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Chapter 10: Problem 38

Can a tessellation be created using only regular ten-sided polygons? Explain your answer.

Short Answer

Expert verified
No, a tessellation cannot be created using only regular ten-sided polygons (decagons), because their interior angles (144 degrees) do not evenly divide into 360 degrees, leaving gaps or overlaps.

Step by step solution

01

Understand the Properties of Regular Polygons

A regular decagon is a ten-sided polygon where all sides and angles are equal. The formula to find each interior angle of a regular polygon is \((n-2) \times \frac{180^{\circ}}{n}\), where \(n\) is the number of sides on the polygon. In this case, \(n=10\). Plug in 10 into the formula to find the measure of each angle.
02

Calculate Each Interior Angle

First subtract 2 from 10, this equals 8. Then multiply 8 by 180, this equals 1440. Finally, divide 1440 by 10 to get 144. So, each angle of a regular decagon measures 144 degrees.
03

Determine if Tessellation is Possible

Tessellation using a single type of regular polygon is possible if the number of polygons meeting at a point can fit around a point to make 360 degrees. In this case, \(\frac{360}{144}\) is not a whole number, implying that regular decagons cannot meet without leaving gaps or overlapping. Thus, tessellation using only regular decagons is not possible.

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