Consider an n-dimensional unit sphere and unit vector from the origin with its tip lying on the surface of sphere. Consider another vector which makes some angle say $\epsilon$ with unit vector. From a 3-dimensional analogy, if we rotate this vector around fixed vector so that angle between two vectors is always constant, we'll get some portion of the area of the sphere covered.
I want to know if there exists some explicit formula for computing such area given $\epsilon$ for an n-dimensional sphere. If not then how should we go about calculating this.
The "area" you are looking for is in fact an $n-1$ dimensional object, as a sphere in $n$ space is an $n-1$ dimensional shell and you are cutting off a piece of it. If you mean to imply that $\epsilon$ is very small, the piece will be an $n-1$ dimensional ball. The analogy with a sphere in 3D is applicable-the piece you cut off is almost a circle because you can't see the curvature of the original sphere. If your fixed vector is along the $x$ axis, the boundary of the cut off piece will be $y^2+z^2+w^2+\dots = \epsilon^2$, with $n-1$ terms on the left. The $n-1$ dimensional volume is then $\frac {\pi^{(n-1)/2}}{\Gamma(\frac n2-1)}\epsilon^{n-1}$