Zelaron Gaming Forum - High school geometry problem finally solved after 272 years

https://i.imgur.com/X1EVwOu.png

Question: How big must r be chosen in the diagram, in order for the red area to equal one half of the area of the circle?

The above problem was first posed in 1748, and it has been known since then that r is approximately equal to 1.1578. Apparently it just took us 272 years to figure out an exact solution to it. Namely,

$\reverse r=2\,\cos\,\left( \frac{1}{2}\cdot\frac{ \oint_{|z-3 \pi / 8|=\pi / 4} z /(\sin z-z \cos z-\pi / 2) \, dz}{\oint_{|z-3 \pi / 8|=\pi / 4} 1 /(\sin z-z \cos z-\pi / 2)\, dz}\right)$

where the integrals are contour integrals.

Good job, humanity.

In all seriousness, it's fascinating how difficult trivial-looking math problems can become once you start to experiment a little with the functions/constants/boundary conditions/shapes/etc. Another good example of this is the calculation of the arc length of a curve.

Specifically, if y = f(x) is a continuously differentiable function on the interval (a,b), then its arc length s between a and b is

$\reverse s=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx.$

There are some simple exact solutions to the above integral, but let's just say that there are reasons why most calculus textbooks stick to examples where f(x) describes either a line, parabola, circle or hyperbolic cosine curve...

https://en.wikipedia.org/wiki/Arc_length

Quote:

In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary.