Math Before Eat

Area of circle = $\pi r^2$
Area of square= $(\sqrt{2}r)^2=2r^2$

Hence grey area= $\pi r^2 - 2r^2$ =56 $\Rightarrow r^2=\dfrac{56}{\pi-2}........(1)$ Required volume= $\pi r^2 h$
= $\pi(\dfrac{56}{\pi-2})$ h ...(Using 1)

= $\dfrac{56}{1-\frac{2}{\pi}}\times 30$
Substituting $\pi=\dfrac{22}{7}$ , we get the desired answer i.e $\boxed{4620}$

Let's find the brown square size, assume the square size is $x$

By using Pythagorean Theorem,

$x^2 = r^2 + r^2$

$x^2 = 2r^2$

$x = r \sqrt {2}$

From the picture, we can conclude that,

Gray area $=$ Circle area $-$ Square Area

$56 = \pi r^2 - (r \sqrt {2})^2$

$56 = \frac {22}{7} r^2 - 2r^2$

$56 = \frac {8}{7} r^2$

$r^2 = 49$

$r = 7$

We have find the radius of the circle, now let's find the area of the circle

$\pi r^2 = \frac{22}{7} \times 7^2 = 154$

Now find the volume of the soupcan

$v = \pi r^2 \times h$

$v = 154 \times 30$

$v = \boxed{4620}$

Which purple is the purple you mean when you say purple in the problem. Please change the square's colour by a few nanometers of wavelength so it contrasts more.