In the school course of stereometry, one of the simplest figures, which has non-zero dimensions along three spatial axes, is a quadrangular prism. Consider in the article what kind of figure this is, what elements it consists of, as well as how to calculate its surface area and volume.
Prism concept
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 pass into each other using the parallel transfer operation to some vector. Such a task of the 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 passes into the cylinder, since its base becomes a circle, and the side parallelograms, connecting, form a cylindrical surface.
Like any polyhedron, a prism is characterized by sides (planes that limit the figure), edges (segments along which any two sides intersect) and vertices (points of meeting of three sides, for the prism, two of them are lateral, and the third is the base). The amounts of these 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 a mathematical representation of Euler's theorem.
Above is a drawing showing two prisms. At the base of one of them (A) lies a regular hexagon, and the sides are lateral perpendicular to the bases. Figure B shows a different 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. The quadrangular prism has its own name - the box.
The number of sides of the box is six, with each side having a parallel parallel to it. Since the bases of the parallelepiped are two sides, the remaining four are lateral.
The number of parallelepiped vertices is eight, which can be easily verified if we recall that the vertices of the prism are formed only at the vertices of the base polygons (4x2 = 8). Using Euler's theorem, we get the number of edges:
P = C + B - 2 = 6 + 8 - 2 = 12
Of the 12 ribs, only 4 are formed independently by the sides. The remaining 8 lie in the planes of the base of the figure.
The rest of the article will focus only on quadrangular prisms.
Types of parallelepipeds
The first type of classification is the particularity of the parallelogram underlying. It can be of the following form:
- ordinary, whose angles are not equal to 90 o ;
- rectangle;
- a square is a regular quadrangle.
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 right, then the prism is called oblique or inclined;
- the angle is 90 o , then such a prism is rectangular or simply straight.
The third type of classification is related to the height of the prism. If the prism is rectangular, and either a square or a rectangle lies at the base, then it is called a rectangular parallelepiped. If the base is a square, 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.
The surface of the prism and its area
The set of all points that lie on the two bases of the 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, it is written like this:
S = 2 * S o + S b
Here, S o and S b are the area of the base and lateral surface, respectively. The number 2 in front of S o appears in view of the fact that 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 parallelogram area S p is calculated by the formula:
S p = a * h
Where the symbols a and h denote the length of one of its sides and the height drawn to this side, respectively.
Square rectangular prism square
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. Denote it by the letter h. Since all the side faces are perpendicular to the bases for the prism type under consideration, the height of the regular quadrangular prism will be equal to the length of its side edge.
There are two terms in the general formula for the surface area of a prism. The base area in this case is simple to calculate, it is equal to:
S o = a 2
To calculate the area of the side surface, we reason 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 S b will be equal to:
S b = 4 * a * h
Note that the product 4 * a 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 lateral surface can be calculated as follows:
S b = P o * h
Where P o 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 = 2 * S o + S b = 2 * a 2 + 4 * a * h = 2 * a * (a + 2 * h)
Square oblique box
Calculating it is somewhat more difficult than for a rectangular one. In this case, the base area of the quadrangular prism is calculated using the same formula as for the parallelogram. The changes relate to the method for determining the lateral surface area.
For this, the same formula is used across the perimeter that is given in the paragraph above. Only now in it will appear somewhat different factors. The general formula for S b in the case of an oblique prism has the form:
S b = P sr * c
Here c is the length of the side edge of the figure. The value of P sr is the perimeter of a rectangular slice. This medium is constructed as follows: it is necessary to intersect all lateral faces with a plane so that it is perpendicular to all of them. The formed rectangle will be the desired slice.
The figure above shows an example of an oblique box. The hatched cross-section with the sides forms right angles. The perimeter of the section is P sr . It is formed by four heights of lateral parallelograms. For this quadrangular prism, the lateral surface area is calculated using the above formula.
The length of the diagonal of a rectangular parallelepiped
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 rectangular box 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 line is its diagonal.
It is very simple to calculate its length if we recall the Pythagorean theorem. Each student can get the desired formula. It has the following form:
D = √ (A 2 + B 2 + C 2 )
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 box with the diagonals of its sides. The figure below shows where the colored diagonals show 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, one should also know their volume. This value for absolutely any prism is calculated by the following formula:
V = S o * h
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, then its volume will be equal to:
V = a 2 * h.
It is easy to see that this formula is converted into an expression for the volume of the cube if the length of the side edge h is equal to the side of the base a.
Rectangular Box Problem
To fix the studied material, we will solve the following problem: there is a rectangular parallelepiped, the sides of which are 3 cm, 4 cm and 5 cm. It is necessary to calculate its surface area, diagonal length and volume.
For definiteness, we assume that the base of the figure is a rectangle with sides 3 cm and 4 cm. Then its area is 12 cm 2 and the period is 14 cm. Using the formula for the surface area of the prism, we get:
S = 2 * S o + S b = 2 * 12 + 5 * 14 = 24 + 70 = 94 cm 2
To determine the length of the diagonal and the volume of the figure, you can directly use the above expressions:
D = √ (3 2 +4 2 +5 2 ) = 7.071 cm;
V = 3 * 4 * 5 = 60 cm 3 .
The oblique box problem
The figure below shows the oblique prism. Its sides are equal: a = 10 cm, b = 8 cm, c = 12 cm. It is necessary to find the surface area of this figure.
First, determine the area of the base. The figure shows that the acute angle is 50 o . Then its area is equal to:
S o = h * a = sin (50 o ) * b * a
To determine the area of the lateral surface, you should find the perimeter of the shaded rectangle. The sides of this rectangle are a * sin (45 o ) and b * sin (60 o ). Then the perimeter of this rectangle is:
P sr = 2 * (a * sin (45 o ) + b * sin (60 o ))
The total surface area of this box is:
S = 2 * S o + S b = 2 * (sin (50 o ) * b * a + a * c * sin (45 o ) + b * c * sin (60 o ))
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 cm 3
From the solution of this problem, it is clear that trigonometric functions are used to determine the areas of oblique figures.