Typical geometric problems on the plane and in three-dimensional space are the problems of determining the surface areas of different shapes. In this article, we give the formula for the area of the lateral surface of a regular pyramid of a quadrangular.
What is a pyramid?
We give a strict geometric definition of a pyramid. Suppose that there is a polygon with n sides and n angles. We choose an arbitrary point in space that will not be in the plane of the indicated n-gon, and connect it to each vertex of the polygon. We get a figure that has some volume, which is called an n-coal pyramid. For example, we will show in the figure below what a pentagonal pyramid looks like.
Two important elements of any pyramid are its base (n-gon) and top. These elements are connected to each other by n triangles, which in the general case are not equal to each other. The perpendicular, lowered from the top to the base, is called the height of the figure. If it crosses the base at the geometric center (coincides with the center of mass of the polygon), then such a pyramid is called a straight line. If, in addition to this condition, the base is a regular polygon, then the whole pyramid is called regular. The figure below shows how the regular pyramids look with triangular, quadrangular, pentagonal and hexagonal bases.
Pyramid surface
Before proceeding to the question of the area of the lateral surface of a regular pyramid of a quadrangular, it is necessary to dwell in more detail on the concept of the surface itself.
As mentioned above and shown in the figures, any pyramid is formed by a set of faces or sides. One side is the base, and n sides are triangles. The surface of the whole figure is the sum of the areas of each of its sides.
It is convenient to study the surface by the example of a figure sweep. The development for a regular quadrangular pyramid is shown in the figures below.
We see that its surface area is equal to the sum of four areas of the same isosceles triangles and the square area.
The total area of all triangles that form the sides of the figure is called the area of the side surface. Next, we show how to calculate it for the correct quadrangular pyramid.
The area of the lateral surface of a quadrangular regular pyramid
To calculate the area of the lateral surface of the indicated figure, we again turn to the above scan. Suppose we know the side of a square base. Denote it by a. It can be seen that each of the four identical triangles has a base of length a. To calculate their total area, you need to know this value for one triangle. From the course of geometry it is known that the triangle area S t is equal to the product of the base and the height, which should be divided in half. I.e:
S t = 1/2 * h b * a.
Where h b is the height of the isosceles triangle drawn to the base a. For a pyramid, this height is an apothem. Now it remains to multiply the resulting expression by 4 to obtain the surface area S b of the side surface for the pyramid under consideration:
S b = 4 * S t = 2 * h b * a.
This formula contains two parameters: apotheme and base side. If the latter is known in most conditions of the problems, then the former has to be calculated, knowing other quantities. We give the formulas for calculating the apothemes h b for two cases:
- when the length of the side rib is known;
- when the height of the pyramid is known.
If we denote the length of the lateral rib (side of the isosceles triangle) by the symbol L, then the apotheme h b is determined by the formula:
h b = √ (L 2 - a 2/4).
This expression is the result of applying the Pythagorean theorem for a triangle of the lateral surface.
If the height h of the pyramid is known, then the apotheme h b can be calculated as follows:
h b = √ (h 2 + a 2/4).
Obtaining this expression is also not difficult if we consider a right triangle inside the pyramid formed by the legs of h and a / 2 and the hypotenuse h b .
We show how to apply these formulas by solving two interesting problems.
Known surface area problem
It is known that the area of the lateral surface of a quadrangular pyramid is equal to 108 cm 2 . It is necessary to calculate the value of the length of its apotheme h b if the height of the pyramid is 7 cm.
We write the formula for the area S b of the lateral surface through the height. We have:
S b = 2 * √ (h 2 + a 2/4) * a.
Here we simply substituted the corresponding apotheme formula into the expression for S b . We square both sides of the equality:
S b 2 = 4 * a 2 * h 2 + a 4 .
To find the value of a, we make the change of variables:
a 2 = t;
t 2 + 4 * h 2 * t - S b 2 = 0.
We substitute the known values and solve the quadratic equation:
t 2 + 196 * t - 11664 = 0.
t ≈ 47.8355.
We have written out only the positive root of this equation. Then the sides of the base of the pyramid will be equal to:
a = √t = √47.8355 ≈ 6.916 cm.
To get the length of the apotheme, just use the formula:
h b = √ (h 2 + a 2/4) = √ (7 2 + 6,916 2/4) ≈ 7,808 cm.
The side surface of the Cheops pyramid
Let us determine the value of the lateral surface area for the largest Egyptian pyramid. It is known that at its base lies a square with a side length of 230.363 meters. The height of the structure was originally 146.5 meters. Substitute these numbers in the corresponding formula for S b , we get:
S b = 2 * √ (h 2 + a 2/4) * a = 2 * √ (146.5 2 +230.363 2/4) * 230.363 ≈ 85860 m 2 .
The found value is slightly larger than the area of 17 football fields.