In the school course of geometry, the properties of various types of prisms are studied, including hexagonal. The latter is often encountered when considering crystal lattices of metals, therefore, knowledge of its characteristics is important in determining the properties of this class of materials. This article is devoted to the issue of the base area of a regular hexagonal prism.
Volume Figure - Prism
In geometry, a prism is understood to mean such a figure, which is formed by two identical polygons located parallel to each other, and a certain number of parallelograms connecting the vertices of these polygons. If the base is formed by a polygon with n vertices, then the number of parallelograms will also be equal to n.
Prisms are characterized by the type of polygon at the base (regular and irregular triangular, quadrangular and so on), which can be concave and convex, and the angle between the side faces (parallelograms) and the base (rectangular and oblique).
The main elements of any prism are its faces (G), edges (P) and vertices (B). The figure above is an example of a triangular prism. As you can see, it has 6 peaks (3 for each base). Below is a scan of this prism. The figure shows that it consists of 5 faces: 2 triangles and 3 rectangles.
To calculate the number of edges of the figure in question, Euler's theorem should be applied:
P = B + G - 2
This expression gives the number of edges for this prism equal to 9. Indeed, if you look at the three-dimensional image of the prism above, you can see that 6 edges form the base of the figure, and 3 more edges are the result of the intersection of the rectangles.
Hexagonal Prism
Before considering the issue of the base area of a regular hexagonal prism, first get acquainted with this figure. From the name and the above classification of prisms, it is clear that we will talk about a figure based on a hexagon. This means that the number of sides in such a prism will be 8 (two bases and six parallelograms), and the number of vertices will be 12 (6 + 6). Then the number of edges will be equal to:
P = 12 + 8 - 2 = 18
Of these 18 ribs, 12 belong to the bases.
If the base has a regular hexagon, and the angles between the sides (parallelograms) and the bases are 90 o , then such a figure will be called a rectangular prism with a regular hexagon in the base, or simply a regular hexagonal prism. Its schematic is shown below.
In a regular hexagonal prism, all edges are equal only if c = a, where c is the height (length of the side edge) and a is the length of the side of the hexagon. In the general case, c ≠ a.
Next, we give formulas for calculating the surface area and volume of the prism under consideration. To do this, you need to know the base area of a regular hexagonal prism.
Hexagon area
We obtain the area formula of a regular hexagon. To do this, consider this flat figure, which is shown in the figure below.
It can be seen that the polygon consists of six identical sides, which form an angle of 120 o . Since these angles are six, their sum will be 720 o .
The figure also shows that the regular hexagon fits harmoniously into the circle. If we connect the center of the circle with each vertex of the figure, we get 6 identical triangles. Since the angular measure of the entire circumference is 360 o , the corresponding angles of the triangle are 60 o (360 o / 6). They are indicated in the figure. Since each gray segment divides the angle of the hexagon in half, the remaining two triangles are also equal to 60 o . This means that the 6 triangles shown are equilateral. The length of each of their sides is equal to the side of the hexagon; we denote it by the letter a.
From the course of geometry it is known that the area S 3 of any triangle is equal to the product of its height h and the side a to which it is drawn, divided in half, that is:
S 3 = h * a / 2
The length h is easy to calculate using the concept of a trigonometric function. It is equal to:
h = a * cos (30 o ) = a * √3 / 2
Then the area of the entire triangle is:
S 3 = √3 * a 2/4
Multiplying this area by 6, we obtain the area formula of a regular hexagon:
S 6 = 6 * S 3 = 3 * √3 * a 2/2
For completeness, it should be noted that there is a formula for the area of a regular polygon with an arbitrary number of sides n. The following is the corresponding expression:
S n = n / 4 * a 2 * ctg (pi / n)
If we substitute the value n = 6 in this expression, then we obtain the formula for the base area of the regular hexagonal prism, which will coincide with the one given above.
Note that dividing the hexagon into 6 equilateral triangles means that the hexagonal prism consists of 6 regular triangular prisms.
Surface area
The total surface area of any prism can be obtained by adding the corresponding areas S o for two bases and for the side surface S b represented by parallelograms:
S = 2 * S o + S b
We study the scan of the considered type of prism, which is shown in the figure below.
We see that the prism consists of two identical hexagons and 6 rectangles. Denote the side of the base by the letter a, and the sides of the rectangles by the letters a and c (side a is common to the hexagon and rectangle). In this case, the total surface area of the hexagonal prism will be:
S = 2 * 3 * √3 * a 2/2 + 6 * a * c = 3 * a * (√3 * a + 2 * c)
Prism volume
This important value for any real object in the case of a prism is simple: you just need to multiply the base area by the height of the figure, that is:
V = S o * h
Since we are considering a rectangular prism, its height is equal to the length of the side edge, i.e., h = c. Then the formula for the volume of the regular hexagonal prism is written in the form:
V = 3 * √3 * a 2 * c / 2
Thus, to determine the area and volume of the figure in question, it is necessary to know the length of its rib at the base and on the side surface.
Why do you need to know the properties of a hexagonal prism?
As mentioned in the introduction, these prisms are found naturally in metals. In particular, the crystalline atomic packing of titanium, zinc, zirconium, magnesium and some other metals has the shape of a hexagonal prism, which is based on 7 atoms (6 at the top and 1 in the center). Many mechanical properties of these metals (deformation and elastic characteristics) depend on the ratio of the length of the rib of this figure to the length of the side of the base.
The above is an example of this packing of atoms, which is abbreviated as hcp (hexagonal tight packing).