Problem 9 - Elementary Algebra

Question

Find the circumference of a circle with a radius of \(0.5 \mathrm{in}\).

Step-by-Step Solution

Answer

The circumference is \( \pi \, \text{in} \).

Step 1: Identify the formula for circumference

The formula to find the circumference of a circle is given by \( C = 2 \times \pi \times r \), where \( C \) represents the circumference and \( r \) is the radius of the circle.

Step 2: Substitute the radius into the formula

Given that the radius \( r \) is \( 0.5 \, \text{in} \), substitute this value into the formula: \( C = 2 \times \pi \times 0.5 \).

Step 3: Calculate the value

Simplify the expression to find the circumference: \( C = 2 \times \pi \times 0.5 = \pi \, \text{in} \).

Key Concepts

Circle Radius

The radius of a circle is one of its most important measurements. It is the distance from the center of the circle to any point on its circumference. The radius is usually denoted by the letter \( r \). For example, if a circle has a radius of \( 0.5 \, \text{in} \), it means every point on the edge of the circle is exactly \( 0.5 \, \text{in} \) away from the center. Understanding the radius helps in calculating other properties of the circle, such as its diameter and circumference.
To find the radius, you can use various methods such as:

• Measuring directly from the center to the edge.
• Given the diameter (which is twice the radius), you can divide by 2.

In our exercise, the radius is already given as \( 0.5 \, \text{in} \), making it straightforward to use this value in further calculations.

Circumference Formula

The circumference is the total distance around a circle. To find the circumference, we use the formula \( C = 2 \times \pi \times r \), where \( C \) stands for circumference, \( \pi \) is a constant (Pi), and \( r \) is the radius. This formula tells us how far we would travel if we walked all the way around the edge of the circle.
Here’s how to apply it:

• Identify the radius of the circle.
• Multiply the radius by \( 2 \).
• Then multiply by \( \pi \).

For example, if the radius is \( 0.5 \, \text{in} \), the circumference calculation would be: \( C = 2 \times \pi \times 0.5 \).
Simplifying, we get \( C = \pi \, \text{in} \). Using this formula, you can easily find out the circumference of any circle if you know its radius.

π (Pi) in Calculations

Pi (\( \pi \)) is a special mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to \( 3.14159 \), but it is typically rounded to 3.14 or symbolized as \( \pi \) in calculations.
Here are some key points about Pi:

• \( \pi \) is an irrational number, meaning it cannot be exactly represented as a simple fraction, and its decimal representation never ends or repeats.
• Using \( \pi \), many circle-related calculations become straightforward.

In our exercise, when we say the circumference is equal to \( \pi \, \text{in} \), we can also approximate it numerically. For practical purposes, multiplying the radius and the circular constant \( 2 \pi \), means \( \pi \) is necessary to arrive at the circumference value.
When doing homework or tests, you might be asked to use different approximations for \( \pi \), but the concept remains the same. Understanding \( \pi \) helps simplify and solve many geometric problems involving circles.