One of the important polyhedra, the properties of which are studied in detail in stereometry, is a prism. In the article, we will consider this figure, revealing in more detail the question that this is the correct prism. We also give formulas to calculate its volume and area.
Prism figure
A prism is a geometric figure that consists of n + 2 faces, where n is the number of sides of a flat polygon. The two faces of the prism are exactly the same polygons. They are located in parallel planes and are called bases. The sides of the bases are interconnected by parallelograms. There are n pieces of these parallelograms; all of them form the side surface of the spatial figure.
In addition to faces, the prism has vertices (2 ร n) and edges (3 ร n). The vertices are all of the same type, but the ribs are of two types: those related to the bases and to the sides.
Any prism can be obtained if parallel to itself transfer the polygon from one plane to another in space. An example of a hexagonal prism having two hexagonal regular bases is shown below. As can be seen from the figure, its lateral surface is formed by six rectangles (a special case of parallelograms). Such a figure is a correct prism. Let's consider it in more detail in the article.
What is the right prism?
There are several different classifications of prisms. So, there are straight or oblique figures, convex and concave, pentagonal and decagonal, but the most symmetrical among all the figures studied is the correct prism. Under it is put such a figure that has the right base and is straight. Let us examine in order each item of this definition.
By the correct base of a prism, we mean a polygon that has all the same sides and all the same angles. The simplest of these polygons is an equilateral triangle, whose angles are 60 o . Further, increasing the number of sides, we get a square, regular pentagon and hexagon, and so on.
As stated in the definition, the correct prism is a direct prism. A prism of a straight line is a figure in which all side parallelograms are rectangles. Moreover, these rectangles are perpendicular to the bases. The latter fact leads to the fact that the length of the side edge of any of the rectangles is equal to the height of the figure. The figure below shows a set of regular prisms (from triangular to octagonal).
Linear shape options
Having understood that this is the correct prism, we give a description of the parameters that are used to describe it. First of all, it is the length of the side of its base a and the height h. As noted, the height is equal to the length of the side rib b.
In addition to these quantities, prisms are also characterized by diagonals. Diagonals are of three types: those lying in the bases, on the sides and inside the figure itself. Calculation of the lengths of the diagonals involves the use of the Pythagorean theorem. For example, for a regular quadrangular prism, the volume diagonals are equal to:
d 1 = โ (2 ร a 2 + b 2 ).
The diagonals of the base d 2 and the side rectangles d 3 are:
d 2 = a ร โ2;
d 3 = โ (a 2 + b 2 ).
We note that the formula for the diagonals d 3 will be the same for any polygonal regular prisms. As for the calculation of the diagonals d 1 and d 2 for other prisms (pentagonal, hexagonal and so on), for this it is necessary to carry out sequential geometric calculations taking into account the properties of the corresponding regular polygons.
Calculation of area and volume
The surface area of โโthe studied figure is the sum of the areas of two n-gons and n rectangles. The area of โโone n-coal base can be calculated by the following formula:
S n = n / 4 ร ctg (pi / n) ร a 2 .
If we denote the side edge by the letter b, of the correct prism the area of โโthe side surface will be:
S b = n ร a ร b.
Then the total area will be equal to:
S = n / 2 ร ctg (pi / n) ร a 2 + n ร a ร b.
The first term in the expression is the area of โโtwo bases at once.
The volume of a prism of any kind is calculated as follows:
V = S o ร h
That is, it is enough to multiply the height h by the base area S o to calculate the desired value. Since we know how to calculate the base area, substituting the corresponding formula in the expression for V, we arrive at the following result:
V = n / 4 ร ctg (pi / n) ร a 2 ร h.
Note that to calculate the area and volume of the studied type of prisms, it is enough to know only two of their linear parameters.
The problem with a triangular prism
It is known that the height of the base of a regular triangular prism is equal to the height of the three-dimensional figure and is 11 cm. It is necessary to find the volume and total surface area for this prism.
From the conditions of the problem, we know two parameters, so any properties of the figure can be uniquely calculated. To find the length of the side of the triangular base, we should recall the properties of an equilateral triangle. In particular, its height is also also a bisector. This allows you to use the definition of the cosine function to write the length of the base height h a in the following form:
h a = a ร cos (30 o ) = โ3 / 2 ร a.
Where do we calculate a:
a = 2 ร h a / โ3 = 12.7 cm.
Since a and h are known, one can use the formulas for area and volume:
S = n / 2 ร ctg (pi / n) ร a 2 + n ร a ร b = โ3 / 2 ร 12.7 2 + 3 ร 12.7 ร 11 = 558.78 cm 2 ;
V = n / 4 ร ctg (pi / n) ร a 2 ร h = โ3 / 4 ร 12.7 2 ร 11 = 768.25 cm 3 .
When using the formula S, we applied the property of equal height and side edges for the correct prism.