Editing 3-sphere (section)

===Elementary properties===
The 3-dimensional surface volume of a 3-sphere of radius {{mvar|r}} is
:<math>SV=2\pi^2 r^3 \,</math>
while the 4-dimensional hypervolume (the content of the 4-dimensional region, or ball, bounded by the 3-sphere) is
:<math>H=\frac{1}{2} \pi^2 r^4.</math>

Every non-empty intersection of a 3-sphere with a three-dimensional [[hyperplane]] is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere that reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.

In a given three-dimensional hyperplane, a 3-sphere can rotate about an "equatorial plane" (analogous to a 2-sphere rotating about a central axis), in which case it appears to be a 2-sphere whose size is constant.