Stereometry is an important part of the general course of geometry, which considers the characteristics of spatial figures. One such figure is a quadrangular prism. In this article, we will expand on the question of how to calculate the volume of a quadrilateral prism .
What is a quadrangular prism?
Obviously, before giving the formula for the volume of a quadrangular prism , it is necessary to give a clear definition of this geometric figure. Such a prism is understood to mean a three-dimensional polyhedron, which is bounded by two arbitrary identical quadrangles lying in parallel planes, and four parallelograms.
The marked quadrangles parallel to each other are called the base of the figure, and the four parallelograms are the sides. It should be clarified here that parallelograms are also quadrangles, however, bases are not always parallelograms. An example of an irregular quadrangle, which may well be the base of a prism, is shown in the figure below.
Any quadrangular prism consists of 6 sides, 8 vertices and 12 edges. There are quadrangular prisms of different types. For example, a figure may be oblique or straight, irregular and regular. Further in the article we show how you can calculate the volume of a quadrangular prism, taking into account its type.
Incorrect prism inclined prism
This is the most asymmetric view of a quadrangular prism, so calculating its volume will be relatively difficult. To determine the volume of the figure allows the following expression:
V = S o * h
The symbol S o here indicates the area of the base. If this base is a rhombus, parallelogram or rectangle, then it is easy to calculate the value of S o . So, for the rhombus and parallelogram, the following formula is true:
S o = a * h a
where a is the side of the base, h a is the length of the height lowered to this side from the top of the base of the height.
If the base is an irregular polygon (see above), then its area should be divided into simpler shapes (for example, triangles), calculate their area and find their sum.
In the formula for volume, the symbol h denotes the height of the prism. It represents the length of the perpendicular segment between the two bases. Since the prism is inclined, the calculation of the height h should be carried out using the length of the side rib b and the dihedral angles between the side faces and the base.
The correct figure and its volume
If the base of the quadrangular prism is a square, and the figure itself is straight, then it is called regular. It should be clarified that a direct prism is called when all its sides are rectangles and each of them is perpendicular to the bases. The correct figure is shown below.
The volume of a regular quadrangular prism can be calculated using the same formula as the volume of an irregular figure. Since the base is a square, its area is calculated simply:
S o = a 2
The height of the prism h is equal to the length of the side rib b (side of the rectangle). Then the volume of the correct quadrangular prism can be calculated by the following formula:
V = a 2 * h = a 2 * b
A regular prism with a square base is called a rectangular parallelepiped. In the case of equality of the sides a and b, this box becomes a cube. The volume of the latter is calculated as follows:
V = a 3
The written formulas for volume V indicate that the higher the symmetry of the figure, the less linear parameters are required to calculate this value. So, in the case of a correct prism, the required number of parameters is two, and in the case of a cube, one.
The task with the correct figure
Having considered the issue of finding the volume of a quadrangular prism from the point of view of theory, we apply the obtained knowledge in practice.
It is known that a regular parallelepiped has a base diagonal length of 12 cm. The diagonal length of its side is 20 cm. The parallelepiped volume must be calculated.
We denote the diagonal of the base by d a and the diagonal of the side face by d b . For the diagonal d a, the following expressions are valid:
d a = a * √2 =>
a = d a / √2
As for the value of d b , it is the diagonal of the rectangle with sides a and b. For it, we can write the following equalities:
d b 2 = a 2 + b 2 =>
b = √ (d b 2 - a 2 )
Substituting the last expression for the found expression for a, we obtain:
b = √ (d b 2 - d a 2/2)
Now you can substitute the resulting formulas in the expression for the volume of the correct figure:
V = a 2 * b = d a 2/2 * √ (d b 2 - d a 2/2)
Replacing d a and d b with numbers from the condition of the problem, we come to the answer: V ≈ 1304 cm 3 .