When stereometry is studied in schools, much attention is paid in the educational process to the study of formulas and methods for calculating the volume of figures in space. In this article, consider what a prism is and give formulas for calculating the volume of a direct prism.
Prism in stereometry
Before we consider the formulas for determining the volume of a prism of a straight line, we will analyze in more detail what class of geometric objects will be discussed. Prism is a vivid representative of perfect polyhedra, in which the number of sides, vertices and edges is connected by Euler equality.
In stereometry, they give the following simple definition of a prism: this is a polyhedron in which the two faces that form it are absolutely identical and are in parallel planes, and the remaining faces are either parallelograms or rectangles (a special case of a parallelogram).
The figure shows a hexagonal prism in which all of the above elements are highlighted. So, she has 12 vertices, 18 edges and 8 sides or faces. The yellow sides together form a side surface, and the red side is the base of the figure.
Geometrically, any prism can be obtained using the following simple technique. It is necessary to take an arbitrary polygon. The only condition is that the polygon must be flat. Then, using a certain segment, which does not belong to the plane of the polygon, transfer all its vertices to a new parallel plane. As a result, we get a prism frame.
Kinds of a figure and a direct prism
In stereometry, several types of prism classifications are distinguished. Figures of this class are distinguished by the number of sides in the base, by the concavity or convexity of the base, by the type of all faces of the side surface and by the correct shape of the base. Here we do not set the task to consider all classification options, but focus only on direct and inclined prisms.
A straight or rectangular prism is a polyhedron in which all sides are rectangles or squares. Another definition of a prism, which is equal with the previous one, says: if all the side ribs are perpendicular to the bases, then a rectangular figure takes place. Both definitions can be used to solve practical problems, since they are sufficient and necessary.
If in a direct prism the bases are equal-angled and equilateral polygons, then they say about such a figure that it is correct. From these definitions it is clear that any regular prism will be rectangular, but not every straight figure is correct.
Direct prism volume formula
In the considered form of the prism, the length of any side edge is equal to the height h of the figure. At the same time, for both a rectangular and an inclined prism, the volume is determined by the following formula:
V = S o × h.
Where the symbol S o denotes the area of one base. If the letter b denotes the length of the side edge, then this expression for a straight figure can be rewritten in the form:
V = S o × b.
As follows from these equalities, the whole question of determining the value of V is to calculate the area S o . Naturally, there is no specific formula for its definition, since the base can be a polygon of a completely arbitrary type. Nevertheless, for some figures, specific formulas can be written.
For example, if the direct prism is correct, then for S o the formula is valid:
S o = n / 4 × ctg (pi / n) × a 2 .
Where n and a are the number of sides and the length of one side, respectively.
Another example where a specific formula can be given is a triangular base. If the length of the side a and the height of the triangle, lowered onto it, h a are known, then the area S o is calculated as follows:
S o = 1/2 × a × h a .
Note that when using the formula for the V figure, you should substitute the area of one rather than two bases in it.
Problem solving example
It is necessary to determine the volume of a prism of a straight line, the base of which is a right triangle. The hypotenuse of a triangle is 12 cm, and one of its legs is 4 cm. The height of the volumetric figure is 10 cm.
From the planimetric course, every student knows that half of the leg work gives the area of the corresponding triangle. Using the Pythagorean theorem, we get the second leg:
a 2 = √ (c 2 - a 1 2 ) = √ (12 2 - 4 2 ) = 11.31 cm.
Then the area S o of the triangle is:
S o = 4 × 11.31 / 2 = 22.62 cm 2 .
Applying the formula for volume, we obtain:
V = 22.62 × 10 = 226.2 cm 3 .
Thus, the volume of the direct prism is 226.2 cm 3 .