Calculating the Radius of a Circle Based on the Sum of Other Circumferences

When dealing with circles, it's often necessary to determine the radius of one circle based on the circumferences of other circles. This is particularly useful in various fields such as geometry, engineering, and architecture. In this article, we will explore how to find the radius of a circle whose circumference is equal to the sum of two given circumferences.

Understanding Circumference and Radius

The circumference of a circle, denoted as (C), is the distance around the circle. It is directly proportional to the radius (r) and is given by the formula:

[text{Circumference} 2pi r]

Example Problem

Consider two circles with radii 19 cm and 9 cm. What is the radius of a new circle whose circumference is equal to the sum of the circumferences of these two circles?

Step-by-Step Solution

Method 1: Using (2pi) Proportionality

First, calculate the circumference of each circle.

[text{Circumference of the first circle} 2pi times 19 38pi]

[text{Circumference of the second circle} 2pi times 9 18pi]

The total circumference of the new circle is the sum of these two circumferences:

[text{Total circumference} 38pi 18pi 56pi]

Let the radius of the new circle be denoted by (R). Using the circumference formula, we have:

[text{Circumference of the new circle} 2pi R]

Setting the total circumference equal to the circumference of the new circle:

[text{Total circumference} 56pi]

[therefore 2pi R 56pi]

[therefore R frac{56pi}{2pi} 28 text{ cm}]

Method 2: Using Pi Approximation ((frac{22}{7}))

Another way to solve this problem is by using the fraction (frac{22}{7}) as an approximation for (pi).

[text{Circumference of first circle} 2 times frac{22}{7} times 19 frac{836}{7}text{ cm}]

[text{Circumference of second circle} 2 times frac{22}{7} times 9 frac{396}{7}text{ cm}]

[text{Total circumference} frac{836}{7} frac{396}{7} frac{1232}{7}text{ cm}]

Equating the total circumference to the new circle's circumference:

[frac{22}{7} times R frac{1232}{7}]

[therefore R frac{1232}{22} 56text{ cm /}2 28 text{ cm}]

Method 3: Direct Proportionality

Given that the circumference is directly proportional to the radius, we can sum the radii of the given circles to find the radius of the new circle.

[text{Sum of radii} 19 9 28 text{ cm}]

Therefore, the radius of the new circle is:

[text{Radius of new circle} 28 text{ cm}]

Conclusion

Through various methods, we have demonstrated how to calculate the radius of a circle whose circumference is equal to the sum of the circumferences of other circles. Whether using (pi) or approximating it as (frac{22}{7}), the radius of the new circle is:

[boxed{28 text{ cm}}]

Understanding the relationship between circumference and radius is crucial in many practical applications. This article provides a clear and concise explanation of the process, ensuring that readers can apply these principles confidently.