SOLUTION: A sign is in the shape of a regular hexagon. If the length of a diagonal of the sign is 10 inches, what is the area of the sign, in square inches?

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Question 172607: A sign is in the shape of a regular hexagon. If the length of a diagonal of the sign is 10 inches, what is the area of the sign, in square inches?
Answer by colliefan(242) (Show Source):
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Draw the regular hexagon and all the diagonals. The diagonals all intersect in the middle of the polygon. The angles formed in the middle of the polygon are all equal and so must be 60 degrees because the sum of all those angles form a circle and so must be 360. 360/6=60.
Pick one of the 6 triangles formed by drawing the diagonals. Draw the perpendicular line that will be the height of the triangle. That perpendicular forms a new triangle that is a right triangle. One of the angles in that right triangle is a 60 deg angle; the other must be 30 deg. This is then a 30-60-90 triangle and the lengths of its sides are in the ratio 1 : sqrt(3) : 2 where 2 is the hypotenuse. The length of the hypotenuse is 5 because it is half of the diagonal which was 10. Using the ratio of the 30-60-90 triangle, the other sides of the triangle have lengths 5/2 and 5/2*sqrt(3).
Using A = 1/2 * b * h gives us the area of one of the small triangles. A = 1/2 * 5/2 * 5/2*sqrt(3). The polygon is made up of 12 of these small triangles so the area of the figure is 12 times the area on one of the triangles.