The calculation of volumes of spatial figures is one of the important tasks of stereometry. In this article, we consider the question of determining the volume of such a polyhedron as a pyramid, and also give the formula for the correct hexagonal volume of a pyramid .
Hexagonal pyramid
To begin, consider what a figure is about, which will be discussed in the article.
Suppose we have an arbitrary hexagon whose sides are not necessarily equal to each other. Also suppose that we have chosen a point in space that is not in the plane of the hexagon. Combining all the corners of the latter with the selected point, we get a pyramid. Two different pyramids with a hexagonal base are shown in the figure below.
It can be seen that in addition to the hexagon, the figure consists of six triangles, the connection point of which is called the vertex. The difference between the depicted pyramids is that the height h of the right of them does not intersect the hexagonal base in its geometric center, and the height of the left figure falls exactly at this center. Due to this criterion, the left pyramid is called a straight line, and the right one is called an inclined one.
Since the base of the left figure in the figure is formed by a hexagon with equal sides and angles, it is called regular. Further in the article we will focus only on this pyramid.
Hexagonal Pyramid Volume
To calculate the volume of an arbitrary pyramid, the following formula is valid:
V = 1/3 * h * S o
Here h is the length of the height of the figure, S o is the area of its base. We use this expression to determine the volume of the correct hexagonal pyramid.
Since the base of the figure under consideration is an equilateral hexagon, then to calculate its area, you can use the following general expression for an n-gon:
S n = n / 4 * a 2 * ctg (pi / n)
Here n is an integer equal to the number of sides (angles) of the polygon, a is the length of its side, the cotangent function is calculated using the corresponding tables.
Using the expression for n = 6, we get:
S 6 = 6/4 * a 2 * ctg (pi / 6) = √3 / 2 * a 2
Now it remains to substitute this expression into the general formula for volume V:
V 6 = S 6 * h = √3 / 2 * h * a 2
Thus, to calculate the volume of the pyramid under consideration, it is necessary to know its two linear parameters: the length of the side of the base and the height of the figure.
Problem solving example
We show how to use the obtained expression for V 6 to solve the following problem.
It is known that the volume of a regular hexagonal pyramid is 100 cm 3 . It is necessary to determine the side of the base and the height of the figure, if it is known that they are related to each other by the following equality:
a = 2 * h
Since the formula for the volume includes only a and h, we can substitute any of these parameters into it, expressed in terms of the other. For example, substitute a, we get:
V 6 = √3 / 2 * h * (2 * h) 2 =>
h = ∛ (V 6 / (2 * √3))
To find the value of the height of the figure, it is necessary to take the root of the third degree from the volume, which corresponds to the dimension of length. We substitute the value of the volume V 6 of the pyramid from the conditions of the problem, we obtain the height:
h = ∛ (100 / (2 * √3)) ≈ 3.0676 cm
Since the base side, in accordance with the condition of the problem, is two times larger than the found value, we obtain the value for it:
a = 2 * h = 2 * 3.0676 = 6.1352 cm
The volume of the hexagonal pyramid can be found not only through the height of the figure and the value of the side of its base. It is enough to know two different linear parameters of the pyramid to calculate it, for example, the apotheme and the length of the side edge.