The study of prisms is carried out by spatial geometry. Their important characteristics are the volume enclosed in them, the surface area and the number of constituent elements. In the article, we consider all these properties for a hexagonal prism.
What prism will we talk about?
A hexagonal prism is a figure formed by two polygons having six sides and six angles, and six parallelograms connecting the marked hexagons into a single geometric entity.
The figure shows an example of this prism.
The hexagon marked in red is called the base of the shape. Obviously, the number of its bases is equal to two, both of which are identical. The yellow-greenish faces of the prism are called its lateral sides. In the figure they are represented by squares, but in the general case they are parallelograms.
The hexagonal prism can be inclined and straight. In the first case, the angles between the base and the sides are not straight, in the second they are equal to 90 o . Also, this prism can be right and wrong. The correct hexagonal prism must be straight and have a regular hexagon at the base. The above prism in the figure satisfies these requirements, therefore it is called correct. Further in the article we will study only its properties, as a general case.
Items
For any prism, its main elements are edges, faces and vertices. The hexagonal prism is no exception. The above figure allows you to calculate the number of these elements. So, we get 8 faces or sides (two bases and six lateral parallelograms), the number of vertices is 12 (6 vertices for each base), the number of edges of the hexagonal prism is 18 (six lateral and 12 for the bases).
In the 1750s, Leonard Euler (Swiss mathematician) established for all polyhedra, which include a prism, a mathematical relationship between the numbers of these elements. This relationship has the form:
number of edges = number of faces + number of vertices - 2.
The above numbers satisfy this formula.
Diagonal prisms
All diagonals of a hexagonal prism can be divided into two types:
- those that lie in the planes of its faces;
- those that belong to the entire volume of the figure.
The figure below shows all of these diagonals.
It can be seen that D 1 is the diagonal of the side, D 2 and D 3 are the diagonals of the entire prism, D 4 and D 5 are the diagonals of the base.
The lengths of the diagonals of the sides are equal. It is easy to calculate them using the well-known Pythagorean theorem. We denote by a the length of the side of the hexagon by a, and by b the length of the side rib. Then the diagonal has a length:
D 1 = √ (a 2 + b 2 ).
Diagonal D 4 is also easily determined. If we recall that a regular hexagon fits into a circle of radius a, then D 4 is the diameter of this circle, that is, we obtain the following formula:
D 4 = 2 * a.
Diagonal D 5 bases are somewhat more difficult to find. To do this, consider the equilateral triangle ABC (see. Fig.). For him, AB = BC = a, the angle ABC is 120 o . If you omit the height from this angle (it will be the bisector and median), then half of the base AC will be equal to:
AC / 2 = AB * sin (60 o ) = a * √3 / 2.
The side AC is the diagonal D 5 , therefore, we obtain:
D 5 = AC = √3 * a.
Now it remains to find the diagonals D 2 and D 3 of the regular hexagonal prism. To do this, you need to see that they are the hypotenuses of the corresponding right-angled triangles. Using the Pythagorean theorem, we get:
D 2 = √ (D 4 2 + b 2 ) = √ (4 * a 2 + b 2 );
D 3 = √ (D 5 2 + b 2 ) = √ (3 * a 2 + b 2 ).
Thus, the largest diagonal for any values of a and b is D 2 .
Surface area
To understand what is at stake, it is easiest to consider the development of this prism. It is shown in the figure.
It can be seen that to determine the area of all sides of the figure in question, it is necessary to calculate the area of the quadrangle and the area of the hexagon separately, then multiply them by the corresponding integers equal to the number of each n-gon in the prism, and add the results. Hexagons 2, rectangles 6.
For the area of the rectangle we get:
S 1 = a * b.
Then the side surface area is equal to:
S 2 = 6 * a * b.
To determine the area of a hexagon, it is easiest to use the appropriate formula, which has the form:
S n = n / 4 * a 2 * ctg (pi / n).
Substituting the number n equal to 6 in this expression, we obtain the area of one hexagon:
S 6 = 6/4 * a 2 * ctg (pi / 6) = 3 * √3 / 2 * a 2 .
This expression should be multiplied by two to obtain the area of the base of the prism:
S os = 3 * √3 * a 2 .
It remains to add S os and S 2 to obtain the total surface area of the figure:
S = S os + S 2 = 3 * √3 * a 2 + 6 * a * b = 3 * a * (√3 * a + 2 * b).
Prism volume
After the formula for the area of the hexagonal base has been obtained, it is easy to calculate the volume enclosed in the prism under consideration. To do this, you only need to multiply the area of one base (hexagon) by the height of the figure, the length of which is equal to the length of the side rib. We get the formula:
V = S 6 * b = 3 * √3 / 2 * a 2 * b.
Note that the product of the base to the height gives the value of the volume of absolutely any prism, including the inclined one. However, in the latter case, the calculation of the height is complicated, since it will no longer be equal to the length of the side rib. As for the hexagonal regular prism, the value of its volume is a function of two variables: sides a and b.