Finding the Area of a Circular Field from Circumferences

This problem involves finding the area of a third circular field whose radius is the sum of the radii of two other circular fields, given their circumferences. To solve this, we first need to determine the radii of the first two fields using their circumferences. Then, we sum these radii to find the radius of the third field, and finally, calculate the area of the third field.

Key Formulas

We will use the following formulas for a circle:

• Circumference (C) = $2\pi R$
• Area (A) = $\pi R^2$

Where R is the radius of the circle and $\pi = 22/7$ as given.

Step-by-Step Solution

Step 1: Find the radius of the first circular field ($R_1$)

The circumference of the first field is given as 396 m.

Using the circumference formula:

$$C_1 = 2\pi R_1$$

Substituting the given values:

$$396 = 2 \times \frac{22}{7} \times R_1$$

$$396 = \frac{44}{7} R_1$$

To find $R_1$, we rearrange the equation:

$$R_1 = \frac{396 \times 7}{44}$$

$$R_1 = \frac{396}{44} \times 7$$

Since $396 \div 44 = 9$:

$$R_1 = 9 \times 7 = 63 \text{ m}$$

The radius of the first circular field is 63 m.

Step 2: Find the radius of the second circular field ($R_2$)

The circumference of the second field is given as 132 m.

Using the circumference formula:

$$C_2 = 2\pi R_2$$

Substituting the given values:

$$132 = 2 \times \frac{22}{7} \times R_2$$

$$132 = \frac{44}{7} R_2$$

To find $R_2$, we rearrange the equation:

$$R_2 = \frac{132 \times 7}{44}$$

$$R_2 = \frac{132}{44} \times 7$$

Since $132 \div 44 = 3$:

$$R_2 = 3 \times 7 = 21 \text{ m}$$

The radius of the second circular field is 21 m.

Step 3: Find the radius of the third circular field ($R_3$)

The radius of the third field is the sum of the radii of the first two fields.

$$R_3 = R_1 + R_2$$

Substituting the values of $R_1$ and $R_2$:

$$R_3 = 63 \text{ m} + 21 \text{ m}$$

$$R_3 = 84 \text{ m}$$

The radius of the third circular field is 84 m.

Step 4: Calculate the area of the third circular field ($A_3$)

Now we find the area of the third field using its radius ($R_3 = 84$ m) and the area formula:

$$A_3 = \pi R_3^2$$

Substituting the values:

$$A_3 = \frac{22}{7} \times (84)^2$$

$$A_3 = \frac{22}{7} \times 84 \times 84$$

We can simplify by dividing 84 by 7:

$$A_3 = 22 \times \left(\frac{84}{7}\right) \times 84$$

$$A_3 = 22 \times 12 \times 84$$

Now, multiply the numbers:

$$A_3 = (22 \times 12) \times 84$$

$$A_3 = 264 \times 84$$

Performing the multiplication:

$$264 \times 84 = 22176$$

So, the area of the third circular field is 22176 m².

Let's quickly summarize the radii and areas calculated:

Conclusion

The radius of the first circular field is 63 m, and the radius of the second circular field is 21 m. The radius of the third circular field is the sum of these radii, which is $63 + 21 = 84$ m. The area of the third circular field with a radius of 84 m is calculated as $\pi \times 84^2 = \frac{22}{7} \times 84^2 = 22176$ m².

Revision Table: Circle Formulas and Calculation Summary

Additional Information: Understanding Circle Properties

The circumference and area of a circle are fundamental properties related to its radius. The radius is the distance from the center of the circle to any point on its edge. The diameter is twice the radius. The circumference is the distance around the circle, and the area is the space enclosed within the circle.

• Relation between Circumference and Radius: The circumference is directly proportional to the radius. If you double the radius, you double the circumference. The constant of proportionality is $2\pi$.
• Relation between Area and Radius: The area is proportional to the square of the radius ($R^2$). If you double the radius, the area increases by a factor of $2^2 = 4$.
• Using $\pi = 22/7$: While $\pi$ is an irrational number, $22/7$ is a common and useful approximation for calculations, especially when the radius or diameter is a multiple of 7. This simplifies calculations significantly, as seen in this problem where radii were 63 and 21 (multiples of 7) and the resulting radius was 84 (a multiple of 7).

These concepts are crucial for solving problems involving circles in geometry and real-world applications like calculating the size of circular fields or objects.