In the school course, stereometry studies the properties of different spatial figures. One of them is a pyramid. This article is devoted to the question of how to find the area of the side surface at the pyramid. The question of determining this area for a truncated pyramid is also disclosed.
What is a pyramid?
Many, having heard the word "pyramid", immediately represent the grandiose structures of Ancient Egypt. Indeed, the tombs of Cheops and Chefren are regular quadrangular pyramids. Nevertheless, the pyramid is also a tetrahedron, figures with a five-, six-, n-coal base.
In geometry, the concept of a pyramid is clearly defined. By this figure we mean an object in space that is formed as a result of connecting some point with the corners of a flat n-gon, where n is an integer. The figure below shows four pyramids with a different number of angles at the base.
The point with which all the vertices of the corners of the base are connected does not lie in its plane. It is called the top of the pyramid. If we draw a perpendicular to the base from it, then we get the height. The figure in which the height crosses the base at the geometric center is called the straight line. Sometimes a straight pyramid has a regular base, for example a square, an equilateral triangle, and so on. In this case, it is called correct.
When calculating the area of the side surface at a pyramid, it is convenient to work with the correct figures.
The surface area of the side shape
How to find the side surface area of a pyramid? You can understand this if you enter the appropriate definition and consider a flat scan for this figure.
Any pyramid is formed by faces that are separated by ribs. A base is a face formed by an n-gon. All other faces are triangles. There are n pieces, and together they form the side surface of the figure.
If along the lateral rib to cut the surface and deploy it on a plane, you get a scan of the pyramid. For example, a scan of a hexagonal pyramid is shown below.
It can be seen that the side surface is formed by six identical triangles.
Now it is not difficult to guess how to find the area of the side surface at the pyramid. To do this, add up the area of all triangles. In the case of an n-coal regular pyramid, the base side of which is equal to a, for the surface under consideration, we can write the formula:
S b = n * a * h b / 2
Here h b is the apothem of the pyramid. That is, the height of the triangle dropped from the top of the figure to the side of the base. If the apothem is unknown, then it can be calculated by knowing the parameters of the n-gon and the value of the height h of the figure.
The truncated pyramid and its surface
As the name suggests, a truncated pyramid can be obtained from an ordinary figure. To do this, cut off the top with a plane parallel to the base. The figure below demonstrates this process for a hexagonal figure.
Its lateral surface is the sum of the areas of the same isosceles trapezoid. The formula for the area of the lateral surface of a truncated pyramid (regular) is:
S b = h b * n * (a 1 + a 2 ) / 2
Here h b is the apothem of the figure, which is the height of the trapezoid. The values of a 1 and a 2 are the lengths of the bases of the sides.
Calculation of the side surface for a triangular pyramid
We show how to find the area of the side surface of the pyramid. Suppose we have a regular triangular one, we will examine an example of a specific problem. It is known that the side of the base, which is an equilateral triangle, is 10 cm. The height of the figure is 15 cm.
The development of this pyramid is shown in the figure. To use the formula for S b , you must first find the apothem h b . Considering the right triangle inside the pyramid, built on the sides h b and h, the equality can be written as follows:
h b = √ (h 2 + a 2/12)
We substitute the data and obtain that h b ≈15.275 cm.
Now you can use the formula for S b :
S b = n * a * h b / 2 = 3 * 10 * 15.275 / 2 = 229.125 cm 2
Note that the base of the triangular pyramid, like its side face, is formed by a triangle. Nevertheless, this triangle is not taken into account when calculating the area S b .