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Chapter 9: Problem 2

Determine whether each regular polygon tessellates the plane. Explain. 30 -gon

Short Answer

Expert verified
A regular 30-gon does not tessellate the plane because its interior angle (168 degrees) is not a divisor of 360 degrees.

Step by step solution

01

Understand the Problem

A regular polygon tessellates the plane if it can cover the plane without gaps or overlaps. For a regular polygon to tessellate, the interior angle of the polygon must be a divisor of 360 degrees.
02

Calculate the Interior Angle of a 30-gon

The formula to find the interior angle of a regular n-gon is \( \frac{(n-2) \times 180}{n} \). For a 30-gon, substitute \( n = 30 \) to get \( \frac{(30-2) \times 180}{30} = \frac{28 \times 180}{30} = 168 \) degrees.
03

Check Divisibility of the Interior Angle

For the 30-gon to tessellate, its interior angle (168 degrees) must divide 360 degrees evenly. Check this by dividing 360 by 168: \( \frac{360}{168} = 2.142857... \). Since it is not an integer, 168 degrees is not a divisor of 360.
04

Conclusion

Since the interior angle of a 30-gon is not a divisor of 360 degrees, a regular 30-gon does not tessellate the plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Regular Polygon
A regular polygon is a shape with all sides and angles equal. This uniformity gives regular polygons some fascinating properties, like the ability to maintain symmetry and balance. Common examples include equilateral triangles, squares, and regular hexagons.
However, the number of sides can increase vastly, leading to shapes like the regular 30-gon. Each vertex of a regular polygon connects in such a way that it can sometimes fill spaces in a plane seamlessly. This possibility depends not only on the uniformity of the angles and sides but also their relationships to known geometric principles.
Thus, to determine if a shape like the regular 30-gon can accomplish a special task, such as plane tessellation, one must analyze its basic attributes.
Interior Angle
The interior angle of a polygon is the angle formed between two sides at a single vertex. For regular polygons, each interior angle remains constant throughout. This uniformity is crucial when considering tasks like tessellation.
The formula to calculate this angle for an n-sided regular polygon is given by \[ \frac{(n-2) \times 180}{n} \]Substituting 30 for n in the case of a 30-gon, you calculate the interior angle as 168 degrees. Breaking this down, \( (30-2) \times 180 = 5040 \), and dividing by 30 gives \( 168 \) degrees.
Understanding this angle is essential because it's a factor in plane tessellation. It determines whether the polygon can fit repeatedly without gaps or overlaps.
Plane Tessellation
Plane tessellation refers to covering a plane with a shape repeatedly, without any gaps or overlaps. This requires that the interior angle of the shape fits evenly into the complete rotation around a point, which is 360 degrees. This means finding if the interior angle of the polygon divides 360 degrees exactly, resulting in an integer.
If we perform this check for a 30-gon, with an interior angle of 168 degrees, we find \( \frac{360}{168} \approx 2.142857 \)..., which isn't an integer.So, this means that the angle doesn't divide 360 perfectly, indicating that such a shape cannot perfectly tessellate a plane.
  • Not all regular polygons can tessellate.
  • The interior angle must divide 360 degrees exactly for perfect fitting.
Understanding these principles helps in visualizing and creating complex tiling patterns in geometry and art.

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