When studying stereometry in high school, the properties of figures in space are considered. One of the main properties is volume, but sometimes geometric problems arise that require the calculation of the surface areas of the figures. In this article, we will consider a specific question: by what formula can the lateral surface area of a triangular prism be found?
Triangular prism
First, let's figure out which figure will be considered in the article. A prism is such a geometric object that consists of two identical and parallel polygonal faces and several arbitrary parallelograms that connect these faces. Building a prism is easy. To do this, it is enough to take the n-gon flat and parallel to itself transfer it to another plane. In the process of transferring the sides of the n-gon, all parallelograms of the figure, the combination of which forms the side surface of the prism, will be described. The n-gons themselves are called its bases.
Here we will not consider all possible types of prisms, but focus on a triangular figure. It is easy to guess that by it we mean a prism whose n-coal bases are triangles. Moreover, triangles can be of very different shapes, including isosceles and equilateral.
Thus, a triangular prism is formed by five faces (2 triangles and 3 parallelograms). The figure has 6 peaks of equal rights and 9 ribs of two kinds: the ribs of the base and the ribs of the side surface. An example of such a prism is shown above.
Types of triangular prisms
The considered figure is the simplest among the prisms, since the triangle is the base with the smallest possible number of sides. Any triangular prism is convex. In the general case, three types of this geometric figure can be distinguished:
- inclined;
- straight;
- right one.
To understand the difference between these types, you should pay attention to the type of base and sides. So, if the sides are parallelograms of a general shape or rhombuses, then the prism will definitely be inclined. If the lateral all faces are formed by rectangles or squares, then we have a direct prism. The latter may also be correct if all three rectangles are the same. Another criterion for the correctness of a straight figure is that it has a foundation that is correct, that is, it is formed by a triangle with equal sides.
Next, we consider the formulas for the area of the lateral surface of a triangular prism regular, straight, inclined, and cut off.
Tilt prism
We are talking about a triangular shape of arbitrary shape. It is most difficult to calculate the side surface area for it, since the height h of the figure (the distance between the bases) does not coincide with the length of the side edge b.
If the problem arises of determining the surface area (side) of such a prism, then proceed as follows: first make an imaginary cut of the figure, which should be perpendicular to all side ribs and faces. Then calculate the perimeter of this slice. In this case, we are talking about the perimeter of a triangle. Assume that it is equal to P sr . The lateral surface area is determined by multiplying the value of P sr by side b, that is, the following formula holds:
S b = P sr × b
Direct prism
As mentioned above, the side surface of this prism is formed by three rectangles. The two sides of these rectangles are the same, they are equal to the length of the side rib b, which is also the height h of the figure. As for the remaining two sides, they may differ. These parties are parties to the grounds. Denote them by the symbol a i , where i = 1, 2, 3. Then the surface area formula of the lateral triangular prism is written as follows:
S b = b × ∑ i = 1 3 a i
Many could notice that this expression does not differ from the analogous one for an inclined prism, because the sum of the three sides a i is the perimeter of the base. This is due to the fact that for a straight figure, the base is a slice perpendicular to the side faces.
Correct figure
The formula for the surface area of the lateral triangular prism is the simplest in comparison with the expressions above. In a regular figure, all side faces are not just rectangles (squares in some cases), but they are also equal to each other. These geometrical facts allow us to write the formula for the surface area of the lateral triangular prism as follows:
S b = 3 × a × b
Here a is the side of the base (triangle). The number 3 appears because the side surface is represented by three equal faces. Recall that in this expression, side b can be replaced by a height h.
Obviously, if the sides are squares, then the formula for S b is written like this:
S b = 3 × a 2
Clipped figure
Such a prism is formed if a part of it is cut off with the help of a plane. If the secant plane is parallel to the bases, then the cut-off prism of the lateral surface of the triangular prism will take one of the forms recorded in the previous paragraphs. Indeed, with a parallel section, we obtain a figure similar in shape to the initial prism.
If the secant plane is not parallel to the bases, then to determine the area of the cut prism, it will be necessary to conduct a special geometric analysis, since its side surface will be represented by irregular quadrangles.