Why is it possible for a regular polygon with more than six sides to tessellate the plane?
A regular polygon is one in which all of the sides and angles are equal. A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular tessellation means a tessellation made up of congruent regular polygons. Vertex is a point at which three or more tiles in a tessellation meet. Two tiles cannot meet in a point, but would have to meet in line. Since the interior angles get larger as the number of sides in a polygon gets larger, no regular polygons with more than six sides can tessellate by themselves. Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.
Well, have you ever seen a shape with six sides try to tessellate the plane? It's like watching a clumsy penguin try to breakdance. Just not meant to happen. So, the polygons with more than six sides, they're like the cool kids who can actually dance. They have enough room and flexibility to groove and tessellate the plane without any trouble. It's all about geometry and some serious funky moves!
It is possible for a regular polygon with more than six sides to tessellate the plane because the angles of these polygons evenly divide the 360° angle around a point.
To tessellate the plane, the interior angles of the polygons at each vertex must add up to 360°. For regular polygons, the interior angle can be calculated using the formula (n-2) * 180° / n, where n is the number of sides.
If we take a regular hexagon as an example, it has interior angles of 120°. To form a tessellation, three hexagons can be placed around a point, with their interior angles totaling 360°.
Similarly, regular polygons with more than six sides can also form tessellations by evenly dividing the 360° angle around a point. For instance, a regular octagon has interior angles of 135°, so three octagons can be placed around a point to tessellate the plane.
Thus, regular polygons with more than six sides can tessellate the plane because their angles allow for a perfect division of the 360° angle required for tessellation.
It is possible for a regular polygon with more than six sides to tessellate the plane because its interior angles evenly divide 360 degrees.
To understand why, we need to consider the sum of the interior angles of a regular polygon. The formula to find the sum of the interior angles of an n-sided polygon is given by (n - 2) * 180 degrees.
For example, in a triangle (a 3-sided polygon), the sum of the interior angles is (3 - 2) * 180 = 180 degrees. In a square (a 4-sided polygon), the sum is (4 - 2) * 180 = 360 degrees.
Now, to tessellate the plane with a regular polygon, we need to fit copies of it together without any gaps or overlaps. This can only happen if the interior angles of the polygon evenly divide 360 degrees.
For a regular hexagon with six sides, each interior angle is 120 degrees, which evenly divides 360 degrees. Therefore, a regular hexagon tessellates the plane.
Similarly, for polygons with more than six sides, their interior angles can also evenly divide 360 degrees. For example, a regular octagon with eight sides has interior angles of 135 degrees, which evenly divides 360 degrees. Thus, a regular octagon can also tessellate the plane.
In summary, a regular polygon with more than six sides can tessellate the plane because its interior angles evenly divide 360 degrees, allowing multiple copies to fit together without gaps or overlaps.