Triangular, quadrangular and hexagonal prisms are the most common volumetric figures among other similar ones that are found in everyday life and nature. Their properties are studied by stereometry, or spatial geometry. In this article, we will reveal the question of how to find the area of the lateral surface of a regular triangular prism, as well as a quadrangular and hexagonal.
What is a prism?
Before calculating the area of the lateral surface of a regular triangular prism and other types of this figure, you should understand what they are. Then we will learn to determine the quantities of interest.
From a geometry point of view, a prism is a volumetric body that is bounded by two arbitrary identical polygons and n parallelograms, where n is the number of sides of one polygon. To draw such a figure is easy, for this you need to draw some kind of polygon. Then draw a line from each of its vertices, which will be equal in length and parallel to all the others. Then you need to connect the ends of these lines together so that you get another polygon equal to the original.
It can be seen above that the figure is limited by two pentagons (they are called the lower and upper bases of the figure) and five parallelograms, which in the figure correspond to rectangles.
All prisms differ from each other in two main parameters:
- the type of polygon lying at the base of the shape;
- angles between parallelograms and bases.
The number of sides of the rectangle gives the name to the prism. From here we get the above-mentioned triangular, hexagonal and quadrangular figures.
They also differ in the amount of slope. As for the marked angles, then if they are equal to 90 o , then such a prism is called a straight line, or rectangular (the angle of inclination is zero). If some of the angles are not straight, then the figure is called oblique. The difference between them is visible at a glance. The figure below demonstrates these varieties.
As can be seen, the height h of the direct prism coincides with the length of its side rib. In the case of oblique, this parameter is always less.
Which prism is called correct?
Since we must answer the question of how to find the lateral surface area of a regular prism (triangular, quadrangular, etc.), we need to define this type of volumetric figure. We will analyze the material in more detail.
A regular prism is a rectangular shape in which a regular polygon forms identical bases. This figure can be an equilateral triangle, a square and others. Any n-gon, all sides of which the lengths and angles are the same, will be correct.
A number of such prisms are shown schematically in the figure below.
Prism side surface
As mentioned in the definition of a prism, this figure consists of n + 2 planes, which, intersecting, form n + 2 faces. Two of them belong to the bases, the rest are formed by parallelograms. The surface area consists of the sum of the areas of the indicated faces. If it does not include the values of two bases, then we get the answer to the question of how to find the area of the side surface of the prism. So, it is possible to determine its significance and grounds separately from each other.
Below is a scan of a triangular prism for which the side surface is formed by three quadrangles.
Consider the calculation process below. Obviously, the surface area of the prism is equal to the sum of n areas of the corresponding parallelograms. Here n is the number of sides of the polygon forming the base of the figure. The area of each parallelogram can be found by multiplying the length of its side by the height lowered onto it. This is regarding the general case.
If the prism under study is direct, then the procedure for determining the area of its lateral surface S b is much easier, since such a surface consists of rectangles. In this case, you can use the following formula:
S b = P o * h,
where h - the height of the figure, P o - the perimeter of its base
The correct prism and its side surface
The formula given in paragraph above in the case of such a figure takes on a very concrete form. Since the perimeter of an n-gon is equal to the product of the number of its sides by the length of one, the following formula is obtained:
S n b = n * a * h
Where a is the side length of the corresponding n-gon.
The lateral surface area of a regular triangular prism, quadrangular and hexagonal
We will use the formula above to determine the necessary values for the marked three types of figures. The calculations will look as follows.
For a triangular formula takes the form:
S 3 b = 3 * a * h
For example, the side of the triangle is 10 cm, and the height of the figure is 7 cm, then:
S 3 b = 3 * 10 * 7 = 210 cm 2
In the case of a quadrangular prism, the desired expression takes the form:
S 4 b = 4 * a * h
If we take the same length values as in the previous example, then we get:
S 4 b = 4 * 10 * 7 = 280 cm 2
The surface area of the hexagonal prism is calculated by the formula:
S 6 b = 6 * a * h
Substituting the same numbers as in the previous cases, we have:
S 6 b = 6 * 10 * 7 = 420 cm 2
Note that in the case of a regular prism of any type, its side surface is formed by identical rectangles. In the examples above, the area of each of them was a * h = 70 cm 2 .
Calculation for oblique prism
Determining the value of the lateral surface area for a given figure is somewhat more complicated than for a rectangular one. Nevertheless, the above formula remains the same, only instead of the perimeter of the base you should take the perimeter of the perpendicular section, and instead of the height - the length of the side rib.
The figure above shows a quadrangular oblique prism. The hatched parallelogram is that perpendicular section whose perimeter P sr must be calculated. The length of the side rib in the figure is indicated by the letter C. Then we obtain the formula:
S b = P sr * C
The cut perimeter can be found if the angles of the parallelograms forming the side surface are known.