The school geometry course is divided into two large sections: planimetry and solid geometry. Stereometry studies spatial figures and their characteristics. In this article, we will look at what a straight prism is and give formulas that describe its properties such as diagonal lengths, volume and surface area.
What is a prism?
When schoolchildren are asked to name the definition of a prism, they answer that this figure is two identical parallel polygons, the sides of which are connected by parallelograms. This definition is as general as possible, since it does not impose conditions on the shape of polygons, on their mutual arrangement in parallel planes. In addition, it implies the presence of connecting parallelograms, the class of which also includes a square, a rhombus, and a rectangle. Below you can see what a quadrangular prism is.
We see that a prism is a polyhedron (polyhedron) consisting of n + 2sides, 2 × n vertices and 3 × n edges, where n is the number of sides (vertices) of one of the polygons.
Both polygons are usually called the bases of the figure, the remaining faces are the sides of the prism.
The concept of a straight prism
There are different kinds of prisms. So, they talk about regular and irregular figures, about triangular, pentagonal and other prisms, there are convex and concave figures, finally, they are inclined and straight. Let's talk about the latter in more detail.
A right prism is such a figure of the studied class of polyhedra, all side quadrilaterals of which have right angles. There are only two types of such quadrilaterals - a rectangle and a square.
The considered form of the figure has an important property: the height of a straight prism is equal to the length of its lateral edge. Note that all side edges of the figure are equal to each other. As for the side faces, in the general case they are not equal to each other. Their equality is possible if, in addition to the fact that the prism is straight, it will also be correct.
The figure below shows a straight figure with a pentagonal base. It can be seen that all its side faces are rectangles.
Prism diagonals and its linear parameters
The main linear characteristics of any prism are its height h and the lengths of the sides of its base ai, where i=1, …, n. If the base is a regular polygon, then it suffices to know the length a of one side to describe its properties. Knowing the marked linear parameters allows us to unambiguouslydefine such properties of a figure as its volume or surface.
The diagonals of a straight prism are segments that connect any two nonadjacent vertices. Such diagonals can be of three types:
- lying in the base planes;
- located in the planes of the side rectangles;
- figures belonging to the volume.
The lengths of those diagonals related to the base should be determined depending on the type of n-gon.
Diagonals of side rectangles are calculated using the following formula:
d1i=√(ai2+ h2).
To determine volume diagonals, you need to know the value of the length of the corresponding base diagonal and height. If some diagonal of the base is denoted by the letter d0i, then the volume diagonal d2i is calculated as follows:
d2i=√(d0i2+ h2).
For example, in the case of a regular quadrangular prism, the length of the volume diagonal will be:
d2=√(2 × a2+ h2).
Note that a right triangular prism has only one of the three named types of diagonals: the side diagonal.
Surface of the studied class of shapes
Surface area is the sum of the areas of all the faces of a figure. To visualize all the faces, you should make a scan of the prism. As an example, such a sweep for a pentagonal figure is shown below.
We see that the number of plane figures is n + 2, and n are rectangles. To calculate the area of the entire sweep, add the areas of two identical bases and the areas of all rectangles. Then the corresponding formula will look like:
S=2 × So+ h × ∑i=1n (ai).
This equality shows that the lateral surface area for the studied type of prisms is equal to the product of the height of the figure and the perimeter of its base.
The base area of So can be calculated by applying the appropriate geometric formula. For example, if the base of a right prism is a right triangle, then we get:
So=a1 × a2 / 2.
Where a1 and a2 are the legs of the triangle.
If the base is an n-gon with equal angles and sides, then the following formula will be fair:
So=n / 4 × ctg (pi / n) × a2.
Volume Formula
Determining the volume of a prism of any kind is not a difficult task if its base area So and height h are known. Multiplying these values together, we get the volume V of the figure, that is:
V=So × h.
Since the parameter h of a straight prism is equal to the length of the side edge, the whole problem of calculating the volume is reduced to calculating the area So. Above wehave already said a few words and given a couple of formulas to determine So. Here we only note that in the case of an arbitrary-shaped base, it should be divided into simple segments (triangles, rectangles), calculate the area of each, and then add all the areas to get So.
