In the seventh century, around 650 A.D., the Indian mathematician Brahmagupta came up with a remarkable formula expressing the area E of a cyclic quadrilateral in terms of the lengths a, b, c, d of its sides. In his formula E = [square root](s-a)(s-b)(s-c)(s-d), s stands for the semiperimeter 1/2(a+b+c+d). The fact that Brahmagupta's formula is symmetric in a, b, c, d suggests that there are different cyclic quadrilaterals having the same area and the same side lengths, though the sides may be in different orders. Since the vertices of a cyclic quadrilateral lie on a circle, then actually there are only three orders for the sides, namely (a, b, c, d), (a, b, d, c), and (a, c, b, d), where a is opposite to c, d and b respectively. The author will show that the three quadrilaterals, corresponding to these orders, can be inscribed in the same circle. He will calculate the lengths of their diagonals and the radius of their circumcircle. Before he does that, he will furnish a proof for Brahmagupta's formula. He applies basic geometric and trigonometric results which would make the content of this article usable in the classroom. (Contains 3 figures.)