User: Find the area of a regular hexagon if its perimeter is 60 cm.

Weegy: This is real easy and is a good example of breaking down some geometry problems into pieces. How many sides does a hexagon have? Correct, six sides. What does it mean to be a regular hexagon? Correct, [ sides of equal length. Connect the corner points of a hexagon to the center of the hexagon using six straight lines. These lines break a regular hexagon into six equilateral triangles (which are triangles where all three sides are the same length). Each equilateral triangle has side lengths of one-sixth of 60 cm or 10 cm. What is the area of each equilateral triangle? An equilateral triangle of side length 2k has area equal to k^2 sqrt(3). (More on that below.) So, the total area of the six equilateral triangles of side length 10 (2k = 10, k = 5) is six times 5^2 sqrt(3) cm = 150 sqrt(3) cm.. Note: In right triangles with a hypotenuse of length 2k and one side of length k, the lenght of the third side is sqrt[(2k)^2 - k^2] = sqrt[3k^2] = k sqrt(3). A rectangle of length k sqrt(3) and width k can be divided into two righ triangles which could be rearranged to form an equilateral triangle and its area would be k^2 sqrt (3). Note: A good formula to remember for the area of triangles is Heron's Formula: Let a, b and c be the lengths of the three sides of a triangle and let s denote the semiperimeter of the triangle (s = one-half of a + b + c). Then the area of the triangle is sqrt [ s(s - a)(s - b)(s - c) ] [Finding the area of an equilateral triangle of side length 2k in the formula shows the are to be sqrt(3k(3k - 2k)(3k - 2k)(3k - 2k)) = sqrt(3k(k)(k)(k)) = sqrt(3k^4) = k^2 sqrt(3).] http://answers.yahoo.com/question/?qid=20080826190547AAzCQnT ]
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