The cosine hexagon is a closed self-intersecting cyclic hexagon formed by joining the adjacent antiparallels
in the construction of the cosine circle (Lachlan
1893, p. 75). The sides 
, 
, and 
are antiparallel to
, 
, and 
, respectively. The remaining sides
satisfy 
,
, and 
. The cosine hexagon is a special case of
a Tucker hexagon.
For an acute triangle as the reference triangle, let 
be its triangle area and let 
, 
,
and 
be its side lengths. The total area 
of the three bounded lobes of the
self-intersecting cosine hexagon and the area 
of the simple hexagon
formed by the convex hull of its vertices
are
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
.
