Three regular hexagons can be
arranged around a point. By considering the size of
the internal angles, explain why this is possible.
Answers
Answer:
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Step-by-step explanation:
Triangles fit effortlessly together, as do squares. When it comes to pentagons,
But of all the blocks designed to lie flat on a table or floor, have you ever seen any shaped like pentagons?
But it was only this year that we finally settled the question of how five-sided polygons “tile the plane.” Why did pentagons pose such a big problem for so long?
To understand the problem with pentagons, let’s start with one of the simplest and most elegant of geometric structures: the regular tilings of the plane. These are arrangements of regular polygons that cover flat space entirely and perfectly, with no overlap and no gaps. Here are the familiar triangular, square and hexagonal tilings. We find them in floors, walls and honeycombs, and we use them to pack, organize and build things more efficiently.
These are the easiest tilings of the plane. They are “monohedral,” in that they consist of only one type of polygonal tile; they are “edge-to-edge,” meaning that corners of the polygons always match up with other corners; and they are “regular,” because the one tile being used repeatedly is a regular polygon whose side lengths are all the same, as are its interior angles. Our examples above use equilateral triangles (regular triangles), squares (regular quadrilaterals) and regular hexagons.
Remarkably, these three examples are the only regular, edge-to-edge, monohedral tilings of the plane: No other regular polygon will work. Mathematicians say that no other regular polygon “admits” a monohedral, edge-to-edge tiling of the plane. And this far-reaching result is actually quite easy to establish using only two simple geometric facts.
First, there’s the fact that in a polygon with n sides, where n must be at least 3, the sum of an n-gon’s interior angles, measured in degrees, is
This is true for any polygon with n sides, regular or not, and it follows from the fact that an n-sided polygon can be divided into (n − 2) triangles, and the sum of the measures of the interior angles of each of those (n − 2) triangles is 180 degrees.
Seven-sided polygon divided into five triangles.
Second, we observe that the angle measure of a complete trip around any point is 360 degrees. This is something we can see when perpendicular lines intersect, since 90 + 90 + 90 + 90 = 360