When a person hears the word "pyramid", he immediately recalls the magnificent Egyptian buildings. Nevertheless, the ancient stone giants are only one of the representatives of the pyramid class. In this article, we will consider from a geometric point of view the properties of a regular quadrangular pyramid.
What is a pyramid in general?
In geometry, it is understood as a volumetric figure, which can be obtained by connecting all the vertices of a flat polygon with one single point lying in a different plane than this polygon. The figure below shows 4 figures that satisfy this definition.
We see that the first figure has a triangular base, the second is quadrangular. The last two are represented by a pentagonal and hexagonal base. However, the lateral surface of all pyramids is formed by triangles. Their number is exactly equal to the number of sides or vertices of the polygon at the base.
A special type of pyramids, which differ from the rest of the class in perfect symmetry, are regular pyramids. For the figure to be correct, the following two prerequisites must be met:
- the base must be a regular polygon;
- the lateral surface of the figure should consist of equal isosceles triangles.
Note that the second prerequisite can be replaced by another: the perpendicular drawn to the base from the top of the pyramid (the intersection point of the side triangles) must intersect this base at its geometric center.
Regular quadrangular pyramid
Now we turn to the topic of the article and consider what properties of a regular quadrangular pyramid characterize it. First, we show in the figure what this figure looks like.
Its base is a square. The sides represent 4 identical isosceles triangles (they can also be equilateral with a certain ratio of the length of the side of the square and the height of the figure). The height omitted from the top of the pyramid will cross the square in its center (the intersection point of the diagonals).
This pyramid has 5 faces (a square and four triangles), 5 vertices (four of them belong to the base) and 8 edges. The axis of symmetry of the fourth order, passing through the height of the pyramid, translates it into itself by turning 90 o .
The Egyptian pyramids at Giza are regular quadrangular.
Next, we give formulas to determine all the characteristics of this figure.
Four main linear parameters
We begin the consideration of the mathematical properties of a regular quadrangular pyramid with the formulas of height, length of the base side, side rib and apothema. We must say right away that all these quantities are related to each other, so it’s enough to know only two of them to uniquely calculate the remaining two.
Suppose that the height h of the pyramid and the length a of the side of the square base are known, then the side edge b will be equal to:
b = √ (a 2/2 + h 2 )
Now we give the formula for the length a b of the apothem (the height of the triangle lowered to the side of the base):
a b = √ (a 2/4 + h 2 )
Obviously, the lateral rib b is always larger than the apotheme a b .
Both expressions can be used to determine all four linear characteristics if the other two parameters are known, for example a b and h.
Area and volume of a figure
These are two more important properties of a regular quadrangular pyramid. The base of the figure has the following area:
S o = a 2
Every student knows this formula. The area of the side surface, which is formed by four identical triangles, can be determined through the apothem a b of the pyramid as follows:
S b = 2 × a × a b
If a b is unknown, then it can be determined by the formulas from the previous paragraph through the height h or edge b.
The total surface area of the figure in question is composed of areas S o and S b :
S = S o + S b = a 2 + 2 × a × a b = a (a + 2 × a b )
The calculated area of all faces of the pyramid is shown in the figure below in the form of its development.
The description of the properties of a regular quadrangular pyramid will not be complete if you do not consider the formula for determining its volume. This value for the pyramid in question is calculated as follows:
V = 1/3 × h × a 2
That is, V is equal to the third part of the product of the height of the figure by the area of its base.
Properties of a regular truncated quadrangular pyramid
You can get this figure from the original pyramid. To do this, cut off the top of the pyramid with a plane. The figure remaining under the cut plane will be called a truncated pyramid.
It is most convenient to study the characteristics of a truncated pyramid if its bases are parallel to each other. In this case, the lower and upper bases will be similar to polygons. Since the base in a quadrangular regular pyramid is a square, the section formed by cutting will also represent a square, but of a smaller size.
The lateral surface of the truncated figure is formed not by triangles, but by isosceles trapezoids.
One of the important properties of this pyramid is its volume, which is calculated by the formula:
V = 1/3 × h × (S o1 + S o2 + √ (S o1 × S o2 ))
Here h is the distance between the bases of the figure, S o1 , S o2 are the areas of the lower and upper bases.