The calculation of the area of figures is one of the priority tasks of geometry on the plane and in space. In this article, we consider a three-dimensional pyramid figure. And we will show which formulas of the base area of the pyramid should be used to calculate this value.
What is a pyramid?
The answer to this question is not as obvious as it may seem to many. When people hear the word "pyramid", a great stone structure of the Egyptian pharaohs pops up in their imagination. However, this is only a special case of figures of this class.
From the point of view of the exact science of geometry, a pyramid is a figure in space formed by an n-gon, each of the vertices of which is connected to one single point. This point in the plane of the n-gon should not be. Here n is an integer equal to the number of angles (sides) of a flat polygon. For a visual representation of the described figure, we present a photograph.
Here is a set of a variety of pyramids. The upper left is called triangular because its base is a triangle. The lower right pyramid is called the diagonal.
This photo allows us to draw some conclusions regarding the pyramids. First, the sides that connect the n-gon to the top of the shape are triangles. Secondly, the number of sides of any pyramid is n + 1 (one n-gon and n triangles), the n-gon is called the base, and the triangles are called side faces. Thirdly, it can be noted that an increase in the sides of the base brings the pyramid closer in shape to a cone. This fact allows us to consider a cone a pyramid with an infinite number of side faces.
Right and wrong figures
We found out what the base of the figure is. Nevertheless, before starting a discussion of the formula for the area of the base of the pyramid, it is necessary to define the correct and incorrect figures of this class.
Every student knows that any flat polygon has a geometric center. If the polygon is made of a homogeneous material, then the geometric center will coincide with the center of mass. For example, the geometric center of a rectangle is the point where its diagonals intersect; for a triangle, it is at the intersection of medians. The concept of a geometric center is associated with the concepts of a regular and an irregular pyramid.
Above was mentioned the top of the pyramid. It corresponds to the point where all triangular side faces of the figure intersect. If the perpendicular is lowered from the top to the base, then the length of the resulting segment will correspond to the distance from the top to the base. This segment is called the height of the figure.
If the height intersects the polygon in its geometric center, then the pyramid is called a straight line. If the base of the straight pyramid is a polygon having sides of the same length and equal angles to each other, then the pyramid is called regular. Accordingly, if any of the above conditions is not met, then they say about the wrong pyramid.
According to the described classification, the Cheops pyramid is regular quadrangular, having a square at the base.
The base area of the regular pyramid
To calculate the area of the base of the pyramid, appropriate formulas for a specific n-gon should be used. For example, in the case of a triangle, this is the product of the height and the base, which is divided in half, in the case of a parallelogram, this is the product of the side and the height dropped to it.
If the n-gon is correct, then the formula for the base area of the pyramid will be universal. We write it down:
S n = n / 4 * a 2 * ctg (pi / n)
Where parameter a is the length of the side of the n-carbon base. This formula is valid regardless of whether a triangular or rectangular pyramid is considered. The cotangent function should be calculated using a calculator, but for a 3-, 4-, and 6-gon, it has a tabular value.
Note that this formula can be used if the base of the pyramid is a regular polygon. The validity of the formula does not depend on whether the pyramid is straight or inclined.
The triangle is triangular correct
An equilateral triangle is the base of a regular triangular pyramid. The base area can be determined by applying the formula in S above for S n . Given that n = 3, we obtain:
S 3 = n / 4 * a 2 * ctg (pi / n) = 3/4 * a 2 * ctg (pi / 3) = √3 / 4 * a 2
Knowing the base length a, we can calculate the corresponding area.
It is curious to note that in the triangular pyramid, which is often called the tetrahedron, all four sides are triangles. In the correct figure, in the general case, only one of the sides is an equilateral triangle (base). The remaining faces are isosceles triangles.
The quadrangular pyramid is right
Perhaps she is the most famous among the class of pyramids. The formula for the square of the base of the pyramid of the quadrangular regular is known to the student in the elementary grades, since we are talking about the square area. Following the general approach, we use the expression for S n . Substituting n = 4, we obtain:
S 4 = 4/4 * a 2 * ctg (pi / 4) = a 2
For example, we find the base area of the quadrangular pyramid of Cheops. The length of the side of its base is approximately 230 meters. This means that the corresponding area is 52.9 thousand m 2 , which is more than the area of 10 football fields.