The problem asks for the sum of the circumference of a circle and the perimeter of a square, given their total area and the circle's diameter.

Circle Radius and Area Calculation

First, find the radius of the circle from its diameter.

• Diameter of the circle, $D = 42$ cm.
• Radius of the circle, $r = \frac{D}{2} = \frac{42 \text{ cm}}{2} = 21$ cm.
• Calculate the area of the circle using the formula $A_c = \pi r^2$. Using $\pi = \frac{22}{7}$:
$A_c = \frac{22}{7} \times (21 \text{ cm})^2$ $A_c = \frac{22}{7} \times 441 \text{ sq. cm}$ $A_c = 22 \times 63 \text{ sq. cm}$ $A_c = 1386 \text{ sq. cm}$

Square Area and Side Length Calculation

Next, determine the area and side length of the square.

• The total area of the circle and the square is given as $A_t = 2611$ sq. cm.
• Calculate the area of the square ($A_s$) by subtracting the circle's area from the total area:
$A_s = A_t - A_c$ $A_s = 2611 \text{ sq. cm} - 1386 \text{ sq. cm}$ $A_s = 1225 \text{ sq. cm}$
• Calculate the side length of the square ($s$) using the formula $s = \sqrt{A_s}$:
$s = \sqrt{1225 \text{ sq. cm}}$ $s = 35 \text{ cm}$

Circumference and Perimeter Calculation

Now, calculate the circumference of the circle and the perimeter of the square.

• Calculate the circumference of the circle using the formula $C_c = 2 \pi r$:
$C_c = 2 \times \frac{22}{7} \times 21 \text{ cm}$ $C_c = 2 \times 22 \times 3 \text{ cm}$ $C_c = 132 \text{ cm}$
• Calculate the perimeter of the square using the formula $P_s = 4s$:
$P_s = 4 \times 35 \text{ cm}$ $P_s = 140 \text{ cm}$

Sum Calculation

Finally, find the sum of the circumference and the perimeter.

• Sum = Circumference of circle + Perimeter of square
$Sum = C_c + P_s$ $Sum = 132 \text{ cm} + 140 \text{ cm}$ $Sum = 272 \text{ cm}$

The sum of the circumference of the circle and the perimeter of the square is 272 cm.