Problem 10 - Elementary Algebra

Question

Find the circumference of a circle with a radius of \(0.7 \mathrm{in}\).

Step-by-Step Solution

Answer

The circumference is approximately 4.396 inches.

Step 1: Identify the Given Values

The radius of the circle is given as 0.7 inches.

Step 2: Recall the Formula for Circumference

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

Step 3: Substitute the Given Radius into the Formula

Substitute the radius value of 0.7 inches into the formula: \( C = 2 \times \pi \times 0.7 \).

Step 4: Calculate the Circumference

Perform the multiplication: \( C = 2 \times \pi \times 0.7 = 1.4 \times \pi \). Using \( \pi \approx 3.14 \), we get \( C \approx 1.4 \times 3.14 = 4.396 \) inches.

Key Concepts

radius

The radius is a fundamental component of a circle. It is the distance from the center of the circle to any point on its edge. A circle's radius is always the same, no matter which direction you measure in. For example, in the given exercise, the radius of the circle is 0.7 inches. This is a crucial piece of information because it allows us to determine other properties of the circle, such as its circumference and area. Always remember, the symbol for radius is usually represented by \(r\). If you have the radius, you can easily find the diameter of the circle by multiplying the radius by 2.

circumference formula

The circumference of a circle is its total perimeter or the distance around it. To find the circumference when the radius is known, we use the circumference formula: \( C = 2 \times \pi \times r \). This formula tells us that we need to multiply the radius by 2 and then by \(\pi\).
This formula is derived from the relationship between the diameter (twice the radius) and pi, since the circumference is essentially how many times the diameter fits around the circle. In the example, the given radius is 0.7 inches, so we substitute in the formula: \( C = 2 \times \pi \times 0.7 \). Understanding this formula is key to solving many problems involving circles.

multiplication with pi

Pi, denoted by the symbol \(\pi\), is an irrational number approximately equal to 3.14159. It represents the ratio of the circumference of any circle to its diameter. When you multiply with \(\pi\), you are scaling your value by this special constant. For practical purposes, we often use the approximation \(\pi \approx 3.14\), which is accurate enough for most calculations.
Let's see how it's done in our exercise. After substituting the radius into the formula \( C = 2 \times \pi \times 0.7 \), we have \( C= 1.4 \times \pi \). Using the approximation, \(1.4 \times 3.14\) gives a circumference of approximately 4.396 inches.

• Always multiply the numeric values first before multiplying by \( \pi\).
• Keep in mind that \( \pi\) can also be kept in terms of itself for more precise answers, for example, 1.4\(\pi\).