Problem 13 - Elementary Algebra

Question

Find the area of a circle with a diameter of \(12 \mathrm{~m}\).

Step-by-Step Solution

Answer

The area is 113.04 m².

Step 1: Understand the formula for the area of a circle

The area of a circle can be calculated using the formula \(\text{Area} = \pi r^{2}\), where \(r\) is the radius of the circle and \(\pi\) is a constant approximately equal to 3.14.

Step 2: Determine the radius

The radius of a circle is half of the diameter. Given that the diameter is \(12 \text{ m}\), the radius \(r\) is \(\frac{12}{2} = 6 \text{ m}\).

Step 3: Substitute the radius into the formula

Now substitute \(r = 6 \text{ m}\) into the area formula: \(\text{Area} = \pi (6)^{2}\).

Step 4: Calculate the area

Square the radius and multiply by \(\pi\): \(\text{Area} = \pi \cdot 36\). Using \(\pi \approx 3.14\), the area is \(\text{Area} = 3.14 \cdot 36 = 113.04 \text{ m}^{2}\).

Key Concepts

circle area formula

To find the area of a circle, you use the formula: \[\text{Area} = \pi r^2\]. This formula requires two main components: \(\text{pi}\) and the radius (\(r\)). The symbol \(\text{pi}\) (π) represents a constant that is used in mathematics to relate the circumference of a circle to its diameter. It is approximately equal to 3.14. The radius (\(r\)) is the distance from the center of the circle to any point on its boundary. Understanding and correctly applying this formula is crucial in geometry. For example, in our exercise, the radius was determined to be 6 meters. When we substitute this into the formula, it helps us find the area efficiently.

radius and diameter relationship

The radius and diameter of a circle have a direct relationship. The diameter is twice the length of the radius. This means if you know the diameter, you can easily find the radius by dividing the diameter by two. In our given problem, we had a circle with a diameter of 12 meters. By dividing this diameter by two, \(d/2\), we find that the radius \(r\) is \(12/2 = 6 \text{ m}\). This step is essential because the radius is used in the circle area formula.

pi value approximation

Pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter. Pi is an irrational number, meaning it has an infinite number of decimal places and doesn't repeat. For most practical purposes, we use the approximation \(\pi ≈ 3.14\). This simplifies calculations and is accurate enough for many uses. However, in some cases, you might use a more accurate approximation like \(\pi ≈ 3.14159\), or use the symbol π itself in equations involving circles. In our problem, we use \(\pi ≈ 3.14\) for finding the area. By multiplying \(π\) with the squared radius value (36), you get: \(3.14 \times 36 = 113.04 \text{ m}^{2}\). This gives us the area of the circle.