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The Area of an (n-1)-Dimensional Subsphere
in Terms of the Arc Distance in the n-Dimensional Sphere

An n-dimensional sphere (hereafter an n-sphere) centered at the origin of a
coordinate system is the set of points
(x_{1}, x_{2}, …, x_{n-1}, x_{n}) such that

x_{1}² + x_{2}² + … + x_{n-1}² +
x_{n}² = R²

where R is a constant called the radius.

Consider the point (0, 0, …, 0, R). As x_{n} decreases from R
a set of (n-1)-spheres are generated. The radius r of the (n-1)-sphere corresponding
to x_{n} is given by

r = (R² − x_{n}²)^{½}

The set of points on the n-sphere such that x_{i}=0 for all i from 1 to (n-2)
is a 2-sphere (circle) of radius R. The set of points from (0, 0, …, 0, R)
to (0, 0, …, r, (R²-r²)^{½}) is the arc of a circle
of radius R. Let s be the length of this arc and θ be the angle subtended
by the arc. Then

s = Rθ
and hence
θ = s/R

The (n-1)-sphere on the n-sphere has a radius of r where

r = Rsin(θ)

and the (n-1)-sphere is centered at (0, 0, …, 0, Rcos(θ)).

The volume of the (n-1)-sphere is a function of its radius r, but r=Rsin(θ)
so its volume is a function of Rsin(s/R) where s is the arc distance from (0, 0, …, 0, R).

For example for n=2 the 1-sphere is a circle on the sphere, as shown below, and its
volume is its circumference of 2πr. In terms of the arc distance s this circumference
is 2πRsin(s/R). For s small compared to R this reduces to 2πs.