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Chapter 9: Problem 29

Determine whether each statement is always, sometimes, or never true. Justify your answers. Regular 16 -gons will tessellate the plane.

Short Answer

Expert verified
Regular 16-gons will never tessellate the plane because 157.5掳 is not a divisor of 360掳.

Step by step solution

01

Understanding Regular 16-gons

A regular 16-gon is a polygon with 16 sides that are all equal in length and where all interior angles are equal. The formula for the interior angle of a regular polygon is \( \frac{(n-2) \times 180^\circ}{n} \), where \( n \) is the number of sides. For a 16-gon, this means each angle is \( \frac{(16-2) \times 180^\circ}{16} = 157.5^\circ \).
02

Understanding Tessellation

A shape can tessellate the plane if it can fit together with copies of itself without any gaps or overlaps. A necessary condition for a regular polygon to tessellate the plane is that its interior angle must be a divisor of 360 degrees, so the angles can fill the space around a point.
03

Checking Tessellation Condition

For a regular 16-gon, each interior angle is 157.5 degrees. To tessellate, \( 157.5^\circ \) must be a divisor of 360. Calculating the number of times the angle can fit into 360 gives \( \frac{360^\circ}{157.5^\circ} \approx 2.2857 \), which is not an integer. Thus, a regular 16-gon cannot tessellate the plane perfectly without gaps.

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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 geometric figure with sides and angles that are all equal. This uniformity in shape allows regular polygons to have symmetrical properties and makes them a popular subject of study in geometry. For any regular polygon with \( n \) sides, each side is the same length, and each interior angle is equivalent. For example, a regular hexagon has six sides and its interior angles are all identical. This symmetry can make regular polygons useful in tiling patterns鈥攖hough not all of them can seamlessly cover a plane.
Interior Angles
Interior angles are the angles found within a polygon that face inwards. In a regular polygon, all these angles are equal. The measure of each interior angle can be calculated with the formula: \[ \text{Interior angle} = \frac{(n-2) \times 180^\circ}{n} \]where \( n \) is the number of sides. For example, in a square (a regular 4-gon), each angle is \( 90^circ \). Understanding interior angles is essential when determining if a shape can tessellate the plane. If the angles can fully encompass 360 degrees around a point, the shape may be able to tessellate.
Tessellate the Plane
To tessellate the plane means to fill an entire flat surface without any gaps or overlaps using copies of a shape. Many shapes can do this, such as squares or equilateral triangles. However, not all polygons can tessellate the plane. A condition for tessellation is that the interior angles must fit around a point without leaving any space. For instance, hexagons can tessellate because their \( 120^circ \) angles perfectly add up to 360 degrees. This feature allows for smooth and seamless covering of a surface, which is why tessellations often appear in tile work and floor designs.
Polygon Divisor Condition
The Polygon Divisor Condition is a crucial guideline when determining a polygon's ability to tessellate. It states that the interior angle of the polygon must be a divisor of 360 degrees. This means you should be able to multiply the interior angle so that the sum equates exactly to 360. For example, because a square has \( 90^circ \) angles and \( 90 \times 4 = 360 \), squares satisfy this condition and can tessellate. In contrast, a regular 16-gon with angles of \( 157.5^circ \) does not meet this condition as \( \frac{360}{157.5} \approx 2.2857 \) is not a whole number, thus making tessellation impossible using perfect copies of the shape.

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Most popular questions from this chapter

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