Methods for calculating $\pi$ that use the sphere?

Question

The area of the unit circle is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as bounding the circumference by sequences of inscribed and outscribed polygons with successively higher edge counts, or randomly sampling points from the unit square and then using the Pythagorean theorem to decide which of them lie within the unit circle to get an approximation of its area.

However, it is also true that the volume of the unit sphere is $\frac{4}{3}\pi$ and that its surface area is $4\pi$.

My question is: Which methods use $\pi$'s role in the geometry of the sphere to calculate or approximate the constant?

Answer 1

You can mimic the way it is done in 2D, by randomly sampling points in the unit cube and calcultate the fraction that lie in the unit sphere inside. The theoretical result is $\dfrac{\frac43\pi}{8} = \dfrac{\pi}{6}$.

However, it don't see it being more efficient than the 2D version, because you need to make $1.5$ times more random samples, and the calculus of the hypothenus length of the triangle is more tedious with the third coordinate.

Answer 2

For the unit circle to obtain circumference and sector area we integrate arc and area respectively in terms of sector surface angle in radians:

$$ c = 2 \pi , A = \pi $$

The constant 2 getting to half its value in higher dimension is related to 2D.

For the unit sphere to obtain area and volume we integrate area and volume in terms of "cone" solid angle in steridians:

$$ A=4 \pi , V = 4 \pi/3 $$

The constant 4 getting to one-third its value in higher dimension is related to 3D.

The elegant Gauss-Bonnet theorem in its simplest terms states that surface and volume angles just add up to a (topological) constant.

Unfortunately imho, these are not associated, that is thought of together, in the manner perhaps intended by Gauss, as it is possible to integratedly imagine 2 and 3 dimensional spaces at high school stage itself.

The unifying role of $\pi$ can be seen in evaluation of two, three and four dimensional geometrical quantities.