Volume is a characteristic of any figure that has nonzero dimensions in all three dimensions of space. In this article, from the point of view of stereometry (geometry of spatial figures), we will consider a prism and show how to find the volumes of prisms of various types.
What is a prism?
Stereometry has the exact answer to this question. A prism in it is understood to mean a figure formed by two polygonal identical faces and several parallelograms. The figure below shows four different prisms.
Each of them can be obtained as follows: you need to take a polygon (triangle, quadrangle, and so on) and a segment of a certain length. Then each vertex of the polygon should be transferred using parallel segments to another plane. In a new plane, which will be parallel to the original one, a new polygon will be obtained, similar to the one selected initially.
Prisms can have a different type. So, they can be straight, inclined and right. If the lateral edge of the prism (the segment connecting the vertices of the bases) is perpendicular to the bases of the figure, then the latter is a straight line. Accordingly, if this condition is not satisfied, then we are talking about an inclined prism. The correct figure is a direct prism with an equiangular and equilateral base.
Later in the article, we will show how to calculate the volume of a prism of each of these types.
Volume of correct prisms
Let's start with the simplest case. We give the formula for the volume of a regular prism having an n-coal base. The volume formula V for any figure of the class in question has the following form:
V = S o * h.
That is, to determine the volume, it is enough to calculate the area of one of the bases S o and multiply it by the height h of the figure.
In the case of a correct prism, we denote the length of the side of its base by the letter a, and the height, which is equal to the length of the side edge, by the letter h. If the base represents the correct n-gon, then it is easiest to use the following universal formula to calculate its area:
S n = n / 4 * a 2 * ctg (pi / n).
Substituting into the equality the value of the number of sides n and the length of one side a, we can calculate the area of the n-coal base. Note that the cotangent function here is calculated for the angle pi / n, which is expressed in radians.
Given the equality written for S n , we obtain the final formula for the prism volume correct:
V n = n / 4 * a 2 * h * ctg (pi / n).
For each specific case, we can write the corresponding formulas for V, but they all uniquely follow from the written general expression. For example, for a regular quadrangular prism , which in the general case is a rectangular parallelepiped, we obtain:
V 4 = 4/4 * a 2 * h * ctg (pi / 4) = a 2 * h.
If in this expression we take h = a, then we get the formula for the volume of the cube.
Volume of direct prisms
We note immediately that for straight figures there is no general formula for calculating the volume, which was given above for regular prisms. When finding the value in question, use the original expression:
V = S o * h.
Here h is the length of the side rib, as in the previous case. As for the base area S o , it can take on a variety of values. The calculation problem for a direct prism of volume is reduced to finding the area of its base.
The calculation of the value of S o should be based on the characteristics of the base itself. For example, if it is a triangle, then the area can be calculated as follows:
S o3 = 1/2 * a * h a .
Here h a is the apothem of the triangle, that is, its height lowered to the base a.
If the base is a quadrangle, then it can be a trapezoid, parallelogram, rectangle or have a completely arbitrary type. For all these cases, use the appropriate planimetry formula to determine the area. For example, for a trapezoid, this formula has the form:
S o4 = 1/2 * (a 1 + a 2 ) * h a .
Where h a is the height of the trapezoid, a 1 and a 2 are the lengths of its parallel sides.
To determine the area for higher-order polygons, you should divide them into simple shapes (triangles, quadrangles) and calculate the sum of the areas of the latter.
Volume of inclined prisms
This is the most difficult case of prism volume calculation. The general formula for such figures is also applicable:
V = S o * h.
However, to the difficulty of finding the area of the base, representing a polygon of arbitrary type, the problem of determining the height of the figure is added. In an inclined prism, it is always less than the length of the side rib.
It is easiest to find this height if any corner of the figure is known (flat or dihedral). If such an angle is given, then with its use it is necessary to construct a rectangular triangle inside the prism that would contain the height h as one of the sides and, using the trigonometric functions and the Pythagorean theorem, find the value of h.
The geometric problem of determining the volume
Given a regular prism with a triangular base, having a height of 14 cm and a side length of 5 cm. What is the volume of a triangular prism?
Since we are talking about the correct figure, we are entitled to use the well-known formula. We have:
V 3 = 3/4 * a 2 * h * ctg (pi / 3) = 3/4 * 5 2 * 14 * 1 / √3 = √3 / 4 * 25 * 14 = 151.55 cm 3 .
A triangular prism is a fairly symmetrical figure, in the form of which various architectural structures are often performed. This glass prism is used in optics.