A circumellipse is a circumconic of a triangle that is an ellipse.

There is an amazing formula for the area of a circumellipse. Let d_A be the length of the chord of the ellipse through the center of the ellipse and parallel to the sideline BC of the reference triangle DeltaABC, and similarly define d_B and d_C. Then


(Chakerian 1979, p. 149), where R is the circumradius of the reference triangle and Delta its area. Explicitly calculating the chord lengths for a circumconic with parameters x:y:z then gives the beautiful formula


(E. W. Weisstein, Dec. 4, 2005).

The following table summarizes the areas of some named circumellipses.

See also

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

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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.

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Cite this as:

Weisstein, Eric W. "Circumellipse." From MathWorld--A Wolfram Web Resource.

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