© Les Bill Gates 2024 The Quadrilateral of Maximum Area - A Different Approach Les Bill Gates Abstract There is a well-known method to calculate the area of a triangle attributed to Heron (Hero of Alexandria, c AD 10-75). For a triangle with side lengths a, b and c, and semiperimeter s =       , Heron’s formul is:   ss  s  s   This formula is independent of the angles of the triangle since the triangle with sides a, b and c is unique. The same is not true for a quadrilateral, however – a quadrilateral with sides a, b, c and d is not unique. As far as I have been able to ascertain, finding a formula for the area of the quadrilateral of maximum area depends on putting the sum of the opposite angles of the quadrilateral equal to 180° in Bretschneider’s formula, which leads directly to Brahmagupta’s formul. This paper investigates a different approach towards proving that the quadrilateral of maximum area is cyclic, using elementary algebra and calculus. Introduction A farmer has four straight lengths of fencing – 60 m, 80m, 90m and 120 m. He wants his sheep to have the maximum grazing area, so he wants to join them into the shape of a quadrilateral such that the area of the quadrilateral is a maximum.