Finding the Greatest Inscribed Circle Area

The question asks for the area of the greatest circle that can fit inside a square with a side length of 21 cm.

Relating Square Side to Circle Diameter

The largest possible circle that can be inscribed within a square will have a diameter equal to the side length of the square.

• Square side length, $s = 21$ cm.
• Diameter of the greatest inscribed circle, $d = s = 21$ cm.

Calculating the Circle's Radius

The radius ($r$) is half the diameter ($d$).

Calculation: $r = \frac{d}{2} = \frac{21 \text{ cm}}{2} = 10.5$ cm.

Computing the Circle's Area

The formula for the area ($A$) of a circle is $A = \pi r^2$. We use the approximation $\pi \approx \frac{22}{7}$.

Calculation: $A = \pi r^2$ $A = \frac{22}{7} \times (10.5 \text{ cm})^2$ $A = \frac{22}{7} \times (\frac{21}{2} \text{ cm})^2$ $A = \frac{22}{7} \times \frac{441}{4} \text{ cm}^2$ $A = \frac{11 \times 441}{7 \times 2} \text{ cm}^2$ $A = \frac{11 \times 63}{2} \text{ cm}^2$ $A = \frac{693}{2} \text{ cm}^2$ $A = 346.5 \text{ cm}^2$

Therefore, the area of the greatest circle that can be inscribed inside the square is 346.5 cm$^2$.