A prolate spheroid is a three-dimensional shape that resembles an elongated sphere, like a rugby ball or an American football. It's essentially an ellipse rotated around its major axis (the longer axis).

Here are some key characteristics of a prolate spheroid:

* Shape: Elongated, with a larger diameter along one axis than the other two.

* Symmetry: Symmetrical about its axis of rotation.

* Volume: Calculated using the formula: (4/3)πa²b, where 'a' is half the length of the major axis and 'b' is half the length of the minor axis.

* Surface area: Calculated using the formula: 2πb² + 2πab * (arcsin(e)/e), where 'e' is the eccentricity of the ellipse.

Examples of prolate spheroids in real life:

* Planetary bodies: Some planets and moons, like Pluto and Saturn's moon Iapetus, are prolate spheroids.

* Sports equipment: Rugby balls and American footballs are designed as prolate spheroids.

* Atomic nuclei: Certain atomic nuclei have a prolate spheroidal shape.

* Aerodynamics: Prolate spheroids are used in aerodynamics for their smooth, low-drag properties.

Difference from an oblate spheroid:

An oblate spheroid, like the Earth, is flattened at the poles and bulging at the equator. The difference lies in the axis of rotation: a prolate spheroid rotates around its longer axis, while an oblate spheroid rotates around its shorter axis.