There are many attractive geometric arguments for irrationality. Here is one for the Golden Ratio.
Construct a golden rectangle. A recipe for doing this can be found in Euclid, and was known to early Pythagoreans.
There should be a diagram to illustrate the idea. Perhaps you can draw it yourself. Say the golden rectangle is $ABCD$, with the vertices as usual enumerated counterclockwise, and let $AB$ be a long side of the rectangle.
We prove that the long side and the short side of a golden rectangle are incommensurable. Suppose to the contrary that sides $AB$ and $BC$ have a common measure $m$. Or else, in more modern language, suppose that $AB$ and $BC$ have lengths that are each an integer multiple of some common number $m$. Or, else, even more arithmetically, suppose that each side is an integer.
Cut off a square $AEFD$ from the rectangle, by finding the point $E$ on $AB$ such that $AE=AD$, and slicing straight up. That leaves a rectangle $EBCF$, which by the definition of golden rectangle, is itself golden. It is clear that $m$ is a common measure of the sides of $EBCF$.
Continue, by cutting off a square from $EBCF$, leaving an even smaller golden rectangle whose sides have common measure $m$. Clearly, this process can be continued forever. But after a while, each side of the little golden rectangle just produced will be less than the hypothesized common measure $m$, and we get our contradiction.
There has been speculation that this was the first irrationality proof. The only problem with this theory is the total lack of evidence.