When solving geometry problems, a pentagonal prism is much less common than triangular, quadrangular, or hexagonal prisms . Nevertheless, it is useful to consider the basic properties of this figure, as well as to learn how to draw it.
What is a pentagonal prism?
We are talking about a three-dimensional figure, the bases of which are pentagons, and the sides are parallelograms. If each of these parallelograms is perpendicular to the parallel bases, then such a prism is called rectangular. The lateral surface of a rectangular pentagonal prism is composed of five rectangles. Moreover, the side of each of them adjacent to the base is equal to the corresponding length of the side of the pentagon.
If the pentagon is regular, that is, all its sides and angles are equal to each other, then such a rectangular prism is called regular. Further in the article we will consider the properties of this particular figure.
Prism elements
For it, as for any prism, the following elements are characteristic:
- faces or sides are parts of the planes that limit the shape in space;
- vertices are the intersection points of three sides;
- edges - segments of the intersection of two sides of the figure.
The numbers of all these elements are related to each other by the following equality:
Number of edges = number of vertices + number of faces - 2
This expression is called the Euler formula for a polyhedron.
In a pentagonal prism, the number of sides is seven (two bases + five rectangles). The number of vertices is 10 (five for each base). The number of edges in this case will be equal to:
Number of ribs = 10 + 7 - 2 = 15
Ten ribs belong to the base of the prism, and five ribs are formed by rectangles.
How to draw a pentagonal prism?
The answer to this question depends on the specific task. If it is necessary to draw an arbitrary prism, then any pentagon should be drawn. After that, draw five parallel segments of equal length from each vertex of the pentagon. Then, connect the upper ends of the segments. The result was a pentagonal arbitrary prism.
If you need to draw the correct prism, then the whole complexity of the problem is reduced to obtaining the correct pentagon. There are several ways to draw this polygon. Here we will consider only two ways.
The first way is to build a circle using a compass. Then an arbitrary diameter of the circle is drawn and five angles of 72 o (5 * 72 o = 360 o ) are counted from it using a protractor. When counting each angle, a notch is made on the circle. To build a rectangle, it remains to connect the marked notches with straight segments.
The second method involves the use of only compass and ruler. It is somewhat complicated compared to the previous one. Below is a video where each step of such a construction is explained in detail.
Note that a pentagon is easy to draw by connecting the ends of a star. If there is no need to draw exactly the right pentagon, then you can use the method with a hand-drawn star.
As soon as the pentagon is depicted, it follows from each of its vertices to draw five identical parallel segments and connect their vertices. Get a pentagonal prism.
Area of the figure
Now consider the question of how to find the area of a pentagonal prism. The figure below shows its sweep. It is seen that the desired area is formed by two identical pentagons and five rectangles equal to each other.
The area of the entire surface of the figure is expressed by the formula:
S = 2 * S o + 5 * S p
Here, the indices o and p mean the base and the rectangle, respectively. Denote the length of the side of the pentagon as a, and the height of the figure as h. Then for the rectangle we write:
S p = a * h
To calculate the area of a pentagon, we use the universal formula:
S n = n / 4 * a 2 * ctg (pi / n)
Where n is the number of sides of the polygon. Substituting n = 5, we obtain:
S 5 = 5/4 * a 2 * ctg (pi / 5) ≈ 1.72 * a 2
The accuracy of the resulting equality is 3 decimal places, which is quite enough to solve any problems.
Now it remains to find the sum of the obtained areas of the base and side surface. We have:
S = 2 * 1.72 * a 2 + 5 * a * h = 3.44 * a 2 + 5 * a * h
It should be remembered that the resulting formula is valid only for a rectangular prism. In the case of an oblique figure, the area of its lateral surface is found based on the knowledge of the perimeter of the slice, which should be perpendicular to all parallelograms.
Figure volume
The formula for calculating the volume of a pentagonal prism is no different from a similar expression for any other prism or cylinder. The volume of the figure is equal to the product of its height and the base area:
V = S o * h
If the prism in question is rectangular, then the height in it is the length of the edge formed by the rectangles. The area of the regular pentagon was calculated above with high accuracy. We substitute this value into the formula for the volume and obtain the necessary expression for the pentagonal regular prism:
V = 1.72 * a 2 * h
Thus, the calculation of the volume and surface area of a pentagonal regular prism is possible if the side of the base and the height of the figure are known.