An important section of geometry, which is studied in high school, is stereometry. The objects of her research are the characteristics and properties of figures in three-dimensional space. This article is devoted to the issue of the total surface area of a prism.
What geometric object will we talk about?
Before considering the surface area at the prism, it is necessary to explain what it is. In stereometry, it is understood as a volumetric body, which is limited to several faces. Two of them lie in parallel planes and are exactly the same, they are called the base of the figure. The remaining faces connect the corresponding sides of the bases with each other and are called lateral. To understand which figure is described above, we give an example of a decagonal prism.
So it is called because of the number of angles at the base. The figure shows that if 10 is the number of corners of the polygonal base, then the number of sides of the figure is 10 + 2 = 12, the number of its vertices is 2 * 10 = 20, and the number of edges is 3 * 10 = 30. In the figure, the side faces are squares. In the general case, these faces are parallelograms.
All representatives of the prism class are classified according to several criteria. Mostly, these features are determined by the type of polygonal base. So, it can be concave and convex, regular and arbitrary shape. If all sides of the side are rectangles or squares, then they speak of straight figures. If some of these sides are parallelograms of an arbitrary type, then the prism is called inclined. A special class is regular geometric objects. In addition to being straight, their bases are equilateral and equiangular planar polygons. The figure below shows a wide range of correct prisms.
Shape surface
The surface of any prism is understood as the totality of all points that lie on the faces and form them. Since the studied polyhedron consists of two types of sides, the surface area of the lateral S b and the base area 2 * S o are distinguished, where the symbol S o denotes one polygonal base.
It is most convenient to study the surface by the example of a flat scan, which is obtained if two bases are cut from the figure, and the side surface is cut along any edge of the side and deployed. For example, a scan of a hexagonal prism is shown in the figure below.
Since the hexagons are regular, and all sides are equal to each other and are rectangles, we have before us a scan of the correct figure.
Full Area Formulas
Above, we found that you can find the total surface area of a prism using the following formula:
S = S b + 2 * S o .
For the base area, an unambiguous formula does not exist, since it can take a completely arbitrary geometric shape. However, if the base is correct and its side is a, then to calculate S o you can use the following expression:
S o = n / 4 * ctg (pi / n) * a 2 .
Where the Latin letter n indicates the number of sides of the base.
To determine the value of S b you can apply the following expressions:
S b = ∑ i = 1 n (a i * h bi );
S b = h * ∑ i = 1 n (a i );
S b = n * a * h.
The first expression is used here when all sides of the side are parallelograms (h bi is the height of the i-th parallelogram), the second formula is used for a direct prism, and the third formula for the correct one.
Task example
It is necessary to calculate the total surface area of a regular triangular prism. The side of its base is 10 cm, and the side is 7 cm.
This prism consists of 5 faces: 3 identical rectangles and 2 equilateral triangles. First, we write the formula for the total area S, we have:
S o = 3/4 * ctg (pi / 3) * a 2 = √3 / 4 * a 2 ;
S b = 3 * a * h.
S = 2 * S o + S b = √3 / 2 * a 2 + 3 * a * h.
Now it remains to substitute the numbers from the conditions of the problem and get the answer: S = 296.6 cm 2 .