Problem 95 - Prealgebra

Question

A circle has a diameter of \(13.96\) inches. Using \(\pi \approx 3.14\), find the area of the circle, correct to the nearest hundredth of a square inch.

Step-by-Step Solution

Answer

The area of the circle is 152.98 square inches.

Step 1: Identify the Radius

The first step in finding the area of a circle is to determine the radius from the given diameter. Since the diameter is given as \(13.96\) inches, we need to divide it by \(2\) to find the radius. Therefore, the radius \(r\) is \(\frac{13.96}{2} = 6.98\) inches.

Step 2: Use the Area Formula

The formula to find the area of a circle is \(A = \pi r^2\). In this formula, \(A\) is the area, \(\pi\) is approximately \(3.14\), and \(r\) is the radius of the circle we calculated in Step 1.

Step 3: Calculate \(r^2\)

Calculate \(r^2\) by squaring the radius. Hence, \(6.98^2 = 48.7204\).

Step 4: Substitute and Calculate the Area

Substitute the approximate value of \(\pi\) and \(r^2\) into the area formula: \(A = 3.14 \times 48.7204\). Calculate: \(A \approx 3.14 \times 48.7204 = 152.982056\) square inches.

Step 5: Round the Area to the Nearest Hundredth

The last step is to round \(152.982056\) to the nearest hundredth. Thus, the area of the circle is \(152.98\) square inches.

Key Concepts

Radius Determination

To find the area of a circle, the very first thing we need is the radius. The radius is half the length of the diameter. When you have a circle with a known diameter, like the one in the exercise which is 13.96 inches, calculating the radius is straightforward. You simply divide the diameter by 2.
For this example, the formula looks like this:

• Diameter: 13.96 inches
• Radius: \( r = \frac{13.96}{2} = 6.98 \) inches

Remember, always ensure your units remain consistent! This means the diameter and radius should always be in the same unit measurement, like inches in this task. Finding the radius is crucial because it is used in the formula to calculate the circle's area.

Area Formula for Circles

Once you know the radius of a circle, you can easily find its area. The area of a circle is the total space that is contained within its circumference. To find this, one can use the formula:

• \( A = \pi r^2 \)

Here, \(\pi\) (pi) is a unique mathematical constant approximately equal to 3.14. It represents the ratio of the circumference to the diameter of any circle. In our task, we use \( \pi \approx 3.14 \).
The formula also includes \( r^2 \), which is the radius squared. This simply means multiplying the radius by itself. Using the radius calculated in the previous section, \(6.98 \), the math looks like this:

• \( r^2 = 6.98 \times 6.98 = 48.7204 \)

Substituting these values into the formula gives us the area of the circle:

• \( A = 3.14 \times 48.7204 \approx 152.982056 \) square inches

Rounding Decimals

Finally, when calculating the area, you might end up with a number that is not a clean, whole number. Most tasks will ask you to round this number to a certain decimal point. Rounding decimals helps make the number easier to read and use.
In this example, the calculated area was approximately 152.982056 square inches. When asked to round to the nearest hundredth, you look at the digits to the right of your hundredth place. Here's how it works:

• The hundreds place is the second digit after the decimal point. Here, it's 8.
• Look at the digit right after the hundredths place, which is 2 in this case. Since 2 is less than 5, you keep the hundreds place as is.
• Thus, round 152.982056 to 152.98 square inches.

Rounding allows you to present your final answer in a format that is tidy and meets the requirements of the exercise. It's an essential skill for accuracy and neatness in math!