Understanding the Problem: Circle, Square, Perimeter, and Area

This problem involves a circle and a square. We are given a relationship between the perimeter of the square and the circumference of the circle, specifically that they are equal. We are also given the radius of the circle and the value of $\pi$. Our goal is to find the ratio of the area of the circle to the area of the square.

Step-by-Step Solution

Let's break down the problem into smaller, manageable steps. We will first calculate the circumference of the circle, then use that to find the side of the square, and finally calculate the areas and their ratio.

Step 1: Calculate the Circumference of the Circle

The formula for the circumference of a circle is $C = 2\pi r$, where $r$ is the radius and $\pi$ is the mathematical constant. We are given the radius $r = 7$ cm and are asked to use $\pi = \frac{22}{7}$.

Substitute the values into the formula:

$$C = 2 \times \frac{22}{7} \times 7$$

$$C = 2 \times 22$$

$$C = 44 \text{ cm}$$

The circumference of the circle is 44 cm.

Step 2: Find the Side Length of the Square

We are told that the perimeter of the square is equal to the circumference of the circle. The formula for the perimeter of a square is $P = 4s$, where $s$ is the length of one side of the square.

Since $P = C$, we have:

$$4s = 44 \text{ cm}$$

To find the side length $s$, divide the perimeter by 4:

$$s = \frac{44}{4}$$

$$s = 11 \text{ cm}$$

The side length of the square is 11 cm.

Step 3: Calculate the Area of the Circle

The formula for the area of a circle is $A_{\text{circle}} = \pi r^2$. We know $r = 7$ cm and $\pi = \frac{22}{7}$.

Substitute the values into the formula:

$$A_{\text{circle}} = \frac{22}{7} \times (7)^2$$

$$A_{\text{circle}} = \frac{22}{7} \times 49$$

$$A_{\text{circle}} = 22 \times 7$$

$$A_{\text{circle}} = 154 \text{ cm}^2$$

The area of the circle is 154 cm$^2$.

Step 4: Calculate the Area of the Square

The formula for the area of a square is $A_{\text{square}} = s^2$, where $s$ is the side length. We found that $s = 11$ cm.

Substitute the value into the formula:

$$A_{\text{square}} = (11)^2$$

$$A_{\text{square}} = 121 \text{ cm}^2$$

The area of the square is 121 cm$^2$.

Step 5: Find the Ratio of the Area of the Circle to the Area of the Square

We need to find the ratio $\frac{A_{\text{circle}}}{A_{\text{square}}}$.

$$\text{Ratio} = \frac{154}{121}$$

To simplify the ratio, we find the greatest common divisor (GCD) of 154 and 121. Both numbers are divisible by 11.

$$154 \div 11 = 14$$

$$121 \div 11 = 11$$

So the simplified ratio is $\frac{14}{11}$.

The ratio of the area of the circle to the area of the square is 14 : 11.

Summary of Calculations

The final ratio of the area of the circle to the area of the square is 14 : 11.

Revision Table: Key Concepts and Formulas

Additional Information: Relating Perimeter and Area

This problem highlights how the perimeter (or circumference) of a shape is related to its area. While the formulas for perimeter and area use the same basic dimensions (radius or side length), they measure different properties. Perimeter/circumference measures the distance around the boundary, while area measures the space enclosed within the boundary.

In this specific case, the equality of the square's perimeter and the circle's circumference allowed us to establish a direct relationship between their respective dimensions ($s$ and $r$). Once this relationship was known, we could easily calculate and compare their areas.

It's interesting to note that for a fixed perimeter/circumference, a circle encloses the maximum possible area compared to any other shape. In this problem, even though the perimeters are equal, the ratio of areas shows that the circle's area is larger than the square's area (154 vs 121).