The correct hexagon: how interesting it is and how to build it

Is there a pencil near you?Take a look at its cross section - it is a regular hexagon or, as it is also called, a hexagon. The nut section, the hexagonal chess field, the crystal lattice of some complex carbon molecules (for example, graphite), the snowflake, the honeycomb and other objects also have this form. Giant regular hexagon was recently discovered in the atmosphere of Saturn. Doesn’t it seem so strange that nature uses for its designs the designs of this particular form? Let's look at this figure in more detail.

A regular hexagon is a polygon with six equal sides and equal angles. From the school course we know that it has the following properties:

• The length of its sides corresponds to the radius of the circumcircle. Of all the geometric shapes, this property has only a regular hexagon.
• The angles are equal to each other, and the size of each is 120 °.
• The perimeter of the hexagon can be found by the formula P = 6 * R,if the radius of the circle circumscribed around it is known, or P = 4 * √ (3) * r, if the circle is inscribed in it. R and r are the radii of the circumscribed and inscribed circle.
• The area occupied by a regular hexagon is defined as follows: S = (3 * √ (3) * R²) / 2. If the radius is unknown, we substitute the length of one of the sides instead - as is known, it corresponds to the radius of the circumcircle.

There is one interesting hexagonthe peculiarity due to which he received such wide distribution in nature - he is able to fill any surface of a plane without overlaps and gaps. There is even the so-called Pala Lemma, according to which a regular hex, the side of which is 1 / √ (3), is a universal tire, that is, it can cover any set with a diameter of one unit.

Now consider building the right one.hexagon. There are several ways, the simplest of which involves the use of compasses, pencil and rulers. First draw a compass arbitrary circle, then in an arbitrary place on this circle make a point. Without changing the compass solution, we put the tip at this point, mark the next notch on the circle, continue this until we get all 6 points. Now it only remains to connect them together in straight line segments, and the desired figure will turn out.

In practice, there are cases when requireddraw a large hexagon. For example, on a two-level gypsum ceiling, around the place of attachment of the central chandelier, you need to install six small lamps on the lower level. It will be very difficult to find a compass of this size. What to do in this case? How do you draw a large circle? Very simple. You need to take a strong thread of the desired length and tie up one of its ends opposite the pencil. Now it remains only to find an assistant who would press the second end of the thread to the ceiling at the desired point. Of course, in this case insignificant errors are possible, but they are unlikely to be visible to a stranger at all.