Suppose $S_n = \{x \in \mathbb{R}^{n + 1} : ||x||_2 = 1\}$ is the $n$-sphere. Let $d : S_n^2 \to \mathbb{R}$ be the angle metric on $S_n$, i.e. $d(x, y) = \arccos(x \cdot y)$, where $\cdot$ is the dot product.
What is the area $A(D_n(\theta))$ of the disk $D_n(\theta) = \{x \in S_n : d(x, p) \leq \theta\}$, where $p \in S_n$?
I'm not sure of the terminology here, and whether I'm missing some information. I suspect this has to do with Riemannian manifolds, but I know little about them. Intuitively, I want to define probability densities on $S_n$, those densities being relative to area. At least for $n \leq 3$ the idea seems clear. Clearly $A(D_1(\theta)) = 2$, and $A(D_2(\theta)) = 2 \theta$. Looking at
https://en.wikipedia.org/wiki/Spherical_cap
we have $A(D_3(\theta)) = 2\pi (1 - \cos(\theta))$.
Edit: The same page in fact gives the general formula. It just has to be transformed to use angle rather than height.
Wikipedia's spherical cap article provides the area of (a boundary of) a spherical cap with height $h(\theta)$ and sphere-radius $r$ as
$$A(D_n(\theta)) = (1/2) A_n r^{n - 1} I(\alpha(\theta), (n - 1) / 2, 1 / 2),$$
where
$$A_n = \frac{2 \pi^{n / 2}}{\Gamma(n / 2)},$$ $$\alpha(\theta) = (2rh(\theta) - h(\theta)^2) / r^2,$$ $$h(\theta) = r - r \cos(\theta),$$ $$I(x, a, b) = \frac{B(x, a, b)}{B(a, b)}.$$
Here $A_n$ is the area of the boundary of the unit $n$-ball, $I$ is the regularized incomplete beta function, and $B$ is the (incomplete) beta function. Expanding the definitions,
$$\alpha(\theta) = 2(1 - \cos(\theta)) - (1 - \cos(\theta))^2.$$
This can further be simplified to
$$\alpha(\theta) = \sin(\theta)^2.$$
The formula for $A(D_n(\theta))$ above is valid only when $0 \leq \theta \leq \pi / 2$, corresponding to $0 \leq h(\theta) \leq r$. When $\pi / 2 \leq \theta$, the formula is given by $A_n r^{n - 1} - A(\pi - \theta)$.