Prism is one of the three-dimensional figures whose properties are studied at school in the course of spatial geometry. In this article, we consider a specific prism - a hexagonal. What is this figure, 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 we have an arbitrary polygon with the number of sides n, which is in some plane. To each vertex of this polygon we construct a vector that will not lie in the plane of the polygon. Using this operation, we get n identical vectors whose vertices form a polygon exactly equal to the original one. A shape 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 lateral n surfaces-parallelograms. The number of edges P of the figure is related to the number of its vertices B and faces G by the Euler formula:
P = B + G - 2
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 a triangular, that is, its base is a triangle.
The classification of prisms is quite diverse. So, they can be right and wrong, rectangular and oblique, convex and concave.
Hexagonal prism
This article is devoted to the question of the volume of a regular hexagonal prism. First, get to know this figure closer.
As the name implies, the base of the hexagonal prism is a polygon with six sides and six corners. In the general case, there are a great many such polygons; however, for practice and for solving geometric problems, one single case is important - a regular hexagon. He has all sides equal, and each of the 6 angles is 120 o . This polygon can be built easily if you divide the circle into 6 equal parts by three diameters (they must intersect at angles of 60 o ).
A regular hexagonal prism implies not only the presence of a regular polygon at its base, but also the fact that all sides of the figure must be rectangles. This is only possible if the side faces are perpendicular to the hexagonal bases.
The correct hexagonal prism is a fairly perfect figure that is found in everyday life and nature. One has only to recall the shape of a honeycomb or a hexagonal wrench. Hexagonal prisms are also common in nanotechnology. For example, the crystal lattices of hcp and C32, which are realized under certain conditions in titanium and zirconium, as well as the lattice of graphite are in the form of hexagonal prisms.
Hexagonal Prism Surface Area
We now turn directly to the question of calculating the area and volume of the prism. First, calculate the surface area of this figure.
The surface area of any prism is calculated using the following equation:
S = 2 * S o + S b
That is, the desired area S is equal to the sum of the areas of the two bases S o and the area of the side surface S b . To determine the value of S o you can do 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 the well-known formula. It is given 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
The lateral surface area is easy to find, for this you should multiply the base of each rectangle a by the height of the prism h, multiply the obtained value by the number of such rectangles, that is 6. As a result:
S b = 6 * a * h
Using the formula for the total surface area, for a regular hexagonal prism we get:
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:
V = S o * h
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 (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.
Hexagonal regular prism volume formula
At the moment, we have considered all the necessary theoretical calculations to obtain an expression for the volume of the prism under consideration. To do this, it is enough to multiply the base area by the length of the side rib, which is the height of the figure. As a result, the volume formula of the hexagonal prism will take the form:
V = 3 * √3 * a 2 * h / 2
Thus, the calculation of the volume of the prism under consideration involves the knowledge of only two quantities: the length of the side of its base and height. These two values uniquely determine the volume of the figure.
Comparison of Hexagonal Prism and Cylinder Volumes
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, then 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 the question posed, it is necessary to calculate the length of the side of the hexagon inscribed in the circle. It can be easily shown that it is equal to the radius. We denote the radius of the circle by the letter R. Suppose that the height of the cylinder and the prism is equal to some value of h. Then the volume of the prism is equal to the following value:
V p = 3 * √3 * R 2 * h / 2
The volume of the cylinder is determined by the same formula as the volume for an arbitrary prism. Given that the area of the circle is equal to pi * R 2 , for the volume of the cylinder we have:
V c = pi * R 2 * h
Find the ratio of the volumes of these figures:
V p / V s = 3 * √3 * R 2 * h / 2 / (pi * R 2 * h) = 3 * √3 / (2 * pi)
The number pi is 3.1416. Substituting it, we get:
V p / V s = 0.827
Thus, the volume of a regular hexagonal prism is about 83% of the volume of the cylinder into which it is inscribed.