The area of the core of a column of cross sectional area A, is A. $$\frac{1}{3}{\text{A}}$$ B. $$\frac{1}{6}{\text{A}}$$ C. $$\frac{1}{{12}}{\text{A}}$$ D. $$\frac{1}{{18}}{\text{A}}$$

The correct answer is $\boxed{\frac{1}{3}{\text{A}}}$.

The area of the core of a column is the area of the column minus the area of the outer shell. The area of the outer shell is a circle with radius $r$, so its area is $\pi r^2$. The area of the column is a rectangle with width $2r$ and height $h$, so its area is $2rh$. Therefore, the area of the core is $2rh – \pi r^2 = \frac{2r^2}{3} – \pi r^2 = \frac{1}{3}{\text{A}}$.

Option A is incorrect because it is the area of a circle with radius $r$. Option B is incorrect because it is the area of a semicircle with radius $r$. Option C is incorrect because it is the area of a quarter circle with radius $r$. Option D is incorrect because it is the area of a sixth of a circle with radius $r$.