A circumellipse is a circumconic of a triangle that is an ellipse.
There is an amazing formula for the area of a circumellipse. Let 
be the length of the chord of the ellipse through the center
of the ellipse and parallel to the sideline 
of the reference triangle 
, and similarly define 
and 
. Then
 |  |  |
(1)
|
 |  |  |
(2)
|
(Chakerian 1979, p. 149), where 
is the circumradius of the
reference triangle and 
its area. Explicitly calculating the chord lengths for a circumconic with parameters
then gives the beautiful formula
![A=(4piabcxyz)/([2(abxy+bcyz+cazx)-(a^2x^2+b^2y^2+c^2z^2)]^(3/2))](/images/equations/Circumellipse/NumberedEquation1.svg) |
(3)
|
(E. W. Weisstein, Dec. 4, 2005).
The following table summarizes the areas of some named circumellipses.
| circumellipse | center | area |
| circumcircle |  |  |
| excentral-hexyl ellipse |  | ![4piabc[((a+b)(b+c)(c+a))/(a+b+c)]^(3/2)Deltaproduct_(cyclic)1/(sqrt(f(a,b,c)))](/images/equations/Circumellipse/Inline18.svg) |
| Macbeath circumconic |  |  |
| Steiner circumellipse |  |  |
