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 from the number to be square rooted. Therefore abcdCos2 θ should equal zero.
To make abcdCos2 θ equal zero, Cos2 has to equal zero, as abcd already have a more than zero value. Cos2 can only equal zero if θ equals ninety as cos-10 is ninety.
To make θ equal 90, the two opposite angles in the shape have to add up to 180degrees. Therefore the quadrilaterals would have to be cyclic quadrilaterals.
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. All the vertices are said to be concyclic. To work out the areas of cylcic quadrilaterals we will use Brahmagupta's formula which is simpler than Bretschneider's formula.
Brahmagupta's formula is .
Brahmagupta’s formula can only be used in cyclic quadrilaterals.
Some examples of cyclic quadrilaterals are below.
Many cyclic quadrilaterals are rectangles, trapeziums or squares, there are also many cyclic quadrilaterals which are not any of the three shapes above but I will not be including them in my investigation because this would be very time consuming as there area an infinite number of these quadrilaterals. I will now investigate the areas of various trapeziums.
I will use a perimeter of 1000m and a fixed base of 200m. I will then set a length for the line parallel to the base, and this length will be changed. The other two sides have the same length so the remaining part of 1000m will be divided into two and this will be the length of one of the sides.
My results show me that the trapezium with the least difference between three unfixed sides has the largest area. I predict that the trapezium with the least difference in the three unfixed lengths would have a bigger area.
I will now work out the area of the trapezium which has no difference between the three unfixed sides. I will work out the length of the three unfixed sides by dividing 800m (1000m – base) by three. The answer is 266.6667m. Using Brahmagupta’s Formula, I have worked out that the area of the trapezium is 61734.20m2. As I predicted, this trapezium has more area than all the other trapeziums of base 200m.