To solve problems on the broad topic of "Stereometry" you need to learn and analyze a lot of elements and subtleties, fully study all the properties of the figures, and also do not forget the properties of all the figures that are included in the course "Planimetry".
Among the tasks on volumetric 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 a regular polygon lies at its base and its vertex is projected into the center of the base. Just when you study this polygon, you will hear about the apothem.
As you already understood, in geometry the concept of apothem is a widespread phenomenon. It is impossible to recognize some dimensions of a pyramid without knowing it. The very word "apothema" is a phenomenon that came to us from the Greek language, and it translates as "put off."
Definition
In planimetry, the apothem is the perpendicular (both itself and its length), which is drawn to the side of the regular polygon from the center. In stereometry, the apothem of the pyramid is the height in the side face that is drawn to the base. Used only for regular pyramids. Accordingly, the apothem of the regular triangular pyramid is the height of its face, which is represented by an isosceles triangle.
What is the role of apothema
Apothem is a very important element of the pyramid, because with its help you can solve a huge number of problems. In particular, the lateral surface of a regular pyramid is equal to the semi-product of the perimeter of the base and the apotheme of the face.
S bp = (P DOS * h) / 2; h - apothem, this is her key role.
Do not confuse with H (the height of the three-dimensional figure in stereometry).
Also, thanks to the knowledge of the apothem, one can find the area of the face as an isosceles triangle.
Apothem Properties
They are few, but still they need to be remembered. In general, these are consequences arising from the definition. So, the apothem in the correct pyramid :
- Lowered to the side of the base at an angle of 90 degrees.
- Divides the side on which it is lowered in half, as it is the height in an isosceles / equilateral triangle and, in combination, is a median.
In the correct pyramid, all apothems are equal, since all its side faces are also the same. When finding the length of the apothem, you will have to use both the properties of the polygon and the properties of the 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 using all the previously acquired knowledge, here are just a few examples:
- If the side rib and the side of the base 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 by the Pythagorean theorem. You can also find the apothem using knowledge of the relationships in a right triangle.
- If the radius of the inscribed circle at the base of the regular pyramid and the height of the whole figure are known. The radius drawn to the point of tangency is perpendicular to the tangent, and the apofem is perpendicular to this 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, by Pythagorean theorem or through relations in a right-angled triangle, you can easily find an apothem.
- Also, if the area of the face is given and the basis is known.
In any case, when finding an 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 to find an apothem. We hope that our article has helped you in mastering such an interesting topic.