Four circles with radius $3$ have centers at the corners of a square, each tangent to its two neighboring circles. What is the total area of the circles and the region bounded by them?

1. The four centers form a square of side $2r$ (adjacent circles tangent). Near each corner, the interior angle of the square ($90^\circ$) is covered twice - once by the square, once by that circle's $90^\circ$ sector - so the total area is the square plus each circle's remaining $270^\circ$ (i.e.\ $\frac34$) that lies outside the square: area $=(2r)^2+4\cdot\frac34\pi r^2$.
2. With $r=3$: $=36+4\cdot\frac34\pi(9)=36+27\pi$.
3. The answer is $\boxed{36+27\pi}$.