Chapter 8.3, Problem 40E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

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

# The length of each side of a regular hexagon measures 6   i n . Find the area of the inscribed regular hexagram shaded in the figure.

To determine

To find:

The area of the regular hexagram.

Explanation

Calculation:

Given,

The length of each side of a regular hexagon is 6Â in.

Calculation:

Consider the regular hexagon,

Area of the regular hexagon is 332a2, where a is the side length of the hexagon.

given each side of a hexagon is 6. i.e., a=6

Therefore,

AABCDEF=332Ã—62=332Ã—36=33Ã—18AABCDEF=543â€‰in2â€‰â€‰â€‰â€‰...(1)

Draw a line ON such that ON is perpendicular to ED and N is the midpoint of ED

Since OND is a right angle triangle and the angle of the triangle is 30âˆ˜âˆ’60âˆ˜âˆ’90âˆ˜.

Therefore by the properties of 30âˆ˜âˆ’60âˆ˜âˆ’90âˆ˜ triangle the hypotenuse is equal to twice the length of the shorter leg, which is the side across from the 30âˆ˜ angle. The longer leg, which is across from the 60âˆ˜ angle, is equal to multiplying the shorter leg by the 3.

Let ON=x

then OD=2x and ND=3x

given ED=6

which implies ND=62=3.

Substitute ND=3 in ND=3x, we get

3=3x

x=33

x=3

i.e

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