Problem 53 - College Algebra

Question

The formula for the circumference of a circle is \(C=2 \pi r\) . Find the circumference of a circle with a diameter of 12 in. (diameter \(=2 r ) .\) Use the symbol \(\pi\) in your final answer.

Step-by-Step Solution

Answer

The circumference is \(12\pi\) inches.

Step 1: Understand the Relationship Between Diameter and Radius

The diameter of a circle is twice the radius, which can be expressed as \(d = 2r\). Therefore, the radius \(r\) can be found by dividing the diameter by 2.

Step 2: Calculate the Radius

Given the diameter \(d = 12\) inches, we find the radius by the formula \(r = \frac{d}{2}\). Substituting the given diameter, we have \(r = \frac{12}{2} = 6\) inches.

Step 3: Use the Circumference Formula

The formula for the circumference \(C\) of a circle is \(C = 2\pi r\). Now that we know the radius \(r = 6\) inches, we can use this formula to find the circumference.

Step 4: Calculate the Circumference

Substitute \(r = 6\) into the circumference formula: \(C = 2\pi \times 6 = 12\pi\). Therefore, the circumference of the circle is \(12\pi\) inches.

Key Concepts

Diameter and Radius Relationship

Understanding the connection between diameter and radius is very important when working with circles. A circle's diameter is simply twice its radius. This is expressed in the formula:

• Diameter, \(d = 2r\)
• Radius, \(r = \frac{d}{2}\)

This means if you know the diameter, you can find the radius by dividing the diameter by two. For example, if a circle's diameter is 12 inches, the radius would be 6 inches, since \(r = \frac{12}{2}\). This relationship is fundamental as it acts as a starting point for more intricate calculations with circles.

Circle Formulas

Formulas related to circles allow us to calculate various properties like circumference, area, and more. The most common formula is for the circumference, which is the distance around the circle. The formula for the circumference is:

• \(C = 2\pi r\)

Here, \(C\) represents the circumference, \(r\) is the radius, and \(\pi\) (pi) is a constant. It's crucial to use the radius in the calculation. For instance, if a circle has a radius of 6 inches, you find the circumference by plugging it into the formula: \(C = 2\pi\times 6\). So, \(C = 12\pi\) inches. This formula allows easy calculations once you know the radius.

Pi in Calculations

Pi, denoted as \(\pi\), is a mathematical constant that approximately equals 3.14159. It represents the ratio of a circle's circumference to its diameter. When dealing with circles, it's common to leave the symbol \(\pi\) in your answers, especially if you're looking for a precise expression.

• For example, the circumference formula should ideally be expressed with \(\pi\) for exactness: \(C = 12\pi\)

This keeps your answer accurate. Remember that \(\pi\) plays a key role not just in circumference calculations but also in finding areas and volumes involving circles and circular shapes. Instead of converting \(\pi\) to 3.14, keeping it in the formula provides a clearer and more mathematically accurate result.