The science of geometry is designed to acquaint schoolchildren with the properties of figures in two-dimensional and three-dimensional space, as well as develop abstract thinking in them. One of the simple figures that help to cope with this task is a prism. What it is and what properties it possesses is considered in the article.
Prism is the perfect polyhedron
To understand that this is a prism, imagine a polygon of arbitrary shape located on a plane. Now let's transfer this polygon parallel to ourselves to a new plane and fix it. The initial and final polygons, together with the region of space that he overcame during the parallel transfer, form a prism shape. To better visualize the described procedure for obtaining this figure, we present the figure.
The hexagonal prism is shown here . The red hexagon at the top is the source, the yellow hexagon at the bottom is the end. Flat figures are equal to each other and lie in parallel planes. They are called prism bases. Six lateral faces that connect the same sides of the bases to each other, are generally parallelograms. However, the sides can also be rectangles, squares, and rhombs.
Prism elements
When considering the answer to the question that this is a prism, it was noted that the figure under study is formed by faces of two types. If there are n-gons in the bases, then the number of side parallelograms will be n, and the total number of sides will be n + 2.
Parties are not the only element of the studied polyhedron. Prism also has several peaks and edges. A vertex is the intersection point of two parallelograms with a base. There are 2 * n such points at the prism. An edge is a segment formed as a result of the intersection of any two faces of a figure. The prism has two types of ribs: those related to the bases — their 2 * n pieces, and those belonging to the sides — their n pieces.
Of the above three elements of the polyhedron, perhaps the faces are the most important, since their type determines the specific type of prism.
What types of prisms are
As already noted, the shape of the prism is uniquely determined by the shape of the two types of faces that form the shape. In geometry, the following types of figures of the considered class are distinguished:
- Concave and convex. Judging by the name, these prisms have concave and convex bases, respectively. In stereometry, the properties of mainly convex prisms are studied. An example of a concave is the figure at the base of which lies a star.
- Direct and inclined. The difference between these types of prisms is in the shape of the sides. For a straight figure, all side quadrangles are rectangular; for an inclined figure, they will be parallelograms of an arbitrary type. An important property of direct prisms is that the length of any lateral rib coincides with the height of the volumetric figure.
- Wrong and right. Any prism is wrong if it does not have the right base and is not straight. The correct base is understood as a polygon, in which all angles and sides are equal. The most famous regular prism is a cube.
The figure above shows two quadrangular prisms. The left of them is straight, and the right is sloping.
Prism surface
The surface of the studied figure is a collection of points that belong to its faces. The main property of a surface is its area. This value is conveniently studied by considering the prism scan. The sweeps of the two quadrangular prisms are shown below.
To find the area of the prism, it is necessary to add the area of its two bases and add the area of the side surface to them. The marked mathematical operations are conveniently performed for the correct prisms, since a specific formula for the total area S can be written for them.
The area of a regular n-gon having side length a is calculated by the following formula:
S n = n / 4 * ctg (pi / n) * a 2
Since the base of the prism are equal to each other, their total area will be:
2 * S o = n / 2 * ctg (pi / n) * a 2
The side surface of a regular figure is represented by n identical rectangles. Denoting the height of the prism by the letter h, we write the formula for the area S b of the side surface:
S b = n * a * h
It remains to take the last step - add up the found areas of the bases and lateral surface:
S = 2 * S o + S b = n / 2 * ctg (pi / n) * a 2 + n * a * h
Figure volume
Considering the answer to the question that this is a prism, we found that we are talking about a figure that has some volume. The prism volume is easy to calculate if the area of its base and height are known. The corresponding formula is as follows:
V = S o * h
If it is necessary to find the volume of the inclined irregular prism, then the problem should be solved sequentially. First determine the area of the base, using the well-known formulas of planimetry, and then proceed to the calculation of the height h. In the latter case, one should use the knowledge of the dihedral angles between parallelograms and the base.
In the case of the correct figure, the formula for the volume takes the form:
V = n / 4 * ctg (pi / n) * a 2 * h
Where a is the base side of the prism.
As can be seen from the written expressions for S and V of the correct figure, two parameters should be known for their calculation - a and h.