The ability to determine the volume of spatial figures is important for solving geometric and practical problems. One such figure is a prism. Consider in the article what it is and show how to calculate the volume of an inclined prism.
What is understood by a prism in geometry?
This is a regular polyhedron (polyhedron), which is formed by two identical bases located in parallel planes, and several parallelograms connecting the marked bases.
The prism bases can be arbitrary polygons, for example, a triangle, a quadrangle, a heptagon, and so on. Moreover, the number of angles (sides) of the polygon determines the name of the figure.
Any prism having an n-gon in its base (n is the number of sides) consists of n + 2 faces, 2 Γ n vertices and 3 Γ n edges. From the above numbers it is seen that the number of prism elements corresponds to Euler's theorem:
3 Γ n = 2 Γ n + n + 2 - 2
The figure below shows what triangular and quadrangular prisms made of glass look like.
Types of figures. Tilt prism
It has already been said above that the name of the prism is determined by the number of sides of the polygon at the base. However, there are other features in its structure that determine the properties of the figure. So, if all the parallelograms forming the side surface of the prism are represented by rectangles or squares, then such a figure is called a straight line. For a direct prism, the distance between the bases is equal to the length of the side edge of any rectangle.
If some or all of the sides are parallelograms, then we are talking about an inclined prism. Its height will already be less than the length of the side rib.
Another criterion by which the classification of the considered figures is carried out is the lengths of the sides and the corners of the polygon at the base. If they are equal to each other, then the polygon will be correct. A straight shape with a regular polygon in the bases is called a regular one. It is convenient to work with it when determining the surface area and volume. The inclined prism in this regard presents some difficulties.
The figure shows two prisms having a quadrangular base. An angle of 90 Β° shows the fundamental difference between a direct and inclined prism.
Formula for determining the volume of a figure
The part of space bounded by the faces of the prism is called its volume. For the considered figures of any type, this value can be determined by the following formula:
V = h Γ S o
Here, the symbol h denotes the height of the prism, which is a measure of the distance between the two bases. Symbol S o - one base area.
The area of ββthe base is easy to find. Given the fact that the polygon is correct or not, and also knowing the number of its sides, you should apply the appropriate formula and get S o . For example, for a regular n-gon with side length a, the area will be equal to:
S n = n / 4 Γ a 2 Γ ctg (pi / n)
Now we turn to the height h. For a direct prism, determining the height does not present any difficulties, but for an inclined prism, this is not an easy task. It can be solved by various geometric methods, starting from specific initial conditions. Nevertheless, there is a universal way to determine the height of a figure. We will describe it briefly.
The idea is to find the distance from a point in space to a plane. Suppose that the plane is given by the equation:
A Γ x + B Γ y + C Γ z + D = 0
Then from the point with coordinates (x 1 ; y 1 ; z 1 ) the plane will be at a distance:
h = | A Γ x 1 + B Γ y 1 + C Γ z 1 + D | / β (A 2 + B 2 + C 2 )
If the coordinate axes are positioned so that the point (0; 0; 0) lies in the plane of the lower base of the prism, then the equation for the base plane can be written as follows:
z = 0
This means that the formula for the height is written like this:
h = z 1
It is enough to find the z coordinate of any point of the upper base to determine the height of the figure.
Problem solving example
The figure below shows a quadrangular prism. The base of the inclined prism is a square with a side of 10 cm. It is necessary to calculate its volume if it is known that the length of the side rib is 15 cm and the acute angle of the front parallelogram is 70 Β°.
Since the height h of the figure is also the height of the parallelogram, we use formulas to determine its area in order to find h. We denote the sides of the parallelogram as follows:
a = 10 cm;
b = 15 cm
Then we can write for him the following formulas for determining the area S p :
S p = a Γ b Γ sin (Ξ±);
S p = a Γ h
Where do we get:
h = b Γ sin (Ξ±)
Here Ξ± is the acute angle of the parallelogram. Since the base is a square, the volume formula of the inclined prism will take the form:
V = a 2 Γ b Γ sin (Ξ±)
We substitute the data into the formula from the condition and get the answer: V β 1410 cm 3 .