Cyclic Quadrilaterals

I was forced to think about cyclic quadrilaterals when looking for information about the number associated with my diurnal age today. 27190 is a member of OEIS A329950:

A329950

Floor of area of quadrilateral with consecutive prime sides configured as a cyclic quadrilateral.

In the case of 27190, the consecutive prime sides are 157, 163, 167 and 173 and the area is given by Brahmagupta's formula: $$ \text{area }=\sqrt{(s-a) \times (s-b) \times (s-c) \times(s-d)}\\ \text{ where } s=\frac{a+b+c+d}{2} \text{ and }a,b,c \text{ and } d \text{ are the four sides}$$Here the area turns out to be 27190.9834504013 which is 27190 when truncated. The initial members of the sequence are (permalink):

13, 30, 70, 130, 214, 310, 461, 627, 874, 1167, 1423, 1750, 2094, 2512, 2995, 3574, 4137, 4603, 5237, 5829, 6526, 7522, 8507, 9478, 10390, 11014, 11650, 12932, 14314, 16053, 17799, 19278, 20698, 22159, 23994, 25403, 27190, 29033, 30595, 32718, 34558, 36255, 38014, 39954

The radius \(R\) of the circumcircle, referred to as the circumradius, is given by the formula:$$R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}$$In the case of the cyclic quadrilateral with sides 157, 163, 167, 173 and truncated area of 27190, the truncated circumradius is 116 and the truncated area of the circumcircle is 42818 square units. 

These types of cyclic quadrilaterals are very close to being square in shape. For example, the quadrilateral with sides of 157, 163, 167 and 173 has an average side length of 165 and the area of a square with this side is 27225 square units and thus very close to 27190.

In the formula for the area given earlier, it can be noted that when \(d=0\), we get a triangle whose area is given by the familiar Heron's formula:$$ \text{area }=\sqrt{s \times (s-a) \times (s-b) \times (s-c) }\\ \text{ where } s=\frac{a+b+c}{2} \text{ and }\\a,b \text{ and } c \text{ are the three sides of the triangle}$$It should be noted that Brahmagupta's formula does not apply to quadrilaterals in general. It only applies to cyclic quadrilaterals. The more general formula is similar but more complex and I won't cover that here. For more information follow this link.