Ellipsoid

High School

Definition

A deformed sphere whose cross sections (no matter where you cut it) is either an ellipse, empty, or just a single point. It has three pairwise perpendicular axes of symmetry that intersect at a center of symmetry. A prominent object that is an ellipsoid is the earth, as it resembles a slightly flattened sphere more than a perfect sphere.

Worked examples

\(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)

The standard equation of an ellipsoid centered at the origin with semi-axes a, b, and c along the x, y, and z axes.

\(\frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{16} = 1\)

This ellipsoid has semi-axes 3, 2, and 4; any plane slice parallel to a coordinate plane gives an ellipse.

Common mistakes

\(\frac{x^2}{4} + \frac{y^2}{4} + \frac{z^2}{4} = 1\) is an ellipsoid → \(x^2 + y^2 + z^2 = 4\) is a sphere When all three denominators are equal, the surface is a sphere, not an ellipsoid.
An ellipsoid has two axes of symmetry → An ellipsoid has three mutually perpendicular axes of symmetry Each semi-axis defines one axis of symmetry; there are always three, even if some have equal length.
Cross sections of an ellipsoid can be circles → Cross sections are ellipses, points, or empty A circle is a special case of an ellipse, so circular cross sections are possible when two semi-axes are equal.

Where you'll use it next

Ellipsoids appear in multivariable calculus when you sketch quadric surfaces, compute volumes with triple integrals, and model planets and error regions in physics and statistics.

Found in 1 StudyPug lesson

Surface area and volume of spheres

Geometry

A basketball is a sphere. Soap bubbles are usually spheres. A vitamin capture is a combination of cylinder and sphere. In this lesson, we will learn how to calculate the surface area and volume of spheres and composite solids.

10th Grade10th