TOPICS

# Circumellipse

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

 (3)

(E. W. Weisstein, Dec. 4, 2005).

The following table summarizes the areas of some named circumellipses.

 circumellipse center area circumcircle excentral-hexyl ellipse MacBeath circumconic Steiner circumellipse

Circle, Circumconic, Excentral-Hexyl Ellipse, Hofstadter Ellipse, Inellipse, MacBeath Circumconic, Steiner Circumellipse

## Explore with Wolfram|Alpha

More things to try:

## References

Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., 1979.Gallatly, W. "The Circum-Ellipse." §1152 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 107-108, 1913.

Circumellipse

## Cite this as:

Weisstein, Eric W. "Circumellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circumellipse.html