Direct triangular prism. Formulas for volume and surface area. Solution of a geometric problem

2024 Author: Angel Austin | [email protected]. Last modified: 2024-01-02 17:39

In high school, after studying the properties of figures on the plane, they move on to the consideration of spatial geometric objects such as prisms, spheres, pyramids, cylinders and cones. In this article, we will give the most complete description of a straight triangular prism.

What is a triangular prism?

Let's start the article with the definition of the figure, which will be discussed further. A 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 line segments. These segments are called lateral ribs. Together with the sides of the base, they form a side surface, which is generally represented by parallelograms.

Two n-gons are the bases of the figure. If the side edges are perpendicular to them, then they speak of a straight prism. Accordingly, if the number of sides n of the polygon at the bases is three, then such a figure is called a triangular prism.

The triangular straight prism is shown above in the figure. This figure is also called regular, since its bases are equilateral triangles. The length of the side edge of the figure, indicated by the letter h in the figure, 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 identical rectangles. In addition to the faces, the prism has six vertices at the bases and nine edges. The numbers of considered elements are related to each other by the Euler theorem:

number of edges=number of vertices + number of sides - 2.

Area of a right 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 full surface of the prism. Their total area is the area of the figure. Below is a triangular prism unfolding, which can be obtained by first cutting off two bases from the figure, and then cutting along one edge and unfolding the side surface.

Let's give formulas for determining the surface area of this sweep. Let's start with the bases of a right triangular prism. Since they represent triangles, the area S_{3 of each of them can be found as follows:}

S₃=1/2ah_a.

Here a is the side of the triangle, h_{a is the height lowered from the vertex of the triangle to this side.}

If the triangle is equilateral (regular), then the formula for S_{3depends on only one parameter a. It looks like:}

S₃=√3/4a^2.

This expression can be obtained by considering a right triangle formed by segments a, a/2, h_a.

The area of bases S_o for a regular figure is twice the value of S_3:

S_o=2S₃=√3/2a^2.

As for the lateral surface area S_{b, it is not difficult to calculate it. To do this, it is enough to multiply by three the area of \u200b\u200bone rectangle formed by sides a and h. The corresponding formula is:}

S_b=3ah.

Thus, the area of a regular prism with a triangular base is found by the following formula:

S=S_o+ S_b=√3/2a^{2+ 3 ah.}

If the prism is straight but irregular, then to calculate its area, you should separately add the areas of rectangles that are not equal to each other.

Determining the volume of a figure

The volume of a prism is understood as the space limited by its sides (faces). Calculating the volume of a right triangular prism is much easier than calculating its surface area. To do this, it is enough to know the area of \u200b\u200bthe base and the height of the figure. Since the height h of a straight figure is the length of its lateral edge, and how to calculate the base area, we have given in the previouspoint, then it remains to multiply these two values \u200b\u200bto each other in order to obtain the desired volume. The formula for it becomes:

V=S_3h.

Note that the product of the area of one base and the height will give the volume of not only a straight prism, but also an oblique figure and even a cylinder.

Problem Solving

Glass triangular prisms are used in optics to study the spectrum of electromagnetic radiation due to the phenomenon of dispersion. It is known that a regular glass 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 contain?

To determine the area, we will use the formula written in the article. We have:

S=√3/2a²+ 3ah=√3/210² + 3 1015=536.6cm^2.

To determine the volume V, we also use the above formula:

V=S₃h=√3/4a²h=√3/410 2^{15=649.5 cm}3^.

Despite the fact that the edges of the prism are 10 cm and 15 cm long, the volume of the figure is only 0.65 liters (a cube with a side of 10 cm has a volume of 1 liter).