Find the area of the shaded region. Assume that the polygon is regular. Round to the nearest tenth.  Figure: http://www.flickr.com/photos/93084714@N07/8616425023/

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embizze | High School Teacher | (Level 1) Educator Emeritus

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First we find the area of the regular hexagon, then subtract the area of teh square to find the shaded area.

We have a regular hexagon with side 14 yd. One formula for the area of a regular polygon is `A=1/2ap` where a is the length of the apothem (the perpendicular distance from the center of the polygon to a side where the center is the center of the circumscribed circle.) and p is the perimeter.

Here the apothem and a radius form a 30-60-90 right triangle with hypotenuse 14. So the radius is 14 and the apothem is `7sqrt(3)` . The perimeter is 6(14)=84 so the area is `A=1/2(7sqrt(3))(84)=294sqrt(3)"yd"^2`

(Alternatively we can dissect the hexagon into 6 equilateral triangles each with area `49sqrt(3)"yd"^2` )

The area of the square is `14^2=196"yd"^2` .

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The area of the shaded region is `294sqrt(3)-196` square yards or approximately 313.2 square yards.

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