The determination of volumes of geometric bodies is one of the important tasks of spatial geometry. This article discusses the question of what is a prism with a hexagonal base, and also gives the formula for the volume of a regular hexagonal prism.
Prism definition
From the point of view of geometry, a prism is a figure in space, which is formed by two identical polygons located in parallel planes. As well as 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 a segment. Moreover, the last plane of the polygon will not belong. Then, locating this segment from each vertex of the polygon, we can obtain a parallel transfer of the latter to another plane. The figure formed in this way will be a prism.
In order to have a clear idea of the class of figures under consideration, we give a drawing of a quadrangular prism.
Many people know this figure called the box. It can be seen that two identical prism polygons are squares. They are called the base of the figure. The other four sides are rectangles, that is, this is a special case of parallelograms.
Hexagonal prism: definition and types
Before giving the formula, how the volume of the hexagonal regular prism is determined, it is necessary to clearly understand which figure will be discussed. The hexagonal prism has a hexagon at its base. That is, a flat polygon with six sides, the same number of angles. The sides of the figure, as well as for any prism, are generally parallelograms. Just note that the hexagonal base can be represented as a regular or irregular hexagon.
The distance between the bases of a figure is its height. Further we will denote it by the letter h. Geometrically, the height h is a segment perpendicular to both bases. If this perpendicular:
- omitted from the geometric center of one of the bases;
- crosses the second base also in 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 prisms can be seen at a glance.
A straight hexagonal prism is a figure with regular hexagons at the base. Moreover, it is direct. Let us consider in more detail 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 figure out what elements the figure consists of, as well as what properties it has. To make it easier to analyze the figure, we will show it in the figure.
Its main elements are faces, edges and vertices. The amounts of these elements obey Euler's theorem. If we denote P - the number of edges, B - the number of vertices and T - faces, then we can write the equality:
P = G + B - 2.
Check it out. The number of faces of the figure in question is 8. Two of them are regular hexagons. 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 ribs should be 18, which is fair. 12 ribs lie in the bases and 6 form parallel sides of the rectangles.
Turning to obtaining the formula for the volume of a regular hexagonal prism, we should focus on one important property of this figure: the rectangles that form the side surface are equal to each other and are perpendicular to both bases. This leads to two important consequences:
- The height of the figure is equal to the length of its side rib.
- Any section of the side surface of a pyramid made using a secant plane that is parallel to the bases is a regular hexagon equal to these bases.
Hexagon area
One can intuitively guess that this area of the base of the figure will appear in the formula for the volume of the regular hexagonal prism. Therefore, in this paragraph of the article we find this area. A regular hexagon divided into 6 identical triangles, the vertices of which intersect at its geometric center, is shown below:
Each of these triangles is equilateral. To prove this is not very difficult. Since the whole circle has 360 o , the angles of the triangles near the geometric center of the hexagon are 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 the isosceles triangles is 60 o , then the other two angles are also equal to 60 o . ((180 o -60 o ) / 2) - triangles are equilateral.
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, the above information should be taken into account. For an arbitrary prism, the volume of space limited by its faces is calculated as follows:
V = h * S o .
That is, V is equal to the product of the base area S o and height h. Since we know that the height h is equal to the length of the side edge b for a regular hexagonal prism, and its base area corresponds to S 6 , then the volume formula 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 correct 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 two times greater than the side of its base. It is necessary to find the volume of the figure.
To find the required value, you need to know the length of the side and side rib. When considering a regular hexagon, it was shown that its geometric center is located in the middle of the circle circumscribed around it. The radius of the latter is equal to the distance from the center to any of the peaks. That is, it is equal to the length of the side of the hexagon. These considerations lead to the following results:
a = r = 10 cm;
b = h = 2 * a = 20 cm.
Substituting these data into the volume formula of the regular hexagonal prism, we get the answer: V 6 ≈5196 cm 3 or about 5.2 liters.