Calculating Radius Using Circumference Ratio In Circles

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In the fascinating world of geometry, circles hold a special place. Their elegant symmetry and constant proportions have captivated mathematicians and thinkers for centuries. One of the fundamental relationships in circles is the connection between their circumference and radius. The circumference of a circle is the distance around it, while the radius is the distance from the center of the circle to any point on its edge. This article delves into a problem involving the ratio of circumferences of two circles and how to determine the radius of the larger circle when the radius of the smaller circle is known. This is a classic problem that illustrates the relationship between the circumference and radius of a circle, and it provides a great opportunity to understand how ratios and proportions can be applied in geometric contexts.

Before we dive into the problem, let's quickly review the basic concepts of circumference and radius. The circumference of a circle is the distance around the circle. It's like the perimeter of a polygon, but for a circle. The radius, on the other hand, is the distance from the center of the circle to any point on its edge. It's a line segment that connects the center to the boundary of the circle. These two measurements are intrinsically linked, and their relationship is described by a fundamental formula. The formula that connects these two measurements is:

Circumference (C) = 2Ï€r

Where:

  • C represents the circumference of the circle.
  • Ï€ (pi) is a mathematical constant approximately equal to 3.14159.
  • r represents the radius of the circle.

This formula tells us that the circumference of a circle is directly proportional to its radius. This means that if you double the radius, you double the circumference. If you triple the radius, you triple the circumference, and so on. This direct proportionality is a key concept in understanding the problem we're about to tackle.

Now, let's consider the problem at hand. We are given that the circumferences of two circles are in the ratio of 2:5. This means that if we call the circumference of the smaller circle C1 and the circumference of the larger circle C2, then:

C1 / C2 = 2 / 5

We are also given that the radius of the smaller circle, which we'll call r1, is 16 inches. Our goal is to find the radius of the larger circle, which we'll call r2. To solve this problem, we'll need to use the relationship between circumference and radius, as well as the concept of ratios and proportions. We know the ratio of the circumferences and the radius of the smaller circle. We can use this information to find the circumference of the smaller circle. Then, we can use the ratio of circumferences to find the circumference of the larger circle. Finally, we can use the formula for circumference to find the radius of the larger circle.

Let's break down the solution into a series of logical steps:

1. Define Variables

First, let's define our variables clearly:

  • r1 = radius of the smaller circle = 16 inches
  • r2 = radius of the larger circle (what we want to find)
  • C1 = circumference of the smaller circle
  • C2 = circumference of the larger circle

2. Express the Ratio of Circumferences

We are given that the ratio of the circumferences is 2:5. This can be written as:

C1 / C2 = 2 / 5

3. Use the Circumference Formula

We know that the circumference of a circle is given by the formula C = 2Ï€r. Therefore, we can write the circumferences of the two circles as:

  • C1 = 2Ï€r1
  • C2 = 2Ï€r2

4. Substitute into the Ratio

Now, let's substitute these expressions for C1 and C2 into our ratio equation:

(2Ï€r1) / (2Ï€r2) = 2 / 5

Notice that the 2Ï€ terms appear in both the numerator and the denominator, so they cancel out:

r1 / r2 = 2 / 5

5. Plug in the Known Radius

We know that r1 = 16 inches, so we can plug that into the equation:

16 / r2 = 2 / 5

6. Solve for the Unknown Radius

To solve for r2, we can cross-multiply:

16 * 5 = 2 * r2

80 = 2 * r2

Now, divide both sides by 2:

r2 = 40

Therefore, the radius of the larger circle is 40 inches.

The radius of the larger circle is 40 inches. This corresponds to option B in the given choices. The solution involves using the formula for the circumference of a circle, understanding ratios and proportions, and applying algebraic manipulation to solve for the unknown radius. By breaking the problem down into smaller steps, we can clearly see how each piece of information contributes to the final answer. This problem showcases the practical application of geometric concepts and their connection to algebraic problem-solving.

Let's recap the key concepts that were crucial in solving this problem:

  • Circumference of a Circle: The distance around a circle, given by the formula C = 2Ï€r.
  • Radius of a Circle: The distance from the center of the circle to any point on its edge.
  • Ratio: A comparison of two quantities, expressed as a fraction or with a colon.
  • Proportion: An equation stating that two ratios are equal.
  • Direct Proportionality: The relationship between circumference and radius, where an increase in one leads to a proportional increase in the other.

Understanding these concepts is fundamental to tackling similar problems involving circles and their properties. The ability to apply these concepts in a step-by-step manner is a valuable skill in mathematics and problem-solving.

To solidify your understanding, try solving these similar problems:

  1. The circumferences of two circles are in the ratio of 3:7. The radius of the smaller circle is 9 cm. What is the radius of the larger circle?
  2. Two circles have radii of 5 inches and 15 inches. What is the ratio of their circumferences?
  3. A circle has a circumference of 88 inches. Another circle has a circumference that is twice as large. What is the radius of the larger circle?

By working through these problems, you'll gain confidence in your ability to apply the concepts discussed in this article. Remember to break down each problem into smaller steps and use the formulas and relationships you've learned.

In conclusion, understanding the relationship between the circumference and radius of a circle is essential for solving geometric problems. By using the formula C = 2Ï€r and applying the concepts of ratios and proportions, we can determine unknown radii or circumferences. The problem discussed in this article provides a clear example of how these concepts can be applied in a practical setting. Remember to break down complex problems into smaller, manageable steps, and don't be afraid to revisit the fundamental concepts when needed. With practice and a solid understanding of the basics, you can confidently tackle a wide range of geometry problems.