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 cross section of the nut, the field of hexagonal chess, the crystal lattice of some complex carbon molecules (for example, graphite), snowflake, bee honeycombs and other objects also have this shape. A giant regular hexagon was recently discovered in the atmosphere of Saturn. Doesn’t it seem strange to use nature so often for its creations of constructions of this particular form? Let's look at this figure in more detail.
A regular hexagon is a polygon with six identical 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 circumscribed circle. Of all the geometric shapes, this property has only a regular hexagon.
- The angles are equal to each other, and the magnitude of each is 120 °.
- The hexagon perimeter can be found by the formula P = 6 * R if the radius of the circle described 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 circles.
- The area occupied by the regular hexagon is defined as follows: S = (3 * √ (3) * R 2 ) / 2. If the radius is unknown, substitute the length of one of the sides instead - as you know, it corresponds to the length of the radius of the circumscribed circle.
The regular hexagon has one interesting feature due to which it has become so widespread in nature - it is able to fill any surface of the plane without overlays and spaces. There is even the so-called Pal lemma, according to which the regular hexagon, whose side is 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 simplest of which involves using a compass, pencil and ruler. First we draw an arbitrary circle with a compass, then at a random place on this circle we make a point. Without changing the solution of the compass, we point the point to this point, mark the next notch on the circle, and continue until we get all 6 points. Now it remains only to connect them together by straight segments, and we get the desired figure.
In practice, there are times when you want to draw a large hexagon. For example, on a two-level plasterboard ceiling, around the place of attachment of the central chandelier, you need to install six small lamps at the lower level. A compass of such sizes will be very, very difficult to find. What to do in this case? How to 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.