Elementary Geometry for College Students (6th Edition)

Published by Brooks Cole
ISBN 10:
ISBN 13:

Chapter 8 - Section 8.3 - Regular Polygons and Area - Exercises - Page 367: 37

Answer

We call half of the long side x. Thus, a side of this hexagon is 2x. In addition, using a 30-60-90 triangle, we find that the apothem is 2x as wel. Thus, its area is: $A=\frac{sp}{2} = 2x^2$ We now consider the inner hexagon. Using the law of sines, we find that its side is $x\sqrt{3}$. In addition, using the same 30-60-90 triangle as before, we find that its apothem is $x\sqrt{3}$. Thus, its area is $1.5x^2$. Dividing these areas, we get .75, which is the same as 3/4.
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