What is the volume of a hexagonal prism? How to find the volume of a regular hexagonal prism (formula)

Determining the volumes of geometric bodies is one of the important problems of spatial geometry. This article discusses the question of what a prism with a hexagonal base is, and also provides a formula for the volume of a regular hexagonal prism.

Definition of a prism

From the point of view of geometry, a prism is a figure in space that is formed by two identical polygons located in parallel planes. And also several parallelograms that connect these polygons into a single figure.

In three-dimensional space, a prism of arbitrary shape can be obtained by taking any polygon and segment. Moreover, the latter will not belong to the plane of the polygon. Then, by placing this segment from each vertex of the polygon, you can obtain a parallel transfer of the latter to another plane. The figure formed in this way will be a prism.

To have a clear idea of ​​the class of figures under consideration, we present a drawing of a quadrangular prism.

Many people know this figure as a parallelepiped. It can be seen that two identical prism polygons are squares. They are called the bases of the figure. Its remaining four sides are rectangles, that is, they are a special case of parallelograms.

Hexagonal prism: definition and types

Before giving the formula for how the volume of a hexagonal regular prism is determined, it is necessary to clearly understand what kind of figure we'll talk. has a hexagon at the base. That is, a flat polygon with six sides and the same number of angles. The sides of the figure, as for any prism, are generally parallelograms. Let us immediately note that the hexagonal base can be represented by both regular and irregular hexagons.

The distance between the bases of the figure is its height. In what follows we will denote it by the letter h. Geometrically, the height h is a segment perpendicular to both bases. If this is perpendicular:

• omitted from the geometric center of one of the bases;
• intersects the second base also at the geometric center.

The figure in this case is called a straight line. In any other case, the prism will be oblique or inclined. The difference between these types of hexagonal prism can be seen at a glance.

A right hexagonal prism is a figure that has regular hexagons at its base. Moreover, it is direct. Let's take a closer look at its properties.

Elements of a regular hexagonal prism

To understand how to calculate the volume of a regular hexagonal prism (the formula is given below in the article), you also need to understand what elements the figure consists of, as well as what properties it has. To make it easier to analyze the figure, we show it in the figure.

Its main elements are faces, edges and vertices. The quantities of these elements obey Euler's theorem. If we denote P - the number of edges, B - the number of vertices and G - faces, then we can write the equality:

Let's check it out. The number of faces of the figure in question is 8. Two of them are regular hexagons. The six faces are rectangles, as can be seen from the figure. The number of vertices is 12. Indeed, 6 vertices belong to one base, and 6 to another. According to the formula, the number of edges should be 18, which is fair. 12 edges lie at the bases and 6 form sides of rectangles parallel to each other.

Moving on to obtaining the formula for the volume of a regular hexagonal prism, you should focus on one important property of this figure: the rectangles forming the lateral surface are equal to each other and perpendicular to both bases. This leads to two important consequences:

1. The height of the figure is equal to the length of its side edge.
2. Any lateral section made using a cutting plane that is parallel to the bases is a regular hexagon equal to these bases.

Hexagon area

You can intuitively guess that this area of ​​the base of the figure will appear in the formula for the volume of a regular hexagonal prism. Therefore, in this paragraph of the article we will find this area. A regular hexagon divided into 6 equal triangles whose vertices intersect at its geometric center is shown below:

Each of these triangles is equilateral. It's not very difficult to prove this. Since the entire circle has 360 o, the angles of the triangles near the geometric center of the hexagon are equal to 360 o /6 = 60 o. The distances from the geometric center to the vertices of the hexagon are the same.

The latter means that all 6 triangles will be isosceles. Since one of the angles of isosceles triangles is equal to 60 o, this means that the other two angles are also equal to 60 o. ((180 o -60 o)/2) - equilateral triangles.

Let us denote the length of the side of the hexagon by the letter a. Then the area of ​​one triangle will be equal to:

S 1 = 1/2*√3/2*a*a = √3/4*a 2 .

The formula is derived from the standard expression for the area of ​​a triangle. Then the area S 6 for the hexagon will be:

S 6 = 6*S 1 = 6*√3/4*a 2 = 3*√3/2*a 2 .

Formula for determining the volume of a regular hexagonal prism

To write down the formula for the volume of the figure in question, you should take into account the above information. For an arbitrary prism, the volume of space limited by its faces is calculated as follows:

That is, V is equal to the product of the base area S o and the height h. Since we know that the height h is equal to the length of the side edge b for a hexagonal regular prism, and the area of ​​its base corresponds to S 6, then the formula for the volume of a regular hexagonal prism will take the form:

V 6 = 3*√3/2*a 2 *b.

An example of solving a geometric problem

A hexagonal regular prism is given. It is known that it is inscribed in a cylinder with a radius of 10 cm. The height of the prism is twice more sides its foundations. You need to find the volume of the figure.

To find the required value, you need to know the length of the side and side edge. When examining a regular hexagon, it was shown that its geometric center is located in the middle of the circle described around it. The radius of the latter is equal to the distance from the center to any of the vertices. That is, he equal to length sides of a hexagon. These arguments lead to the following results:

a = r = 10 cm;

b = h = 2*a = 20 cm.

Substituting these data into the formula for the volume of a regular hexagonal prism, we get the answer: V 6 ≈5196 cm 3 or about 5.2 liters.

Prism is one of the volumetric figures, the properties of which are studied at school in the course of spatial geometry. In this article we will consider a specific prism - a hexagonal one. What kind of figure is this, how to find the volume of a regular hexagonal prism and its surface area? The answers to these questions are contained in the article.

Prism figure

Suppose that we have an arbitrary polygon with the number of sides n, which is located in some plane. For each vertex of this polygon we will construct a vector that will not lie in the plane of the polygon. Using this operation, we will obtain n identical vectors, the vertices of which form a polygon exactly equal to the original one. A figure bounded by two identical polygons and parallel lines connecting their vertices is called a prism.

The faces of the prism are two bases, represented by polygons with n sides, and n side parallelogram surfaces. The number of edges P of a figure is related to the number of its vertices B and faces G by Euler’s formula:

For a polygon with n sides, we get n + 2 faces and 2 * n vertices. Then the number of edges will be equal to:

P = B + G - 2 = 2 * n + n + 2 - 2 = 3 * n

The simplest prism is triangular, that is, its base is a triangle.

The classification of prisms is quite diverse. So, they can be regular and irregular, rectangular and oblique, convex and concave.

Hexagonal prism

This article is devoted to the issue of the volume of a regular hexagonal prism. First, let's take a closer look at this figure.

As the name suggests, the base of a hexagonal prism is a polygon with six sides and six angles. In the general case, a great variety of such polygons can be made, but for practice and for solving geometric problems, one single case is important - a regular hexagon. All its sides are equal to each other, and each of the 6 angles is 120 o. This polygon can be easily constructed by dividing the circle into 6 equal parts with three diameters (they should intersect at angles of 60 o).

A regular hexagonal prism requires not only the presence of a regular polygon at its base, but also the fact that all the sides of the figure must be rectangles. This is only possible if side faces will be perpendicular to the hexagonal bases.

A regular hexagonal prism is a fairly perfect figure that is found in everyday life and nature. One has only to think about the shape of a honeycomb or a hex wrench. Hexagonal prisms are also common in the field of nanotechnology. For example, the crystal lattices of HCP and C32, which are realized under certain conditions in titanium and zirconium, as well as the graphite lattice, have the shape of hexagonal prisms.

Surface area of ​​a hexagonal prism

Let us now move directly to the issue of calculating the area and volume of the prism. First, let's calculate the surface area of ​​this figure.

The surface area of ​​any prism is calculated using the following equation:

That is, the required area S is equal to the sum of the areas of the two bases S o and the area of ​​the lateral surface S b . To determine the value of S o, you can proceed in two ways:

• Calculate it yourself. To do this, the hexagon is divided into 6 equilateral triangles. Knowing that the area of ​​one triangle is equal to half the product of the height and the base (the length of the side of the hexagon), you can find the area of ​​the polygon in question.
• Use a known formula. It is shown below:

S n = n / 4 * a 2 * ctg(pi / n)

Here a is the side length of a regular polygon with n vertices.

Obviously, both methods lead to the same result. For a regular hexagon, the area is:

S o = S 6 = 3 * √3 * a 2 / 2

It’s easy to find the lateral surface area; to do this, multiply the base of each rectangle a by the height of the prism h, multiply the resulting value by the number of such rectangles, that is, by 6. As a result:

Using the formula for the total surface area, for a regular hexagonal prism we obtain:

S = 3 * √3 * a 2 + 6 * a * h = 3 * a * (√3 * a + 2 * h)

How to find the volume of a prism?

Volume is a physical quantity that reflects the area of ​​space occupied by an object. For a prism, this value can be calculated using the following formula:

This expression answers the question of how to find the volume of a prism of arbitrary shape, that is, it is necessary to multiply the base area S o by the height of the figure h (the distance between the bases).

Note that the above expression is valid for any prism, including concave and oblique figures formed by irregular polygons at the base.

Formula for the volume of a hexagonal regular prism

On this moment we have considered all the necessary theoretical calculations to obtain an expression for the volume of the prism in question. To do this, it is enough to multiply the area of ​​the base by the length of the side edge, which is the height of the figure. As a result, the hexagonal prism will take the form:

V = 3 * √3 * a 2 * h / 2

Thus, calculating the volume of the prism in question requires knowledge of only two quantities: the length of the side of its base and the height. These two quantities uniquely determine the volume of the figure.

Comparison of volumes and cylinder

It was said above that the base of a hexagonal prism can be easily constructed using a circle. It is also known that if you increase the number of sides of a regular polygon, its shape will approach a circle. In this regard, it is of interest to calculate how much the volume of a regular hexagonal prism differs from this value for a cylinder.

To answer this question, you need to calculate the side length of a hexagon inscribed in a circle. It can be easily shown that it is equal to the radius. Let us denote the radius of the circle by the letter R. Let us assume that the height of the cylinder and the prism is equal to a certain value h. Then the volume of the prism is equal to the following value:

V p = 3 * √3 * R 2 * h / 2

The volume of a cylinder is determined by the same formula as the volume for an arbitrary prism. Considering that the area of ​​the circle is equal to pi * R 2, for the volume of the cylinder we have:

Let's find the ratio of the volumes of these figures:

V p / V с = 3 * √3 * R 2 * h / 2 / (pi * R 2 * h) = 3 * √3 / (2 * pi)

Pi is 3.1416. Substituting it, we get:

Thus, the volume of a regular hexagonal prism is about 83% of the volume of the cylinder into which it is inscribed.

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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 basis of the correct triangular prism lies equilateral triangle, and at the base of a regular hexagonal prism lies 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 height equal to the distance between edge AC and vertex B. But in 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:

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 ABDE.

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.

P.S: I would be grateful if you tell me about the site on social networks

Regular hexagonal prism- a prism, at the bases of which there are two regular hexagons, and all the side faces are strictly perpendicular to these bases.

• A B C D E F A1 B1 C1 D1 E1 F1 - regular hexagonal prism
• a- length of the side of the base of the prism
• h- length of the side edge of the prism
• Smain- area of ​​the prism base
• Sside .- area of ​​the lateral face of the prism
• Sfull- total surface area of ​​the prism
• Vprisms- prism volume

Prism base area

At the bases of the prism there are regular hexagons with sides a. According to the properties of a regular hexagon, the area of ​​the bases of the prism is equal to

This way

Smain= 3 3 √ 2 ⋅ a2