In the school course of solid geometry, one of the simplest figures that has non-zero dimensions along three spatial axes is a quadrangular prism. Consider in the article what kind of figure it is, what elements it consists of, and also how you can calculate its surface area and volume.
The concept of a prism
In geometry, a prism is a spatial figure, which is formed by two identical bases and side surfaces that connect the sides of these bases. Note that both bases are transformed into each other using the operation of parallel translation by some vector. This definition of a prism leads to the fact that all its sides are always parallelograms.
The number of sides of the base can be arbitrary, starting from three. When this number tends to infinity, the prism smoothly turns into a cylinder, since its base becomes a circle, and the side parallelograms, connecting, form a cylindrical surface.
Like any polyhedron, a prism is characterized bysides (planes that bound the figure), edges (segments along which any two sides intersect) and vertices (meeting points of three sides, for a prism two of them are lateral, and the third is the base). The quantities of the named three elements of the figure are interconnected by the following expression:
P=C + B - 2
Here P, C and B are the number of edges, sides and vertices, respectively. This expression is the mathematical notation of Euler's theorem.
The picture above shows two prisms. At the base of one of them (A) lies a regular hexagon, and the side sides are perpendicular to the bases. Figure B shows another prism. Its sides are no longer perpendicular to the bases, and the base is a regular pentagon.
What is a quadrangular prism?
As is clear from the description above, the type of prism is primarily determined by the type of polygon that forms the base (both bases are the same, so we can talk about one of them). If this polygon is a parallelogram, then we get a quadrangular prism. Thus, all sides of this type of prism are parallelograms. A quadrangular prism has its own name - a parallelepiped.
The number of sides of a parallelepiped is six, and each side has a similar parallel to it. Since the bases of the box are two sides, the remaining four are lateral.
The number of vertices of the parallelepiped is eight, which is easy to see if we remember that the vertices of the prism are formed only at the vertices of the base polygons (4x2=8). Applying Euler's theorem, we get the number of edges:
P=C + B - 2=6 + 8 - 2=12
Out of 12 ribs, only 4 are formed independently by the sides. The remaining 8 lie in the planes of the bases of the figure.
Further in the article we will talk only about quadrangular prisms.
Types of parallelepipeds
The first type of classification is the features of the parallelogram underlying. It may look like this:
- regular, whose angles are not equal to 90o;
- rectangle;
- a square is a regular quadrilateral.
The second type of classification is the angle at which the side crosses the base. Two different cases are possible here:
- this angle is not straight, then the prism is called oblique or oblique;
- the angle is 90o, then such a prism is rectangular or just straight.
The third type of classification is related to the height of the prism. If the prism is rectangular, and the base is either a square or a rectangle, then it is called a cuboid. If there is a square at the base, the prism is rectangular, and its height is equal to the length of the side of the square, then we get the well-known cube figure.
Prism surface and area
The set of all points that lie on two bases of a prism(parallelograms) and on its sides (four parallelograms) form the surface of the figure. The area of this surface can be calculated by calculating the area of the base and this value for the side surface. Then their sum will give the desired value. Mathematically, this is written as follows:
S=2So+ Sb
Here So and Sb are the area of the base and side surface, respectively. The number 2 before So appears because there are two bases.
Note that the written formula is valid for any prism, and not just for the area of a quadrangular prism.
It is useful to recall that the area of a parallelogram Sp is calculated by the formula:
Sp=ah
Where the symbols a and h denote the length of one of its sides and the height drawn to this side, respectively.
The area of a rectangular prism with a square base
In a regular quadrangular prism, the base is a square. For definiteness, we denote its side by the letter a. To calculate the area of a regular quadrangular prism, you should know its height. According to the definition for this quantity, it is equal to the length of the perpendicular dropped from one base to another, that is, equal to the distance between them. Let's denote it by the letter h. Since all side faces are perpendicular to the bases for the type of prism under consideration, the height of a regular quadrangular prism will be equal to the length of its side edge.
BThe general formula for the surface area of a prism is two terms. The area of \u200b\u200bthe base in this case is easy to calculate, it is equal to:
So=a2
To calculate the area of the lateral surface, we argue as follows: this surface is formed by 4 identical rectangles. Moreover, the sides of each of them are equal to a and h. This means that the area of Sb will be equal to:
Sb=4ah
Note that the product 4a is the perimeter of the square base. If we generalize this expression to the case of an arbitrary base, then for a rectangular prism the side surface can be calculated as follows:
Sb=Poh
Where Po is the perimeter of the base.
Returning to the problem of calculating the area of a regular quadrangular prism, we can write the final formula:
S=2So+ Sb=2a2+ 4 ah=2a(a+2h)
Area of an oblique parallelepiped
Calculating it is somewhat more difficult than for a rectangular one. In this case, the area of the base of a quadrangular prism is calculated using the same formula as for a parallelogram. The changes concern the way the lateral surface area is determined.
To do this, use the same formula through the perimeter as given in the paragraph above. Only now it will have slightly different multipliers. The general formula for Sb in the case of an oblique prism is:
Sb=Psrc
Here c is the length of the side edge of the figure. The value Psr is the perimeter of the rectangular slice. This environment is built as follows: it is necessary to intersect all the side faces with a plane so that it is perpendicular to all of them. The resulting rectangle will be the desired cut.
The figure above shows an example of an oblique box. Its cross-hatched section forms right angles with the sides. The perimeter of the section is Psr. It is formed by four heights of lateral parallelograms. For this quadrilateral prism, the lateral surface area is calculated using the above formula.
The length of the diagonal of a cuboid
The diagonal of a parallelepiped is a segment that connects two vertices that do not have common sides that form them. There are only four diagonals in any quadrangular prism. For a cuboid with a rectangle at its base, the lengths of all diagonals are equal to each other.
The figure below shows the corresponding figure. The red segment is its diagonal.
Calculating its length is very simple, if you remember the Pythagorean theorem. Each student can get the desired formula. It has the following form:
D=√(A2+ B2 + C2)
Here D is the length of the diagonal. The remaining characters are the lengths of the sides of the box.
Many people confuse the diagonal of a parallelepiped with the diagonals of its sides. Below is a picture where the coloredthe segments represent the diagonals of the sides of the figure.
The length of each of them is also determined by the Pythagorean theorem and is equal to the square root of the sum of the squares of the corresponding side lengths.
Prism volume
In addition to the area of a regular quadrangular prism or other types of prisms, to solve some geometric problems, you should also know their volume. This value for absolutely any prism is calculated by the following formula:
V=Soh
If the prism is rectangular, then it is enough to calculate the area of its base and multiply it by the length of the edge of the side to get the volume of the figure.
If the prism is a regular quadrangular prism, then its volume will be:
V=a2h.
It is easy to see that this formula is converted into an expression for the volume of a cube if the length of the side edge h is equal to the side of the base a.
Problem with a cuboid
To consolidate the studied material, we will solve the following problem: there is a rectangular parallelepiped whose sides are 3 cm, 4 cm and 5 cm. It is necessary to calculate its surface area, diagonal length and volume.
For definiteness, we will assume that the base of the figure is a rectangle with sides of 3 cm and 4 cm. Then its area is 12 cm2, and the period is 14 cm. Using the formula for the surface area of the prism, we get:
S=2So+ Sb=212 + 514=24 + 70=94cm 2
To determine the length of the diagonal and the volume of the figure, you can directly use the above expressions:
D=√(32+42+52)=7. 071 cm;
V=345=60cm3.
Problem with an oblique parallelepiped
The figure below shows an oblique prism. Its sides are equal: a=10 cm, b=8 cm, c=12 cm. You need to find the surface area of this figure.
First, let's determine the area of the base. The figure shows that the acute angle is 50o. Then its area is:
So=ha=sin(50o)ba
To determine the area of the lateral surface, you should find the perimeter of the shaded rectangle. The sides of this rectangle are asin(45o) and bsin(60o). Then the perimeter of this rectangle is:
Psr=2(asin(45o)+bsin(60o))
The total surface area of this box is:
S=2So+ Sb=2(sin(50o)ba + acsin(45o) + bcsin(60o))
We substitute the data from the condition of the problem for the lengths of the sides of the figure, we get the answer:
S=458, 5496 cm3
It can be seen from the solution of this problem that trigonometric functions are used to determine the areas of oblique figures.
