Area of a sphere bounded by hyperplanes Say we have a sphere in d-dimensional space, and k hyperplanes (d-1 dimensional) all passing through the origin. Is there a way to calculate (or approximate) the area of the surface of the sphere enclosed by the half-spaces \begin{align*}w_1 \cdot x &\leq 0 \\
w_2 \cdot x &\leq 0\\ \vdots& \\w_k \cdot x &\leq 0\end{align*}

Seven years late here. But this citation might be useful if someone (probably someone else) is looking into this problem.
Cho, Y., & Kim, S. (2020). Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities. In The Electronic Journal of Linear Algebra (Vol. 36, Issue 36, pp. 228–255). University of Wyoming Libraries. https://doi.org/10.13001/ela.2020.5085
https://journals.uwyo.edu/index.php/ela/article/download/5085/5047
