In high school, after studying the properties of figures on the plane, they go on to consider spatial geometric objects such as prisms, spheres, pyramids, cylinders and cones. In this article we give the most complete description of a straight triangular prism.
What is a triangular prism?
We begin the article with the definition of the figure, which will be discussed later. The prism from the point of view of geometry is a figure in space formed by two identical n-gons located in parallel planes, the same angles of which are connected by straight segments. These segments are called side ribs. Together with the sides of the base, they form a side surface, which in the general case is represented by parallelograms.
Two n-gons are the bases of the figure. If the lateral ribs are perpendicular to them, then they speak of a direct prism. Accordingly, if the number of sides n of the polygon in the bases is three, then such a figure is called a triangular prism.
The triangular direct prism is shown above in the figure. This figure is also called regular, since equilateral triangles lie at its bases. The length of the side edge of the figure, indicated in the figure by the letter h, is called its height.
The figure shows that a prism with a triangular base is formed by five faces, two of which are equilateral triangles, and three are the same rectangles. In addition to the faces, the prism has six vertices at the bases and nine edges. The quantities of the elements considered are related to each other by Euler's theorem :
number of edges = number of vertices + number of sides - 2.
The area of a straight triangular prism
We found out above that the figure in question is formed by five faces of two types (two triangles, three rectangles). All these faces form the complete surface of the prism. Their total area is the area of the figure. Below is a scan of a triangular prism, which can be obtained if two bases are first cut from the figure, and then cut along one edge and unfold the side surface.
We give formulas for determining the surface area of this sweep. Let's start with the bases of a straight triangular prism. Since they represent triangles, the area S 3 of each of them can be found as follows:
S 3 = 1/2 * a * h a .
Here a is the side of the triangle, h a is the height dropped from the top of the triangle to this side.
If the triangle is equilateral (regular), then the formula for S 3 depends on only one parameter a. It has the form:
S 3 = √3 / 4 * a 2 .
This expression can be obtained by considering a right-angled triangle formed by the segments a, a / 2, h a .
The base area S o for the correct figure is twice as large as S 3 :
S o = 2 * S 3 = √3 / 2 * a 2 .
As for the lateral surface area S b , it is not difficult to calculate. To do this, just multiply by three the area of one rectangle formed by the sides a and h. The corresponding formula is:
S b = 3 * a * h.
Thus, the area of a regular prism with a triangular base is found by the following formula:
S = S o + S b = √3 / 2 * a 2 + 3 * a * h.
If the prism is straight, but incorrect, then to calculate its area, you should separately add the areas of the rectangles that will not be equal to each other.
Determining the volume of a figure
By the volume of a prism is understood the space limited by its sides (faces). Calculating the volume of a direct triangular prism is much simpler than its surface area. To do this, it is enough to know the area of the base and the height of the figure. Since the height h of the straight figure is the length of its lateral edge, and how to calculate the base area, we cited in the previous paragraph, it remains to multiply these two values by each other to obtain the desired volume. The formula for it takes the form:
V = S 3 * h.
Note that the product of the area of one base and the height will give the volume not only of a direct prism, but also of an inclined figure and even a cylinder.
The solution of the problem
Glass triangular prisms are used in optics to study the spectrum of electromagnetic radiation due to the dispersion phenomenon. It is known that a glass regular prism has a base side length of 10 cm and an edge length of 15 cm. What is the area of its glass faces and what volume does it enclose?
To determine the area, we use the formula written in the article. We have:
S = √3 / 2 * a 2 + 3 * a * h = √3 / 2 * 10 2 + 3 * 10 * 15 = 536.6 cm 2 .
To determine the volume V, we also use the formula written above:
V = S 3 * h = √3 / 4 * a 2 * h = √3 / 4 * 10 2 * 15 = 649.5 cm 3 .
Despite the fact that the edges of the prism have lengths of 10 cm and 15 cm, the volume of the figure is only 0.65 liters (a cube with a side of 10 cm has a volume of 1 liter).