26.11.2019
The formula for the volume of a polyhedron is that all dihedral angles are right angles. How to find the volume of a polyhedron
As part of the exam in mathematics, there are a number of tasks for 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 in order to immediately see all the faces and parallelepipeds of which the polyhedra "consist". Even if you know how to do it 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 to be attentive and attentive again, like this.
So, if there is a question about the surface area, then on a sheet in a cage, build all the faces of the polyhedron, 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, then the error will be excluded.
We use the specified method. He is visual. On a sheet in a cage, we build all the elements (faces) on a scale. If the edge lengths 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).
Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).
Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).
More tasks. They provide solutions in a different way (without construction), try to figure out what came from where. Also solve in the way already presented.
* * *
If you want to find the volume of a composite polyhedron. We divide the polyhedron into its constituent parallelepipeds, carefully write down the lengths of their edges and calculate.
The volume of the polyhedron shown in the figure is 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).
First of all, let's define what a polyhedron is. This is a three-dimensional geometric figure, the faces of which are presented in the form of flat polygons. There is no single formula for finding the volume of a polyhedron, since polyhedra are different shapes. In order to find the volume of a complex polyhedron, it is conditionally divided into several simple ones, such as a parallelepiped, prism, pyramid, and then the volumes of simple polyhedra are added up and the desired volume of the figure is obtained as a result.
How to find the volume of a polyhedron - a 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 figures.
- The simplest cuboid is a cube. All edges of a cube are equal. 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 cuboid whose sides have different dimensions is calculated using the following formula:
where a, b and с are the lengths of the edges.
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, and 4 more are the side faces. An inclined parallelepiped differs from a straight one in that its lateral faces are not at a right angle with respect to the base. The volume of such a figure is calculated as the product between the area of \u200b\u200bthe base and the height:
where S is the area of the quadrilateral at the base, h is the height of the desired figure.
How to find the volume of a polyhedron - a 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 base area 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 a polygon is located at the base of the figure, and the side faces are presented in the form of triangles that meet 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, it has a top. To find the volume of a pyramid, multiply its base by its height and divide the result by 3.
Polygons are flat geometric shapes. Volumetric (three-dimensional) geometric shapes include.
Definition. A polyhedron is a geometric spatial body bounded on all sides by a finite number of flat polygons (faces).
The cuboid is . The simplest rectangular parallelepiped is a cube. All sides are equal
For a cuboid, each face is a rectangle that has a common side and two common vertices with an adjacent face.
The box has 8 vertices, 4 side rectangles and 2 rectangles at the bases. A cube has all b faces - equal squares. A rectangular parallelepiped has side shapes and bases that are rectangles. These rectangles are pairwise equal (the rectangles of the bases and two pairs of opposite rectangles constituting the side faces are equal). Consequently, the faces of a cuboid are rectangles of three types, differing in size.
Three rectangles with different sizes have
one common point- the top 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 upper picture of the box are marked with a bold line.
Volume is the amount of liquid or bulk material that can be placed inside the figure (between boundary planes).
Volume is one of the characteristics of three-dimensional geometric shapes.
Volume denoted by a capital Latin 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 in a cube:
1 cm 3 \u003d 1,000 mm 31 dm 3 \u003d 1,000 cm 3 \u003d 1,000,000 mm 3
1 m 3 \u003d 1,000 dm 3 \u003d 1,000,000 cm 3 - 1,000,000,000 mm 31 km 3 - 1,000,000,000 m 3
- b) special unit of volume (litre):
1 l \u003d 1 dm 3 \u003d 1,000 cm 3.
The 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 the 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 \u003d 6 * 4 * 8 \u003d 192 (m 3)
Answer: 192 m 3.
Calculate the volume of a cube with a base of 10 cm.
Solution. We substitute the numerical value of the side of the cube into the formula for calculating the volume V = a 3 and calculate:
V \u003d 10 * 10 * 10 \u003d 10 3 \u003d 1,000 (cm 3) - 1 l.
Answer: 1,000 cm 3, or 1 liter.
Dear friends! For you, another article with prisms. There is a type of tasks in the exam in which it is required to determine the volume of the polyhedron. Moreover, it is not given in a “pure form”, but first it needs to be built. I would put it this way - it needs to be "seen" in another given body.
An article on with such tasks was already on the blog,. In the tasks below, straight regular prisms are given - triangular or hexagonal. If you completely forgot what a prism is, then.
IN right prism the base is a regular polygon. Therefore, at the base of the correct triangular prism lies an equilateral triangle, and at the base of a regular hexagonal prism lies a regular hexagon.
When solving problems, the pyramid volume formula is used, I recommend looking at the information.It will also be useful with parallelepipeds, the principle of solving tasks is similar.Look again at the formulas you need to know.
Prism volume:
The 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 In 1 With 1 , whose base area is 2 and whose side edge is 3.
We got a pyramid with the base ABC and the top A 1 . The area of its base is equal to the area of the base of the prism (the base is common). 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, of a regular triangular prism ABCA 1 B 1 C 1, whose base area is 3, and the side edge is 2.
Let's build the specified polyhedron on the sketch:
This is a pyramid with AA base 1 from 1 C and a height equal to the distance between edge AC and vertex B. But in this case, calculating the area of \u200b\u200bthis base and the indicated height is too long a way to the result. It's easier to do this:
To get the volume of the specified polyhedron, it is necessary from the volume of the given ABCA prism 1 In 1 With 1 subtract the volume of the pyramid BA 1 In 1 With 1 . Let's write:
Answer: 4
245342. Find the volume of a polyhedron whose vertices are points A 1, B 1, B, C, of a regular triangular prism ABCA 1 B 1 C 1, whose base area is 4, and the side edge is 3.
Let's build the specified polyhedron on the sketch:
To get the volume of the indicated polyhedron, it is necessary from the volume of the ABCA prism 1 In 1 With 1 subtract the volumes of two bodies - pyramids ABCA 1 and pyramids CA 1 B 1 C 1 . Let's write:
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 , whose base area is 4, and whose side edge is 3.
Let's build the specified 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:
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 whose base area is 6 and whose side edge is 3.
Let's build the specified 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 equal to three (this is the length of the side edge). It remains to determine the area of \u200b\u200bthe base, that is, the triangle ABC.
Since the prism is regular, then at its base lies a regular hexagon. The area of the triangle ABC is equal to one sixth of this hexagon, more on this (item 6). So the area of ABC is 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 whose base area is 6 , and the side edge is 2.
Let's build the specified polyhedron on the sketch:
The height of the original prism and the resulting one is equal to two (this is the length of the side edge). It remains to determine the area of \u200b\u200bthe base, that is, the quadrilateral ABDE.
Since the prism is regular, then at its base lies a regular hexagon. The area of quadrilateral ABDE is equal to four sixths of that 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 whose base area is 6 , and the side edge is 2.
Let's build the specified polyhedron on the sketch:
The resulting polyhedron is a straight prism.
The height of the original prism and the resulting one is equal to two (this is the length of the side edge). It remains to determine the area of \u200b\u200bthe base, that is, the quadrilateral ABCD. The segment AD connects the diametrically opposite points of a regular hexagon, which means that it divides it into two equal trapezoids. Therefore, the area of the 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 with a base area of 6 and a side edge of 3.
Let's build the specified 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 equal to three (this is the length of the side edge).
It remains to determine the area of \u200b\u200bthe base of the pyramid, that is, the triangle ABC. It is equal to one sixth of the area of a regular hexagon, which is the base of the prism. We calculate:
Answer: 1
245357. Find the volume of a regular hexagonal prism, all edges of which 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 is already given to us - this is the root of three. Calculate the area of the regular hexagon lying at the base. Its area is equal to six areas equal to each other regular triangles, whereby the side of such a triangle is equal to the edge of the hexagon:
* We used the triangle area formula - the area of \u200b\u200ba triangle is equal to half the product of adjacent sides by the sine of the angle between them.
Calculate the volume of the prism:
Answer: 13.5
What can be noted especially? Carefully build a polyhedron, not mentally, but draw it on a piece of paper. Then the probability of an error due to inattention will be excluded. Remember the properties of a regular hexagon. Well, it is important to remember the volume formulas that were used.
Solve two volume problems yourself:
27084. Find the volume of a regular hexagonal prism with base sides equal to 1 and side edges equal to √3.
27108. Find the volume of a prism whose bases are regular hexagons with sides 2, and side edges equal to 2√3 and inclined to the plane of the base at an angle of 30 0 .
That's all. Good luck!
Sincerely, Alexander.
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