The study of the characteristics of three-dimensional geometric figures is engaged in stereometry. One of the well-known three-dimensional figures that appears in geometry problems is a direct prism. Consider in this article what it is, and also describe in detail a prism with a triangular base.
Prism and its types
By a prism is meant a figure that is formed as a result of the parallel transfer of a polygon in space. As a result of this geometric operation, a figure is formed consisting of several parallelograms and two identical polygons parallel to each other. Parallelograms are the sides of a prism, and polygons are its bases.
Any prism has n + 2 sides, 3 * n edges and 2 * n vertices, where n is the number of angles or sides of the polygonal base. The image shows a pentagonal prism, which consists of 7 sides, 10 vertices and 15 edges.
The considered class of figures is represented by prisms of several types. We list them briefly:
- concave and convex;
- inclined and straight;
- wrong and right.
Each figure refers to one of these three types of classification. When solving geometric problems, it is easiest to carry out calculations for regular and direct prisms. The latter will be considered in more detail in the following paragraphs of the article.
What is a direct prism?
A straight line is a concave or convex, regular or irregular prism, in which all sides are represented by quadrangles with angles of 90 °. If at least one of the quadrangles of the sides is not a rectangle or square, then the prism is called inclined. You can also give another definition: a direct prism is such a figure of this class in which any side edge is equal to the height. Under the height h of the prism is the distance between its bases.
Both of the above definitions of the fact that this is a direct prism are equal and self-sufficient. It follows from them that all dihedral angles between any of the bases and each side are equal to 90 °.
It was said above that it is convenient to work with direct figures when solving problems. This is due to the fact that the height coincides with the length of the side rib. The latter fact facilitates the process of calculating the volume of the figure and the area of its lateral surface.
Direct prism volume
Volume - a value characteristic of any spatial figure, which numerically reflects a part of the space enclosed between the surfaces of the object in question. The prism volume can be calculated by the following general formula:
V = S o * h.
That is, the product of the height and the base area will give the desired value V. Since the bases are equal for the direct prism, any of them can be taken to determine the area S o .
The advantage of using the above formula for a direct prism in comparison with its other types is that it is very easy to find the height of the figure, since it coincides with the length of the side rib.
Lateral surface area
It is convenient to calculate not only the volume for a straight figure of the class in question, but also its side surface. Indeed, any side of it is either a rectangle or a square. Each student knows how to calculate the area of these flat figures, for this it is necessary to multiply the adjacent sides by each other.
Suppose that at the base of the prism lies an arbitrary n-gon whose sides are equal to a i . Index i runs from 1 to n. The area of one rectangle is calculated as follows:
S i = a i * h.
The surface area of the side S b can be easily calculated if all the areas S i of the rectangles are added. In this case, we obtain the final formula for S b of the direct prism:
S b = h * ∑ i = 1 n (a i ) = h * P o .
Thus, in order to determine the lateral surface area for a direct prism, it is necessary to multiply its height by the perimeter of one base.
Triangular prism problem
Suppose a direct prism is given. The base is a right triangle. The legs of this triangle are 12 cm and 8 cm. It is necessary to calculate the volume of the figure and its total area if the height of the prism is 15 cm.
First, we calculate the volume of the direct prism. A triangle (rectangular) located at its base has an area of:
S o = a 1 * a 2/2 = 12 * 8/2 = 48 cm 2 .
As you might guess, a 1 and a 2 in this equality are legs. Knowing the base area and height (see the condition of the problem), we can use the formula for V:
V = S o * h = 48 * 15 = 720 cm 3 .
The total area of the figure is formed by two parts: the areas of the bases and the lateral surface. The areas of the two bases are equal:
S 2o = 2 * S o = 48 * 2 = 96 cm 2 .
To calculate the lateral surface area, you need to know the perimeter of a right triangle. We calculate by the Pythagorean theorem its hypotenuse a 3 , we have:
a 3 = √ (a 1 2 + a 2 2 ) = √ (12 2 + 8 2 ) = 14.42 cm.
Then the perimeter of the triangle of the base of the direct prism is:
P = a 1 + a 2 + a 3 = 12 + 8 + 14.42 = 34.42 cm.
Applying the formula for S b , which was written in the previous paragraph, we obtain:
S b = h * P = 15 * 34.42 = 516.3 cm.
Adding the areas S 2o and S b , we obtain the total surface area of the studied geometric figure:
S = S 2o + S b = 96 + 516.3 = 612.3 cm 2 .
The triangular prism, which is made from special types of glass, is used in optics to study the spectra of light-emitting objects. Such prisms are able to decompose light into frequency components due to the phenomenon of dispersion.