The Given Area and Its Implications

Oct 13, 2025 by windailys.com 84 views

Introduction

When dealing with geometric shapes, particularly circles, it's essential to understand the relationships between their various properties. One such property is the area of a circle, which is given by the formula $A = \pi r^2$ , where $A$ is the area and $r$ is the radius. In this article, we will explore how to simplify the expression for the radius of a circle with a given area.

The Given Area and Its Implications

The area of the circle is given as 60 square centimeters. Using the formula for the area of a circle, we can set up the equation $\pi r^2 = 60$ . To find the radius, we need to isolate $r$ by taking the square root of both sides of the equation. This gives us $r = \sqrt{\frac{60}{\pi}}$ centimeters.

Simplifying the Expression for the Radius

Now, let's simplify the expression for the radius. We can start by rationalizing the denominator by multiplying both the numerator and the denominator by $\sqrt{\pi}$ . This gives us $r = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{<br/>

Introduction

In our previous article, we explored how to simplify the expression for the radius of a circle with a given area. We started with the formula for the area of a circle, $A = \pi r^2$ , and used it to find the radius of a circle with an area of 60 square centimeters. We then simplified the expression for the radius by rationalizing the denominator.

Q&A: Simplifying the Expression for the Radius

Q: What is the simplified expression for the radius of a circle with an area of 60 square centimeters?

A: The simplified expression for the radius is $\sqrt{\frac{60}{\pi}}$ centimeters.

Q: How do I rationalize the denominator of the expression for the radius?

A: To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt{\pi}$ . This gives us $r = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}}$ .

Q: What is the relationship between the area and the radius of a circle?

A: The area of a circle is given by the formula $A = \pi r^2$ . This means that the area is proportional to the square of the radius.

Q: How do I find the radius of a circle with a given area?

A: To find the radius, use the formula $r = \sqrt{\frac{A}{\pi}}$ .

Q: What is the significance of the expression $\sqrt{\frac{60}{\pi}}$ ?

A: The expression $\sqrt{\frac{60}{\pi}}$ represents the radius of a circle with an area of 60 square centimeters.

Q: Can I simplify the expression $\sqrt{\frac{60}{\pi}}$ further?

A: Yes, we can simplify the expression by rationalizing the denominator. This gives us $r = \sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}}$ .

Q: What is the final simplified expression for the radius?

A: The final simplified expression for the radius is $\sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}}$ .

Conclusion

In this article, we explored how to simplify the expression for the radius of a circle with a given area. We started with the formula for the area of a circle, $A = \pi r^2$ , and used it to find the radius of a circle with an area of 60 square centimeters. We then simplified the expression for the radius by rationalizing the denominator. We also answered some common questions related to simplifying the expression for the radius.

Final Answer

The final simplified expression for the radius is $\sqrt{\frac{60\pi}{\pi^2}} = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}}$ .

Additional Resources

Formula for the area of a circle: $A = \pi r^2$
Formula for the radius of a circle: $r = \sqrt{\frac{A}{\pi}}$
Rationalizing the denominator: $r = \sqrt{\frac{60}{\pi}} \cdot \frac{\sqrt{\pi}}{\sqrt{\pi}} = \sqrt{\frac{60\pi}{\pi^2}}$