Elementary Geometry for College Students (5th Edition)

Published by Brooks Cole
ISBN 10: 1439047901
ISBN 13: 978-1-43904-790-3

Chapter 11 - Section 11.2 - The Cosine Ratio and Applications - Exercises - Page 511: 43

Answer

$A=5r^2cos36^{\circ}sin36^{\circ}$

Work Step by Step

When we break a regular pentagon into 10 separate right triangles, each triangle has an angle of 36 degrees in it. Using this angle and the fact that the radius is r, we find: $sin36^{\circ}=\frac{s/2}{r} \\ s/2 = rsin36^{\circ} \\ s = 2rsin36^{\circ}$ Since there are 5 sides, we find: $P = 5s = 10rsin36^{\circ}$ The area is equal to the 1/2 times perimeter times the apothem. Thus, we find the apothem: $cos36^{\circ}= \frac{a}{r} \\ a = rcos36^{\circ}$ Thus, we obtain: $A = \frac{1}{2} rcos36^{\circ}10rsin36^{\circ} =5r^2cos36^{\circ}sin36^{\circ}$
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