   Chapter 8.3, Problem 32E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

In terms of the apothem a and length of side s of the smaller regular pentagon, find an expression for the shaded area. To determine

To find:

An expression for the shaded area in given pentagon.

Explanation

1) The perimeter of a regular polygon is given by P = ns, where n is the number of sides and s is the length of any side.

2) The area of a regular polygon with apothem a and perimeter P is given by A=12(aP).

3) If the side of the regular polygon in s then the apothem a is

a=s2tan(180n), where n is the number of sides.

Calculation:

Given,

The smaller regular pentagon with side s and apothem a and the large regular pentagon with side 2s.

Then the area of the shaded portion is equal to the area of the larger regular pentagon - area of the smaller pentagon.

To find the area of the smaller pentagon

Use the formula for the perimeter of regular polygon, i.e., perimeter = ns

Substitute n = 5 and s = s in P = ns.

P=5s

Use the formula of area of regular pentagon, A=12(aP)

Substitute a = a and P = 5s in A=12(aP).

A=12(a5s)=52as=2.5as

The area of the smaller pentagon is 2.5as

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