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Find the area of a regular hexagon if its perimeter is 60 cm. ...
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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 ...
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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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Education|No Subcategories|Auto answered|6/18/2010 2:15:36 PM
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Perimeter, P = 60 cm
Since a hexagon has six sides, each sector can be found as follows;

6s = P
s = P / 6
s = 10 cm

Each sector is an equilateral triangle.

Height of each equilateral triangle equals the length of the line segment from the center of the hexagon to middle of each side.

The bisector of each side is 5 cm from the vertex.

Height of each triangle can be found as follows;

h^2 + 5^2 = 10^2
h^2 + 25 = 100
h^2 = 75
h = 5 sqrt(3)

The area of each equilateral triangle is

At = 1/2 s h
At = 1/2 10 * 5 sqrt(3)
At = 25 sqrt(3) cm^2

Since each hexagon is made up of 6 equilateral triangles, the area of the hexagon Ax is 6 times greater than the area of the triangle At.

Ax = 6 At
Ax = 6 (25 sqrt(3))
Ax = 150 sqrt(3) cm^2

sqrt(3) = 1.73,
--> Ax = 150*1.73 = 259.5 cm^2

6/19/2010 2:38:35 AM
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