Is there a pencil near you? Take a look at its section - it is a regular hexagon or, as it is also called, a hexagon. The section of a nut, the field of hexagonal chess, the crystal lattice of some complex carbon molecules (for example, graphite), a snowflake, honeycombs and other objects also have this shape. A giant regular hexagon was recently discovered in Saturn's atmosphere. Doesn't it seem strange that nature so often uses structures of this particular form for its creations? Let's take a closer look at this figure.
A regular hexagon is a polygon with six identical sides and equal angles. We know from the school course that it has the following properties:
- The length of its sides corresponds to the radius of the circumscribed circle. Of all geometric shapes, only a regular hexagon has this property.
- The angles are equal to each other, and the value of each is120°.
- The perimeter of a hexagon can be found using the formula Р=6R, if the radius of the circumscribed circle around it is known, or Р=4√(3)r, if the circle is inscribed in it. R and r are the radii of the circumscribed and inscribed circles.
- 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 of it - as you know, it corresponds to the length of the radius of the circumscribed circle.
A regular hexagon has one interesting feature, thanks to which it has become so widespread in nature - it is able to fill any surface of a plane without overlaps and gaps. There is even the so-called Pal lemma, according to which a regular hexagon whose side is equal to 1/√(3) is a universal tire, that is, it can cover any set with a diameter of one unit.
Now consider the construction of a regular hexagon. There are several ways, the easiest of which involves the use of a compass, pencil and ruler. First, we draw an arbitrary circle with a compass, then we make a point in an arbitrary place on this circle. Without changing the solution of the compass, we put the tip at this point, mark the next notch on the circle, and continue until we get all 6 points. Now it remains only to connect them to each other with straight segments, and the desired figure will turn out.
In practice, there are times when you need to draw a large hexagon. For example, on a two-level plasterboard ceiling, around the attachment point of the central chandelier, you need to install six small lamps at the lower level. It will be very, very difficult to find a compass of this size. How to proceed in this case? How do you draw a big circle? Very simple. You need to take a strong thread of the desired length and tie 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 right point. Of course, in this case, minor errors are possible, but they are unlikely to be noticeable to an outsider at all.
