One of the geometric figures whose properties are studied in schools in the course of stereometry is a pyramid. Consider what this figure is, as well as describe in detail an important linear parameter - the side edge of the pyramid.
Pyramid as a geometry figure
Before considering the concept of the lateral edge of a pyramid, a definition of this spatial figure should be given. In short, the pyramid is a surface bounded by one n-gon and n triangles. The figure below shows one of the possible options for this figure.
From a geometric point of view, you can get a pyramid in this way: take an n-gon and connect all its angles with some point in space, which should not lie in the plane of the n-gon.
Note that, regardless of the number of sides n in the original polygon, always when connecting its angles with a single point, triangles are obtained. Their combination forms the side surface of the pyramid, and the original polygon is its base. The point at which all the triangles connect is called the top of the pyramid.
Pyramid elements
Each pyramid is formed by three main elements:
The faces or sides of the figure are always n + 1. This is easy to see in the figure given in the previous paragraph. The hexagonal base is one face. The remaining 6 sides are triangles resting on the sides of the base and intersecting at the top of the pyramid.
Edges are a set of intersection points of adjacent faces. A figure has two types of these elements:
- base ribs;
- side edges of the pyramid.
Their numbers, regardless of the number of sides n of the base, are always equal to each other, that is, the figure has 2 × n edges. If everything is clear with the edges of the base (they are the sides of the n-gon), then for the side ribs it should be clarified that they are segments connecting the corners of the base with the height of the figure in question.
Finally, the third type of pyramid elements will be vertices. The shape has n + 1 vertices. However, n of them are formed by the base and two side faces. Only one single peak is not connected with the base. It plays an important role in the study of the quantitative characteristics of the pyramid, for example, its height or apothem.
Regular pyramids
Pyramids can be inclined and straight, regular and irregular, convex and concave. All these types of figures differ from each other in a polygonal base and features of height behavior.
Suppose there is a pyramid in which the height (the perpendicular dropped from the top to the base) falls on the polygon exactly at its geometric center. In this case, the figure is called a straight line. If the polygon is equilateral, then in addition to the straight line, the pyramid will also be regular. Recall that the center of a geometric plane figure is similar to the center of mass in physics. For a square, it coincides with the intersection point of the diagonals, and for a triangle, it coincides with the point where the medians intersect.
The correct pyramids are conveniently studied due to their symmetry. So, the side edges of the regular pyramid and its side faces are equal to each other. A special case is the situation when the side faces will be formed by equilateral triangles.
Next, we consider what formulas should be used to determine the sizes of the side edges of the pyramids - the regular quadrangular and triangular.
Triangular pyramid
There are four linear parameters that describe the dimensions of a regular pyramid. These include the base side a, the lateral rib b, the height h and the apothem h b . Below are the formulas that allow you to calculate the length of the side rib for the correct triangular pyramid. The base of this figure is a triangle with equal sides, which allows us to write the following equalities:
b = √ (h b 2 + a 2/4);
b = √ (h 2 + a 2/3).
Both formulas are a consequence of the Pythagorean theorem for triangles in which the side edge b is a hypotenuse.
Quadrangular Pyramid
This figure is perhaps the most famous among the other pyramids due to the majestic ancient Egyptian structures. The lateral edge of the correct quadrangular pyramid can be determined by the following formulas:
b = √ (h b 2 + a 2/4);
b = √ (h 2 + a 2/2).
As in the previous case, these expressions are a consequence of the properties of the legs and the hypotenuse of a right triangle.
We note that the formula for calculating the lateral edge of a regular pyramid of a quadrangular shape through its apothem and side of the base is similar to that for a triangular figure. This coincidence is not accidental, since the side faces of both pyramids are isosceles triangles.
The task of determining the lateral edge of the Cheops pyramid
Everyone knows that the first miracle of the world - the pyramid of Cheops, has a dizzying size. It is the largest of all the pyramids located in the Egyptian Giza. The sides of its base form a square with an accuracy of several tens of centimeters. The average length of the side of the pyramid is estimated at 230.363 meters. The height of the pyramid is currently about 137 meters, but the initial height of the stone giant was 146.50 meters.
We will use the above figures to determine what the side edge of the regular quadrangular pyramid dedicated to the Pharaoh Cheops is equal to.
Since we know the height h and the length of the side a of the monument, we should apply the following formula for b:
b = √ (h 2 + a 2/2).
Substituting the known data into it, we find that the lateral edge of the regular quadrangular pyramid is 273 meters, which is slightly less than the perimeter of the football field (300 meters).