fencing problem part 2/8

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This is a graph of the areas of isosceles triangles. It contains more details than the table above and there are more results shown.

My results show that the triangle with the largest area for a perimeter is the equilateral triangle. This also follows my previous observation but with a slight change. This triangle has no difference between all three sides, not just two sides. As there is only one form of equilateral triangle for each perimeter, I will now work out the area of an equilateral triangle with a perimeter of a thousand metres.

To find out the area of this equilateral triangle, I will use Heron’s Formula. The formula is

Where S stands for semiperimeter.

 (Half of the total perimeter).

48112.52m2

Now that I know that equilateral triangles are the best shape to use for three sided shapes, I will now investigate what is the best shape to use for four sided shapes (quadrilaterals).

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To work out the areas of quadrilaterals, I will use Bretschneider's formula. The formula is

. In this formula a, b, c and d are the lengths of the four sides of the quadrilateral. The s stands for the semiperimeter and θ is half the sum of two opposite angles.

As we want the maximum area for a quadrilateral, the number which has to be square rooted should be as high as it can be. This can only be achieved if the ‘abcdCos2θ   ’ part of the formula is a low as possible because this number is taken away ...

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