Volume and surface area are two important characteristics of any body having finite dimensions in three-dimensional space. In this article, we consider a well-known class of polyhedra - prisms. In particular, the question will be revealed how to find the surface area of a direct prism.
What is a prism?
A prism is any polyhedron that is bounded by several parallelograms and two identical polygons located in parallel planes. These polygons are considered the bases of the figure, and its parallelograms are the sides. The number of sides (angles) of the base determines the name of the figure. For example, the figure below shows a pentagonal prism.
The distance between the bases is called the height of the figure. If the height is equal to the length of any side rib, then such a prism will be straight. The second sufficient sign for a direct prism is that all of its sides are rectangles or squares. If at least one side is a parallelogram of a general form, then the figure will be inclined. Below you can see how the direct and inclined prisms differ visually using the example of quadrangular figures.
Direct prism surface area
If the geometric figure has an n-coal base, then it consists of n + 2 faces, n of which are rectangles. We denote the lengths of the sides of the base by the symbol a i , where i = 1,2, ..., n, and the height of the figure, which is equal to the length of the side rib, is denoted by h. To determine the surface area (S) of the surface of all faces, it is necessary to add the area S o of each of the bases and all the areas of the sides (rectangles). Thus, the general formula for S can be written as follows:
S = 2 * S o + S b
Where S b is the lateral surface area.
Since the base of the direct prism can be absolutely any flat polygon, a single formula for calculating S o cannot be given, and to determine this value in the general case, a geometric analysis should be carried out. For example, if the base is a regular n-gon with side a, then its area is calculated by the formula:
S o = n / 4 * ctg (pi / n) * a 2
As for the value of S b , the expression for its calculation can be given. The surface area of the direct prism is:
S b = h * ∑ i = 1 n (a i )
That is, the value of S b is calculated as the product of the height of the figure on the perimeter of its base.
Problem solving example
We apply the obtained knowledge to solve the following geometric problem. A prism is given, the base of which is a right-angled triangle with sides at a right angle of 5 cm and 7 cm. The height of the figure is 10 cm. It is necessary to find the surface area of a straight triangular prism.
First, we calculate the hypotenuse of the triangle. It will be equal to:
c = √ (5 2 + 7 2 ) = 8.6 cm
Now we will do one more preparatory mathematical operation - calculate the perimeter of the base. He will make:
P = 5 + 7 + 8.6 = 20.6 cm
The lateral surface area of the figure is calculated as the product of the value of P by the height h = 10 cm, that is, S b = 206 cm 2 .
To find the area of the entire surface, two foundations should be added to the found value. Since the area of a right-angled triangle is determined by half the product of legs, we get:
2 * S o = 2 * 5 * 7/2 = 35 cm 2
Then we get that the surface area of the direct triangular prism is 35 + 206 = 241 cm 2 .