Surface area of a sphere patch cut out by a regular tetrahedron Description: consider a regular tetrahedron (with height 1), construct a sphere centering at one of the tetrahedron's vertex, with radius 1 also. Then what's the surface area of the sphere's portion that gets cut out by the tetrahedron? Following are links to some illustrations I made using Mathematica. 
$\hskip 1.3in$ 
$$\text{a side view}$$
$\hskip 1.3in$ 
$$\text{a bottom view: you are viewing the unknown area directly}$$

One useful tool in getting this kind of information is the solid angle of an object. Simply multiplying such a solid angle by $r^2$ will give you the surface area cut out from a sphere of radius $r$ centered around the apex of the angle. Since your sphere has radius $1$, you're in fact simply asking for the solid angle at the top of a tetrahedron. Wikipedia has a section on its computation. That section is for general, i.e. not neccessarily regular tetrahedra. You could apply those computations to a regular tetrahedron. Or you have a look at the article on (the regular) tetrahedron. There you find the solid angle at a vertex to be
$$\Omega = \arccos\frac{23}{27} \approx 0.55129\,\text{sr}$$
Since your radius is $1$, you can ignore the unit of sr and interpret this number directly as the surface area you asked about. Note that the size of the tetrahedron is irrelevant in all of this, as long as one of its vertices is centered at the center of the sphere and the opposite side lies completely outside the sphere.
