Finding the Radius of the Circle

Oct 13, 2025 by recoveryshake.com 144 views

Introduction

In geometry, a square inscribed in a circle is a common problem that involves understanding the properties of both shapes. When a square is inscribed in a circle, its vertices touch the circle, and the sides of the square are chords of the circle. In this problem, we are given a square with sides of $$3 \sqrt{2}$$ inscribed in a circle. We need to find the area of one of the sectors formed by the radii to the vertices of the square.

Understanding the Problem

To solve this problem, we need to understand the properties of the square and the circle. The square has sides of $$3 \sqrt{2}$$, and it is inscribed in a circle. This means that the vertices of the square touch the circle, and the sides of the square are chords of the circle. We need to find the area of one of the sectors formed by the radii to the vertices of the square.

Drawing a Diagram

Let's draw a diagram to visualize the problem. We can draw a square with sides of $$3 \sqrt{2}$$ inscribed in a circle. The vertices of the square touch the circle, and the sides of the square are chords of the circle. We can also draw the radii to the vertices of the square, which form the sectors.

Finding the Radius of the Circle

To find the area of one sector, we need to find the radius of the circle. Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle. The diagonal of the square is $$3 \sqrt{2} \times \sqrt{2} = 6$$, which is equal to the diameter of the circle. Therefore, the radius of the circle is $$\frac{6}{2} = 3$$.

Finding the Area of One Sector

Now that we have the radius of the circle, we can find the area of one sector. The area of a sector is given by the formula $$\frac{\theta}{360} \pi r^2$$, where $$\theta$$ is the central angle of the sector, and $$r$$ is the radius of the circle. In this case, the central angle of the sector is $$90^\circ$$, and the radius of the circle is $$3$$. Therefore, the area of one sector is $$\frac{90}{360} \pi (3)^2 = \frac{1}{4} \pi (9) = 2.25 \pi$$.

Conclusion

In this problem, we were given a square with sides of $$3 \sqrt{2}$$ inscribed in a circle. We needed to find the area of one of the sectors formed by the radii to the vertices of the square. We found that the radius of the circle is $$3$$, and the area of one sector is $$2.25 \pi$$. Therefore, the correct answer is $$2.25 \pi$$.

Final Answer

The final answer is $$2.25 \pi$$.

References

[1] Geometry textbook by [Author]
[2] Online resource for geometry problems

Additional Information

This problem is a classic example of a geometry problem that involves understanding the properties of a square and a circle.
The problem requires the use of the formula for the area of a sector, which is $$\frac{\theta}{360} \pi r^2$$.
The problem also requires the use of the Pythagorean theorem to find the radius of the circle.

Tips and Tricks

When solving geometry problems, it's essential to draw a diagram to visualize the problem.
Use the formula for the area of a sector, which is $$\frac{\theta}{360} \pi r^2$$.
Use the Pythagorean theorem to find the radius of the circle.

Common Mistakes

Failing to draw a diagram to visualize the problem.
Not using the formula for the area of a sector.
Not using the Pythagorean theorem to find the radius of the circle.

Conclusion

$In conclusion, this problem is a classic example of a geometry problem that involves understanding the properties of a square and a circle. We found that the radius of the circle is $$3$$, and the area of one sector is $$2.25 \pi$$. Therefore, the correct answer is $$2.25 \pi$$.<br/> **A Square with Sides of $$3 \sqrt{2}$$ Inscribed in a Circle: Q&A**$

Q: What is the relationship between the square and the circle in this problem?

A: In this problem, the square is inscribed in the circle, which means that the vertices of the square touch the circle, and the sides of the square are chords of the circle.

Q: How do we find the radius of the circle?

A: To find the radius of the circle, we use the fact that the diagonal of the square is equal to the diameter of the circle. The diagonal of the square is $$3 \sqrt{2} \times \sqrt{2} = 6$$, which is equal to the diameter of the circle. Therefore, the radius of the circle is $$\frac{6}{2} = 3$$.

Q: What is the formula for the area of a sector?

A: The formula for the area of a sector is $$\frac{\theta}{360} \pi r^2$$, where $$\theta$$ is the central angle of the sector, and $$r$$ is the radius of the circle.

Q: How do we find the area of one sector in this problem?

A: In this problem, the central angle of the sector is $$90^\circ$$, and the radius of the circle is $$3$$. Therefore, the area of one sector is $$\frac{90}{360} \pi (3)^2 = \frac{1}{4} \pi (9) = 2.25 \pi$$.

Q: What is the significance of the Pythagorean theorem in this problem?

A: The Pythagorean theorem is used to find the radius of the circle. The diagonal of the square is $$3 \sqrt{2} \times \sqrt{2} = 6$$, which is equal to the diameter of the circle. Therefore, the radius of the circle is $$\frac{6}{2} = 3$$.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

Failing to draw a diagram to visualize the problem.
Not using the formula for the area of a sector.
Not using the Pythagorean theorem to find the radius of the circle.

Q: What are some tips and tricks for solving this problem?

A: Some tips and tricks for solving this problem include:

Drawing a diagram to visualize the problem.
Using the formula for the area of a sector.
Using the Pythagorean theorem to find the radius of the circle.

Q: How does this problem relate to real-world applications?

A: This problem relates to real-world applications in geometry and trigonometry. It can be used to solve problems involving the area of sectors and the properties of circles and squares.

Q: What are some similar problems to this one?

A: Some similar problems to this one include:

A square with sides of $$4 \sqrt{2}$$ is inscribed in a circle. Find the area of one of the sectors formed by the radii to the vertices of the square.
A circle with a radius of $$5$$ has a square inscribed in it. Find the area of one of the sectors formed by the radii to the vertices of the square.

Q: How can I practice solving problems like this one?

A: You can practice solving problems like this one by:

Drawing diagrams to visualize the problem.
Using the formula for the area of a sector.
Using the Pythagorean theorem to find the radius of the circle.
Practicing with similar problems.

Conclusion

In conclusion, this problem is a classic example of a geometry problem that involves understanding the properties of a square and a circle. We found that the radius of the circle is $$3$$, and the area of one sector is $$2.25 \pi$$. Therefore, the correct answer is $$2.25 \pi$$.