SOLUTION: Find the area of a hexagon with a square having an area of 72 sq. cm. inscribed in a circle which is inscribed in a hexagon. a. 124.71 sq. cm. b. 150.26 sq. cm. c. 150.35 sq. cm.

Algebra ->  Finance -> SOLUTION: Find the area of a hexagon with a square having an area of 72 sq. cm. inscribed in a circle which is inscribed in a hexagon. a. 124.71 sq. cm. b. 150.26 sq. cm. c. 150.35 sq. cm.      Log On


   

Question 1165480: Find the area of a hexagon with a square having an area of 72 sq. cm. inscribed in a circle which is inscribed in a hexagon.
a. 124.71 sq. cm. b. 150.26 sq. cm. c. 150.35 sq. cm. d. 130.77 sq. cm.
Answer by greenestamps(13200) About Me  (Show Source):
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Note the statement of the problem should specify a REGULAR hexagon; without that, the problem is not defined well enough to get an answer.

Area of square which is inscribed in the circle: 72

Side of square: sqrt%2872%29+=+6%2Asqrt%282%29

Diagonal of square = diameter of circle: %286%2Asqrt%282%29%29%2Asqrt%282%29+=+12

The circle is inscribed in the hexagon; the diameter of the circle is the distance from the middle of one side of the hexagon to the middle of the opposite side.

View the hexagon as being composed of 6 equilateral triangles. The radius of the circle, length 12/2=6, is the long leg of a 30-60-90 right triangle in which the hypotenuse is the length of one side of one of those equilateral triangles.

Side length of hexagon = side length of one of the equilateral triangles = 6%2A%282%2Fsqrt%283%29%29+=+12%2Fsqrt%283%29+=+4%2Asqrt%283%29

Area of hexagon = area of 6 equilateral triangles = 6%28%28s%5E2%2Asqrt%283%29%29%2F4%29+=+6%28%2848sqrt%283%29%29%2F4%29+=+72sqrt%283%29 = 124.71 to 2 decimal places.

ANSWER: a. 124.71 sq cm