounded on all sides by a finite number of flat polygons (faces).

A rectangular parallelepiped is . The simplest rectangular parallelepiped is a cube. All its edges are equal

Each face of a rectangular parallelepiped is a rectangle, which has a common side and two common vertices with the adjacent face.

A parallelepiped has 8 vertices, 4 side rectangles and 2 base rectangles. A cube has all six faces - equal squares. A rectangular parallelepiped has side figures and bases that are rectangles. These rectangles are equal in pairs (the rectangles of the bases and two pairs of opposite rectangles that make up the side faces are equal). Consequently, the faces of a rectangular parallelepiped are three types of rectangles, differing in size.

Three rectangles with different sizes have
one common point - the vertex of the parallelepiped.

At each vertex the box has a common point for three segments, which are called the dimensions of the box (length, width and height). The three dimensions in the top drawing of the parallelepiped are highlighted with a thick line.

Volume is the amount of liquid or bulk material that can be placed inside the figure (between the boundary planes).

Volume is one of the characteristics of three-dimensional geometric shapes.

Volume denoted by a capital letter V(“ve”) Volume values are interrelated (one cubic unit of volume can be replaced by another).

Rule. Volume of a rectangular parallelepiped is equal to the product of its three dimensions.

Units of measurement volume serve:

a) standard units of length cubed:
1 cm 3 = 1,000 mm 3
1 dm 3 = 1,000 cm 3 = 1,000,000 mm 3
1 m 3 = 1,000 dm 3 = 1,000,000 cm 3 - 1,000,000,000 mm 3
1 km 3 - 1,000,000,000 m 3
b) special unit of volume (liter):
1 l = 1 dm 3 = 1,000 cm 3.

Formula for calculating the volume of a rectangular parallelepiped:

Where A- length, b- width, With- height.

Since all dimensions of a cube are equal (a = b = c), the formula for calculating the volume of a cube is V = a 3.

Calculate the volume of a rectangular parallelepiped 6 m long, 4 m wide and 8 m high.

Solution. Since length, width and height are measured by the same unit of length (m), we substitute them into the formula V=a*b*c and calculate the volume:

V = 6 * 4 * 8 = 192 (m3)
Answer: 192 m 3.

Calculate the volume of a cube with a base side of 10 cm.

Solution. Let's substitute the numerical value of the side of the cube into the formula for calculating the volume V = a 3 and calculate:
V = 10 * 10 * 10 = 10 3 = 1,000 (cm 3) - 1 l.

Answer: 1,000 cm 3, or 1 liter.

Dear friends! Another article with prisms for you. The exam includes this type of task in which you need to determine the volume of a polyhedron. Moreover, it is not given in “pure form”, but first it needs to be built. I would put it this way: he needs to be “seen” in another given body.

There was already an article on such tasks on the blog. In the tasks presented below, straight regular prisms are given - triangular or hexagonal. If you have completely forgotten what a prism is, then...

A regular prism has a regular polygon at its base. Therefore, the base of a regular triangular prism is an equilateral triangle, and the base of a regular hexagonal prism is a regular hexagon.

When solving problems, the formula for the volume of a pyramid is used, I recommend looking at the information.It will also be useful with parallelepipeds; the principle of solving problems is similar.Look again at the formulas you need to know.

Prism volume:

Volume of the pyramid:

245340. Find the volume of a polyhedron whose vertices are points A, B, C, A 1 regular triangular prism ABCA 1 B 1 C 1 , the base area of which is 2 and the side edge is 3.

We got a pyramid with a base ABC and apex A 1 . The area of its base is equal to the area of the base of the prism (common base). The height is also common. The volume of the pyramid is:

Answer: 2

245341. Find the volume of a polyhedron whose vertices are points A, B, C, A 1, C 1, a regular triangular prism ABCA 1 B 1 C 1, the base area of which is 3, and the side edge is 2.

Let's construct the indicated polyhedron on the sketch:

This is a pyramid with base AA 1 C 1 C and a height equal to the distance between edge AC and vertex B. But in this case, calculating the area of this base and the indicated height is too long a path to the result. It's easier to do this:

To obtain the volume of the specified polyhedron, it is necessary from the volume of the given prism ABCA 1 B 1 C 1 subtract the volume of the pyramid BA 1 B 1 C 1 . Let's write down:

Answer: 4

245342. Find the volume of a polyhedron whose vertices are points A 1, B 1, B, C, a regular triangular prism ABCA 1 B 1 C 1, the base area of which is 4, and the side edge is 3.

Let's construct the indicated polyhedron on the sketch:

To obtain the volume of the specified polyhedron it is necessary from the volume of the ABCA prism 1 B 1 C 1 subtract the volumes of two bodies - pyramid ABCA 1 and pyramids CA 1 B 1 C 1. Let's write down:

Answer: 4

245343. Find the volume of a polyhedron whose vertices are points A, B, C, D, E, F, A 1 of a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1, the base area of which is 4, and the lateral edge is equal to 3.

Let's construct the indicated polyhedron on the sketch:

This is a pyramid having a common base with a prism and a height equal to the height of the prism. The volume of the pyramid will be equal to:

Answer: 4

245344. Find the volume of a polyhedron whose vertices are points A, B, C, A 1 , B 1 , C 1 of a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 , the base area of which is 6 and the lateral edge is 3.

Let's construct the indicated polyhedron on the sketch:

The resulting polyhedron is a straight prism. The volume of a prism is equal to the product of the area of the base and the height.

The height of the original prism and the resulting one is common; it is equal to three (this is the length of the side edge). It remains to determine the area of the base, that is, of triangle ABC.

Since the prism is regular, there is a regular hexagon at its base. The area of triangle ABC is equal to one sixth of this hexagon, more on this (point 6). Therefore, the area ABC is equal to 1. We calculate:

Answer: 3

245345. Find the volume of a polyhedron whose vertices are points A, B, D, E, A 1, B 1, D 1, E 1 of a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1, the base area of which is 6 , and the side edge is 2.

Let's construct the indicated polyhedron on the sketch:

Since the prism is regular, there is a regular hexagon at its base. The area of the quadrilateral ABDE is equal to four-sixths of this hexagon. Why? See more about this (point 6). Therefore, the area ABDE will be equal to 4. We calculate:

Answer: 8

245346. Find the volume of a polyhedron whose vertices are points A, B, C, D, A 1, B 1, C 1, D 1 of a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1, the base area of which is 6 , and the side edge is 2.

Let's construct the indicated polyhedron on the sketch:

The resulting polyhedron is a straight prism.

The height of the original prism and the resulting one are common; it is equal to two (this is the length of the side edge). It remains to determine the area of the base, that is, the quadrilateral ABCD. Line segment AD connects diametrically opposite points of a regular hexagon, which means that it divides it into two equal trapezoids. Therefore, the area of quadrilateral ABCD (trapezoid) is equal to three.

We calculate:

Answer: 6

245347. Find the volume of a polyhedron whose vertices are points A, B, C, B 1 of a regular hexagonal prism ABCDEFA 1 B 1 C 1 D 1 E 1 F 1 , the base area of which is 6 and the lateral edge is 3.

Let's construct the indicated polyhedron on the sketch:

The resulting polyhedron is a pyramid with base ABC and height BB 1.

*The height of the original prism and the resulting one is common, it is equal to three (this is the length of the side edge).

It remains to determine the area of the base of the pyramid, that is, triangle ABC. It is equal to one sixth of the area of the regular hexagon, which is the base of the prism. We calculate:

Answer: 1

245357. Find the volume of a regular hexagonal prism, all of whose edges are equal to the root of three.

The volume of a prism is equal to the product of the area of the base of the prism and its height.

The height of a straight prism is equal to its side edge, that is, it has already been given to us - this is the root of three. Let's calculate the area of the regular hexagon lying at the base. Its area is equal to six areas of equal regular triangles, and the side of such a triangle is equal to the edge of the hexagon:

*We used the formula for the area of a triangle - the area of a triangle is equal to half the product of adjacent sides and the sine of the angle between them.

We calculate the volume of the prism:

Answer: 13.5

Anything special to note? Carefully build the polyhedron, not mentally, but draw it on a piece of paper. Then the possibility of errors due to inattention will be eliminated. Remember the properties of a regular hexagon. Well, it’s important to remember the volume formulas that we used.

Solve two volume problems yourself:

27084. Find the volume of a regular hexagonal prism whose base sides are equal to 1 and whose side edges are equal to √3.

27108. Find the volume of a prism, the bases of which contain regular hexagons with sides of 2, and the side edges are equal to 2√3 and are inclined to the plane of the base at an angle of 30 0.

That's all. Good luck!

Sincerely, Alexander.

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