Question

Find the area of each figure to nearest hundredth. circle: diameter, 7 meters

Short answer

The area of the circle is approximately 38.48 square meters.

Step-by-step solution

Identify the Radius

The diameter of the circle is given as 7 meters. The radius, which is half of the diameter, is calculated by dividing the diameter by 2: \[ r = \frac{7}{2} = 3.5 \text{ meters} \]

Use the Formula for Area of a Circle

The area \( A \) of a circle is given by the formula: \[ A = \pi r^2 \] where \( r \) is the radius.

Plug in the Radius Value

Substitute the radius value from Step 1 into the area formula: \[ A = \pi (3.5)^2 \]

Calculate the Area

First, square the radius: \[ (3.5)^2 = 12.25 \] Then, multiply by \( \pi \): \[ A = \pi \times 12.25 \approx 38.484510006 \]

Round to the Nearest Hundredth

Round the area calculated in Step 4 to the nearest hundredth: \[ A \approx 38.48 \]

Key concepts

Understanding Radius and Diameter

When dealing with circles, it's important to recognize the role that both the radius and diameter play. The **diameter** of a circle is the straight line that passes through the center and touches two points on the edge of the circle. In every circle, the diameter is twice the length of the radius. Think of the diameter as the full width of the circle.

The **radius** is the distance from the center of the circle to any point on its edge. It is half of the diameter. In mathematical terms, if the diameter of the circle is given as a certain value (let's say 7 meters as in the problem), the radius would be the diameter divided by two. So, you can calculate the radius as:

• r = diameter ÷ 2
• Example: If the diameter is 7 meters, then r = 7 ÷ 2 = 3.5 meters

This relationship helps in finding other properties of the circle, like its area, using the radius.

Exploring the Area of a Circle Formula

The area of a circle represents the space contained within its boundary. To find the area, you use the formula:

\[ A = \pi r^2 \]

Here, **\( \pi \)** (pi) is a constant approximately equal to 3.14159, and **\( r \)** is the radius of the circle. The formula highlights that the area is proportional to the square of the radius. This means that as the radius increases, the area increases rapidly. In our example with a radius of 3.5 meters, here's how you would plug in the values:

• Use the formula: A = \pi r^2
• Substitute the radius: A = \pi (3.5)^2

It's a practical formula that allows for quick calculation of the space inside a circle, essential in many geometry and real-world applications.

The Magic of Squaring Numbers

Squaring a number is a fundamental mathematical operation that means multiplying that number by itself. This operation becomes crucial when calculating the area of a circle since the formula includes \( r^2 \), the square of the radius.

Let's take our radius of 3.5 meters. To square this number, you perform the following calculation:

• 3.5 × 3.5 = 12.25

After finding the square of the radius, the next step is to multiply it by \( \pi \) to find the area:

• So: A = \pi \times 12.25 ≈ 38.48 (rounded to nearest hundredth)

Understanding squaring numbers helps in many areas of mathematics beyond finding areas of circles, making it a versatile tool in calculations.