To solve problems on the extensive topic "Stereometry", you need to learn and analyze a lot of elements and subtleties, fully study all the properties of figures, and also do not forget the properties of all figures that are included in the "Planimetry" course.
Among the problems of three-dimensional figures, the correct pyramid is very often found, in order to easily solve them, you need to get to know it well. A pyramid is called regular if it has a regular polygon at its base and its vertex is projected into the center of the base. Just as you study this polygon, you will hear about the apothem.
As you have already understood, in geometry the concept of apothem is a widespread phenomenon. It is impossible to know some of the dimensions of the pyramid without knowing it. The very word "apothem" is a phenomenon that came to us from the Greek language, and it is translated as "I postpone".
Definition
In planimetry, apothem is a perpendicular (both itself and its length), which is drawn to the side of a regular polygon from the center. In stereometrythe apothem of a pyramid is the height in the side face, which is drawn to the base. Used only for regular pyramids. Accordingly, the apothem of a regular triangular pyramid is the height of its face, which is represented by an isosceles triangle.
What is the role of apothem
Apothem is a very important element of the pyramid, because it can be used to solve a huge number of problems. In particular, the lateral surface of a regular pyramid is equal to the half product of the perimeter of the base and the apothem of the face.
Sbp =(Pmainh)/2; h is an apothem, this is its key role.
Do not confuse with H (the height of a three-dimensional figure in stereometry).
Also, thanks to the knowledge of the apothem, you can find the area of a face as an isosceles triangle.
Apothem properties
They are few, but still they need to be remembered. In general, these are consequences following from the definition. So, the apothem in the correct pyramid:
- Lowered to the side of the base at a 90 degree angle.
- Divides the side on which it is lowered in half, as it is the height in an isosceles / equilateral triangle and, in combination, the median.
In a regular pyramid, all apothems are equal, since all its side faces are also the same. When finding the length of an apothem, you will have to use both the properties of a polygon and the properties of a polyhedron. How to find the numerical value of the apothem in the correct pyramid?
How to find the apothem of the pyramid
It can be found by applying all previously acquired knowledge, that's alljust a few examples:
- If the side edge and base side are known. Since the apothem divides the side of the base in half and forms an angle of 90 degrees with it, it will not be difficult for you to find it from a right triangle using the Pythagorean theorem. You can also find apothem using knowledge of the ratios in a right triangle.
- If you know the radius of the inscribed circle at the base of a regular pyramid and the height of the entire figure. The radius drawn to the tangent point is perpendicular to the tangent, and the apothem is perpendicular to that side of the base (which is tangent to the inscribed circle). The height of the figure is perpendicular to the base and falls into the center of the circle inscribed in the base of the pyramid. Consequently, the radius and height of the figure are legs and form a right angle, and together with the apothem, a right triangle. And again, using the Pythagorean theorem or through the ratios in a right triangle, you can easily find the apothem.
Also if the face area is given and the base is known
In any case, when finding the apothem, you will have to remember all the basic laws and rules of planimetry. If some elements from this list are unknown, then you can operate with these parameters, and, gradually finding the above data, it will not be difficult for you to find an apothem. We hope that our article has helped you in mastering such an interesting topic.
