This question will be easier if you draw an OUTER square to the largest circle (like how the ones inside are drawn).

You only need this largest square (the new one not in the picture), the largest circle in the diagram and the one square inside it. Do not draw the rest.

Area of the square inside the circle = 900 cm²
Area of the square outside the circle = 1800 cm²

(Think why. If you divide the square inside the circle to four equal parts, then there will be four equal triangles. The square outside the circle will then be divided into eight equal parts if you see carefully, which is twice the number of identical triangles found in the square inside the circle).

Since the area of the square outside the circle is 1800 cm²,

Length x Length = 1800 cm²
Length = Square root of 1800 cm

This means that the diameter of the largest circle in the diagram is square root of 1800 cm, which is 42.42640687 cm, and its radius is half of this, or 21.21320344 cm.

Area of circle
= pi x radius x radius
= pi x 21.21320344 x 21.21320344
= pi x 450
= 1413.716694 cm²

Remember that the area of the square inside the circle is 900 cm².

Area of shaded region
= Area of circle - Area of square inside circle
= 1413.716694 - 900
= 513.7166941 cm²
~ 513.72 cm²

Let me know if you need me to draw this on paper - the large circle, the square immediately inside it and the square immediately outside it (not drawn in the picture shown).

Hi Eric, thank you so much for your explanatiion. Yes, would it be possible to draw for me.

Hang on