Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

The circumference of a circle is \(16 \pi\) inches. Find the circle's radius and diameter.

Step-by-Step Solution

Answer

The radius of the circle is \(8\) inches, and the diameter is \(16\) inches.

Step 1: Determine the Radius from the Circumference

The formula for the circumference of a circle is \(C = 2\pi r\). Given that the circumference, \(C\) is \(16 \pi\) inches, we can substitute this value into the formula and solve for the radius (\(r\)). This gives us \(r = \frac{C}{2\pi} = \frac{16\pi}{2\pi} = 8\) inches.

Step 2: Determine the Diameter from the Radius

The diameter of a circle is twice its radius. Therefore, the diameter (\(d\)) can be determined by multiplying the radius by 2: \(d = 2r = 2(8) = 16\) inches.

Key Concepts

Understanding the Radius of a Circle

The radius of a circle is a fundamental part of circle geometry. It is the distance from the center of the circle to any point on its boundary. This is a very important measurement because it helps in calculating other properties of the circle.
For instance, the circumference and the area of the circle are directly related to the radius.

• To find the circumference (\(C\)), you use the formula: \(C = 2\pi r\), where \(r\) is the radius.
• For the area (\(A\)), the formula is \(A = \pi r^2\).

In the given exercise, once you know that the circumference is \(16\pi\) inches, finding the radius involves simple substitution into the formula \(C = 2\pi r\), rearranging to solve for \(r\). This gives us \(r = \frac{16\pi}{2\pi} = 8\) inches.

Determining the Diameter of a Circle

The diameter of a circle is another crucial aspect, which is always twice the radius. Think of it as the longest straight line that can be drawn across the circle, passing through the center. The relationship between the diameter (\(d\)) and the radius (\(r\)) is expressed simply as \(d = 2r\).
This is very useful when you want to quickly determine one from the other since knowing just the radius or the diameter can help calculate other properties of the circle.

• If the radius is known, you simply multiply it by 2 to find the diameter.
• If only the diameter is known, divide it by 2 to get the radius.

From the exercise, since \(r\) was found to be 8 inches, by applying \(d = 2(8)\), the diameter is readily calculated as 16 inches.

Applying Geometry Formulas

In geometry, there are specific formulas for calculating different aspects of a circle, and understanding these can make complex problems easier. These formulas help convert between circumference, radius, and diameter effortlessly.
For circles:

• The circumference formula, \(C = 2\pi r\), links the perimeter of the circle to its radius.
• The area formula, \(A = \pi r^2\), shows how to calculate the space inside the circle.
• The relationship between the diameter and the radius, \(d = 2r\), provides straightforward conversions between different circle measurements.

Understanding and memorizing these formulas allows students to solve circle-related problems efficiently. By practicing these formulas, it becomes intuitive to switch between calculating the radius, diameter, and other properties as needed. This skill is essential not only for academic exercises but also for practical applications in various fields, such as engineering and architecture.