Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 10 - Geometry - 10.3 Polygons, Perimeter, and Tessellations - Exercise Set 10.3 - Page 637: 38

Answer

A tessellation cannot be created using only regular ten-sided polygons.

Work Step by Step

A tessellation is a type of art that is used to define a relationship between geometry and the visual arts. Tessellations are created by repeated use of same figures that will leave no gap and no overlaps and thus cover the whole plane. To create a tessellation, the primary requirement is the sum of the measures of the angles of a regular polygon that are together at each vertex must be\[360{}^\circ \]. To determine whether a tessellation can be created or not, use the formula\[\left( n-2 \right)\times 180{}^\circ \], which computes the sum of the measures of the angle of a regular decagon and then divides it by the sides of a decagon to find the measure of each angle. Compute a measure of each angle: \[\begin{align} & \text{Measure of angle}=\frac{\left( n-2 \right)\times 180{}^\circ }{n} \\ & =\frac{\left( 10-2 \right)\times 180{}^\circ }{10} \\ & =\frac{8\times 180{}^\circ }{10} \\ & =144{}^\circ \end{align}\] The measure of the angles at each vertex must be \[360{}^\circ \]to form a tessellation. The measure of each angle of a regular polygon is\[140{}^\circ \]. Hence, the two regular decagons fill in \[2\times 144{}^\circ =288{}^\circ \]and leave a gap of \[360{}^\circ -288{}^\circ =72{}^\circ \], which does not fulfill the requirement of forming a tessellation by regular ten-sided polygons. Hence, A tessellation cannot be created using only regular ten-sided polygons.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.