The word apothem has its origin in a Greek word that, when translated into Spanish, is understood as “lower” or “deposit”. In the field of geometry, this term is used to name the shortest path that separates the center point of regular polygons from any of their respective sides.
It can be said, therefore, that the apothem of regular polygons constitutes a segment that extends from the central axis of the figure to the middle of one of its sides. According to DigoPaul, the apothem, in short, is in all cases perpendicular to the side in question. It can also be taken into account that polygons are closed geometric figures that are made up of straight line segments and consecutive character (but that are not aligned), which are called sides. When all the sides and the respective angles of the figure are identical, we are talking about a regular type polygon.
It should be noted that the apothem is complemented by the sagite (as the fragment of a line that arises from the central point of the arc of a circle and that of its corresponding chord is known) to compose the radius. The radius, on the other hand, identifies all the segments that go from the central axis to any point on the circumference.
To understand these three concepts graphically, it is first necessary to imagine a circumference; then, locate within it (and formed with four of its own points) a square, so that if it were drawn larger it would exceed the surface of the circumference. With these two figures in mind, if you start from the center of the first to trace its radius and go through the midpoint of one of the four sides of the square, then three segments will be appreciated: one from the center to the side, which is called apothem; another, from the side to the limit of the circumference, or the sagita; and finally, the sum of both results in the segment called radius.
It is interesting to know that the apothem, the sagite and the radius allow various measurements to be carried out to obtain data linked to the polygons. For this, different formulas are used to define the variables.
In regular pyramids, the apothem constitutes the height of their triangular faces. It is, according to experts in the field, the segment that joins the vertex with the central part of any of the sides of the polygon that constitute its base. The apothem, therefore, comes to coincide with the height of each of the triangular faces.
When dealing with a problem with regular polygons it is very common to overlook the way in which the apothem relates to the side, which can result in a mistake of various importance. However, just by using the apothem table, it is possible to perform the calculation simply taking into account the chosen side. The formula shown in the image shows the trigonometric relationship in question.
First of all, it should be noted that n equals the number of sides that the polygon in question has. Therefore, it is possible to deduce that the value of α is obtained simply by dividing 360 ° by n. If we take as an example a side that is equal to the unit, then we can easily find a list of numbers that help to calculate the apothem of any regular polygon, just starting from the value of one side. The image also shows the necessary angles for some of the most common polygons.
After solving the equation in this way, a table is obtained that returns the value of the apothem for each type of regular polygon (triangle, square, etc.) whose sides are equal to unity. Thus, to calculate any apothem, simply multiply the value corresponding to the type of polygon by the measure of the side in question.