Problem 93 - Prealgebra

Question

A circle has a diameter of \(10.75\) 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 approximately 90.72 square inches.

Step 1: Find the Radius of the Circle

The radius of a circle is half of its diameter. Given that the diameter is 10.75 inches, we calculate the radius as follows: \[ r = \frac{10.75}{2} = 5.375 \text{ inches} \]

Step 2: Use the Formula for the Area of a Circle

The formula for the area of a circle is \( A = \pi r^2 \), where \( r \) is the radius. We will plug the radius we found into this formula.

Step 3: Calculate the Circle's Area

Substitute the radius and \( \pi \) into the formula: \[ A = 3.14 \times (5.375)^2 \]Next, calculate \( 5.375^2 \) which equals 28.890625. Substitute this back into the area calculation: \[ A = 3.14 \times 28.890625 = 90.7155625 \]

Step 4: Round the Answer to the Nearest Hundredth

The area calculation is 90.7155625. To round this number to the nearest hundredth, we look at the third decimal place. The number is 5, so we round up: \[ A \approx 90.72 \text{ square inches} \]

Key Concepts

Circle Geometry

When exploring circle geometry, it's important to understand the basic components that make up a circle.
Circles are defined by their geometric properties, which include radius, diameter, and of course, the circle itself.

• The **radius** is the distance from the center of the circle to any point on its boundary.
• The **diameter** is twice the length of the radius and stretches across the circle through its center.
• The **circumference** is the distance around the circle, calculated as \( C = \pi d \) or \( C = 2\pi r \).
• The **area** of a circle, which we often need to find, is the space contained within its boundaries and is given by the formula \( A = \pi r^2 \).

Understanding these concepts and relationships is critical in solving geometry problems related to circles. Geometry provides a visual understanding that helps in grasping complex algebraic computations, making it easier to select the correct formulas and apply them correctly.

Diameter and Radius

The relationship between diameter and radius is a fundamental concept in circle geometry.
These two distinct measurements are directly linked by a simple yet crucial relationship:

• The **diameter** of a circle is always twice the length of its **radius**.
• Conversely, the radius is half of the diameter.

Mathematically, this is represented as: \[ d = 2r \text{ and } r = \frac{d}{2} \]Knowing the diameter or the radius allows you to find the other easily.
This was evident in the problem where the given diameter allowed us to find the radius by dividing it by two.
These relationships are foundational and appear frequently in practical problems such as finding the area or circumference of a circle.

Rounding Numbers

Rounding numbers is a useful mathematical skill, especially in geometry problems where precision is crucial.
In the given example, after calculating an area, we needed to round to the nearest hundredth.
Here's how rounding works, particularly to the nearest hundredth:

• Identify the digit in the hundredth place, which is two places to the right of the decimal.
• Look at the digit directly after the hundredth place.
• If this digit is 5 or greater, increase the hundredth digit by 1.
• If it's less than 5, simply leave the hundredth digit unchanged.

For instance, when rounding 90.7155625 to the nearest hundredth, we note the third decimal number is 5.
Thus, we round up, giving us 90.72.
Learning to round correctly ensures your results are both accurate and easy to communicate.