The concept of a prism. Volume formulas for prisms of different types: regular, straight and oblique. The solution of the problem

2024 Author: Angel Austin | [email protected]. Last modified: 2023-12-17 05:14

Volume is a characteristic of any figure that has non-zero dimensions in all three dimensions of space. In this article, from the point of view of stereometry (the 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 as a figure formed by two identical polygonal faces and several parallelograms. The picture below shows four different prisms.

Each of them can be obtained as follows: you need to take a polygon (triangle, quadrilateral, 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 the new plane, which will be parallel to the original one, a new polygon will be obtained, similar to the one chosen initially.

Prisms can be of different types. So, they can be straight, oblique and correct. If the lateral edge of the prism (segment,connecting the vertices of the bases) perpendicular to the bases of the figure, then the latter is a straight line. Accordingly, if this condition is not met, then we are talking about an inclined prism. A regular figure is a right prism with an equiangular and equilateral base.

Later in the article we will show how to calculate the volume of each of these types of prisms.

Volume of regular prisms

Let's start with the simplest case. We give the formula for the volume of a regular prism with an n-gonal base. The volume formula V for any figure of the class under consideration is as follows:

V=S_oh.

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 regular prism, let's denote the length of the side of its base with the letter a, and the height, which is equal to the length of the side edge, with the letter h. If the base of the n-gon is correct, then the easiest way to calculate its area is to use the following universal formula:

S=n/4a²ctg(pi/n).

Substituting the value of the number of sides n and the length of one side a, we can calculate the area of the n-gonal 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, we obtain the final formula for the volume of a regular prism:

V=n/4a²hctg(pi/n).

For each specific case, you can write the corresponding formulas for V, but they alluniquely follow from the written general expression. For example, for a regular quadrangular prism, which in the general case is a rectangular parallelepiped, we get:

V₄=4/4a²hctg(pi/4)=a² h.

If we take h=a in this expression, then we get the formula for the volume of the cube.

Volume of direct prisms

We note right away that for straight figures there is no general formula for calculating volume, which was given above for regular prisms. When finding the value in question, the original expression should be used:

V=S_oh.

Here h is the length of the side edge, as in the previous case. As for the base area S_o, it can take on a variety of values. The task of calculating a straight prism of volume is reduced to finding the area of its base.

The calculation of the value of S_oshould be carried out based on the characteristics of the base itself. For example, if it is a triangle, then the area can be calculated like this:

S_o3=1/2ah_a.

Here h_a is the apothem of the triangle, that is, its height lowered to the base a.

If the base is a quadrilateral, then it can be a trapezoid, a parallelogram, a rectangle, or a completely arbitrary type. For all these cases, you should use the appropriate planimetry formula to determine the area. For example, for a trapezoid, this formula looks like:

S_o4=1/2(a₁+ a₂)h _a.

Where h_a is the height of the trapezoid, a₁ and a₂ are the lengths of its parallel sides.

To determine the area for polygons of a higher order, you should split them into simple shapes (triangles, quadrangles) and calculate the sum of the areas of the latter.

Tilted Prism Volume

This is the most difficult case of calculating the volume of a prism. The general formula for such figures also applies:

V=S_oh.

However, to the complexity of finding the area of the base representing an arbitrary type of polygon, the problem of determining the height of the figure is added. It is always less than the length of the side edge in an inclined prism.

The easiest way to find this height is if you know any angle of the figure (flat or dihedral). If such an angle is given, then one should use it to construct a right-angled triangle inside the prism, which would contain the height h as one of the sides and, using trigonometric functions and the Pythagorean theorem, find the value h.

Geometric volume problem

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 the triangular prism?

Since we are talking about the correct figure, we have the right to use the well-known formula. We have:

V₃=3/4a²hctg(pi/3)=3/45²141/√3=√3/42514=151.55 cm³.

A triangular prism is a fairly symmetrical figure, in the form of which various architectural structures are often made. This glass prism is used in optics.