Intuition behind $(-\frac{1}{2})! = \sqrt{\pi}$

Consider the area of the surface of the $n-$Ball with radius $1$ . It is given by:

$$ A_{n}=2\frac{\pi^{n/2}}{\Gamma(n/2)} $$

Our intuition tells us that for $n=1$ the surface "area" (or to be mathematically more precise the, Hausdorff measure as @Michael Galuza pointed out correctly)should be 2, because it consist of two points. To make this consistent with the above formula we have to demand that

$$\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$$

Is there an intuitive explanation to this ?

Yes. There is an umbilical connection between $\bigg(\dfrac1n\bigg){\large!}$ or $\Gamma\bigg(\dfrac1n\bigg)$, and geometric shapes
of the form $X^n+Y^n=R^n\iff x^n+y^n=1\iff y=\sqrt[n]{1-x^n}$, whose area is
$\displaystyle\int_0^1\sqrt[n]{1-x^n}~dx$, which is nothing else than the beta function in disguise. Ultimately, it's
all related to Newton's binomial theorem. The latter expands the power of a sum into a
sum of powers, with the help of binomial coefficients, or beta functions, which are then
expressed in terms of factorials, or $\Gamma$ functions.