All figures that are limited by faces located in different planes in space have some volume. A special geometric section, stereometry, is engaged in calculating this quantity. In this article, we give the formula for the volume of a hexagonal prism.
What is a prism?
Obviously, before you find the volume of a geometric figure, you should get acquainted with it and understand what properties it has. In this case, we are talking about a prism. The following definition is given in stereometry for this figure: a prism is any spatial geometric object that is bounded by two n-gons located in parallel planes and n parallelograms. Here n is any positive integer starting from three.
Building a shape is easy. To do this, take an arbitrary polygon and use the same parallel segments to move it to another plane. The resulting figure will be a prism. Note that, unlike a cone, a cylinder, and a sphere, it is not a rotation figure, that is, it cannot be obtained by rotation around the axis of any plane figure.
The above figure shows an example of a box, which is a quadrangular prism.
Hexagonal prism and its types
Further in the article we give the formula for the volume of a hexagonal prism . What is this figure? Any prism having a hexagon in its base is called a hexagonal.
It is formed by two hexagons in the bases and six parallelograms, the totality of which is the lateral area of the figure. This prism has 12 peaks, 8 faces or sides and 18 edges, 2/3 of which belong to the bases.
The above description of the elements corresponds to several types of hexagonal prism. Firstly, this figure can be convex or concave, which depends on the hexagon in the bases, and secondly, the prism can be inclined and straight. The difference between them is that in a straight figure, any side will be perpendicular to the bases, and in an inclined figure, the sides will intersect the bases at certain angles that are different from 90 ° . Both prisms are shown.
Note that the condition of perpendicularity of the sides and bases leads to the fact that parallelograms of a direct prism become rectangles.
Finally, thirdly, the hexagonal prism is right and wrong. The last will be any figure that is not straight and does not have a regular hexagonal base. Next, we will focus on the correct prism.
Regular hexagon
To determine the volume of geometric shapes of many classes, it is necessary to know the value of the area of their base. This fact is true for pyramids, cylinders, cones. Prisms are no exception.
To find the base area of a hexagonal prism, calculate the area of the hexagon. The easiest way to do this is for the right figure. For clarity, we show what a regular hexagon is.
It can be seen that it represents a polygon formed by six identical sides that intersect at angles of 120 o . It is also seen that a circle of some radius can be inscribed in a hexagon, and its circle can also be described.
The calculation of the base area of the hexagonal correct prism comes down to determining the area of the above figure. If the hexagon is divided into equilateral triangles as shown in the figure, then its area will be equal to 6 times the area of one triangle. Denote the length of the side of the hexagon by the letter a, then for the area S of the hexagon we get:
S = 6 * 1/2 * a * √3 / 2 * a = 3 * √3 / 2 * a 2 .
For any other hexagon that is not correct, this formula will be unfair.
Hexagonal Prism Volume Formula
It is not difficult to calculate the volume of any prism, for this you should know only two of its parameters: height h and base area S. Volume V is calculated using the following formula:
V = h * S.
An important thing to note: the recorded expression is valid for all types of prisms, including concave and inclined ones. Nevertheless, for an arbitrary prism, despite the simplicity of the formula, it can be difficult to apply. The difficulty is associated with the determination of both parameters in the expression.
In connection with the foregoing, we consider a specific regular prism with a regular hexagonal base. If its height is equal to h, and the length of the side is equal to a, then the formula for the volume of the hexagonal prism will take the correct form:
V = 3 * √3 / 2 * h * a 2 .
When writing this expression, the formula for S given in the previous paragraph was substituted.
Next, we will solve two problems in which we show how to find the volume of a hexagonal prism for specific cases.
The task with a known diagonal
The figure below shows the correct prism. It is known that the side of its base is 9 cm. What is the volume of the hexagonal prism, if the diagonal AB is 21 cm long.
It is not difficult to guess by looking at the figure that the triangle ABC is rectangular, and side AB is the hypotenuse. The leg of AC is the height h of the figure. To calculate the prism volume, we need to find the length of this leg. Note that the second leg of CB has twice the length than the side of the base, that is, 18 cm. We apply the Pythagorean theorem and obtain:
h = AC = √ (AB 2 -CB 2 ) = √ (21 2 -18 2 ) ≈ 10.82 cm.
We rounded the height value to hundredths of a centimeter.
Since we know the height h and the side of the base a, we can apply the formula for V. We get:
V = 3 * √3 / 2 * h * a 2 = 3 * √3 / 2 * 10.82 * 9 2 = 2277 cm 3 .
Thus, the prism considered has a volume of almost 2.3 liters.
The problem with a cylinder inscribed in a prism
It is known that a cylinder with a radius of 12 cm is inscribed in a regular hexagonal prism. The volume of the cylinder is 1360 cm 3 . What is the volume of a prism?
As was shown, it is possible to determine the volume of a prism if you know its height and side of the base. Let's start by defining the side. Since the radius r of the circle inscribed in the hexagon is known, it means that the length of the side a can be calculated as follows:
a = 2 * r / √3.
To understand where this formula came from, you can take into account that the radius r is the height of one of the six equilateral triangles of the hexagon.
Now we calculate the height h of the prism. According to the condition of the problem, it must coincide with the height of the cylinder. The volume of the cylinder is calculated by the same formula as for the prism. We have:
V c = S o * h = pi * r 2 * h =>
h = V c / (pi * r 2 ).
We substitute the expressions for a and h in the formula for V prisms, we obtain:
V = 3 * √3 / 2 * h * a 2 = 3 * √3 / 2 * V c / (pi * r 2 ) * (2 * r / √3) ** 2 = 2 * √3 * V c / pi.
We came to an interesting result: it turns out that the volume of the hexagonal prism does not depend on the radius of the inscribed cylinder, but is uniquely determined by its volume. Substituting the value of V c , we obtain the volume of the prism equal to approximately 1500 cm 3 .