A Square Is Inscribed In A Circle With A Diameter Of $12 \sqrt{2}$ Millimeters. What Is The Area Of The Shaded Region?Recall That In A $45^{\circ}-45^{\circ}-90^{\circ}$ Triangle, If The Legs Each Measure \$x$[/tex\]

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Introduction

In this article, we will explore the problem of finding the area of the shaded region when a square is inscribed in a circle. The circle has a diameter of $12 \sqrt{2}$ millimeters, and we need to determine the area of the shaded region. To solve this problem, we will use the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle and the formula for the area of a square.

Understanding the Problem

The problem involves a square inscribed in a circle with a diameter of $12 \sqrt{2}$ millimeters. The square is symmetrically positioned within the circle, and we need to find the area of the shaded region. To approach this problem, we need to understand the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle and how it relates to the square.

Properties of a $45{\circ}-45{\circ}-90^{\circ}$ Triangle

A $45{\circ}-45{\circ}-90^{\circ}$ triangle is a special right triangle with two legs of equal length and a hypotenuse that is $\sqrt{2}$ times the length of each leg. In this triangle, the legs each measure $x$, and the hypotenuse measures $x \sqrt{2}$. This property is crucial in solving the problem, as the square inscribed in the circle is formed by connecting the midpoints of the circle's diameter.

Finding the Side Length of the Square

To find the side length of the square, we need to consider the properties of the $45{\circ}-45{\circ}-90^{\circ}$ triangle. Since the diameter of the circle is $12 \sqrt{2}$ millimeters, the radius of the circle is half of the diameter, which is $6 \sqrt{2}$ millimeters. The diagonal of the square is equal to the diameter of the circle, which is $12 \sqrt{2}$ millimeters. Using the properties of the $45{\circ}-45{\circ}-90^{\circ}$ triangle, we can find the side length of the square.

Calculating the Side Length of the Square

Let's denote the side length of the square as $s$. Since the diagonal of the square is $12 \sqrt{2}$ millimeters, we can use the Pythagorean theorem to find the side length of the square:

s2+s2=(122)2s^2 + s^2 = (12 \sqrt{2})^2

Simplifying the equation, we get:

2s2=2882s^2 = 288

Dividing both sides by 2, we get:

s2=144s^2 = 144

Taking the square root of both sides, we get:

s=12s = 12

Therefore, the side length of the square is $12$ millimeters.

Finding the Area of the Square

Now that we have found the side length of the square, we can find the area of the square. The area of a square is given by the formula:

A=s2A = s^2

Substituting the value of $s$, we get:

A=122A = 12^2

Simplifying the equation, we get:

A=144A = 144

Therefore, the area of the square is $144$ square millimeters.

Finding the Area of the Circle

To find the area of the shaded region, we need to find the area of the circle. The area of a circle is given by the formula:

A=Ï€r2A = \pi r^2

Substituting the value of $r$, we get:

A=Ï€(62)2A = \pi (6 \sqrt{2})^2

Simplifying the equation, we get:

A=72Ï€A = 72 \pi

Therefore, the area of the circle is $72 \pi$ square millimeters.

Finding the Area of the Shaded Region

The area of the shaded region is the difference between the area of the circle and the area of the square. Therefore, we can find the area of the shaded region by subtracting the area of the square from the area of the circle:

Ashaded=Acircle−AsquareA_{\text{shaded}} = A_{\text{circle}} - A_{\text{square}}

Substituting the values, we get:

Ashaded=72π−144A_{\text{shaded}} = 72 \pi - 144

Simplifying the equation, we get:

Ashaded=72π−144A_{\text{shaded}} = 72 \pi - 144

Therefore, the area of the shaded region is $72 \pi - 144$ square millimeters.

Conclusion

In this article, we have found the area of the shaded region when a square is inscribed in a circle with a diameter of $12 \sqrt{2}$ millimeters. We used the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle and the formula for the area of a square to solve the problem. The area of the shaded region is $72 \pi - 144$ square millimeters.

References

  • [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [2] "Mathematics for the Nonmathematician" by Morris Kline

Glossary

  • **$45{\circ}-45\circ}-90^{\circ}$ triangle** A special right triangle with two legs of equal length and a hypotenuse that is $\sqrt{2$ times the length of each leg.
  • Diameter: The distance across a circle passing through its center.
  • Radius: The distance from the center of a circle to its edge.
  • Square: A four-sided shape with equal sides and equal angles.
  • Shaded region: The area between the circle and the square.
    A Square Inscribed in a Circle: Q&A =====================================

Introduction

In our previous article, we explored the problem of finding the area of the shaded region when a square is inscribed in a circle. We used the properties of a $45{\circ}-45{\circ}-90^{\circ}$ triangle and the formula for the area of a square to solve the problem. In this article, we will answer some frequently asked questions related to the problem.

Q: What is the diameter of the circle?

A: The diameter of the circle is $12 \sqrt{2}$ millimeters.

Q: What is the side length of the square?

A: The side length of the square is $12$ millimeters.

Q: How do you find the area of the square?

A: To find the area of the square, you need to use the formula $A = s^2$, where $s$ is the side length of the square.

Q: How do you find the area of the circle?

A: To find the area of the circle, you need to use the formula $A = \pi r^2$, where $r$ is the radius of the circle.

Q: What is the area of the shaded region?

A: The area of the shaded region is the difference between the area of the circle and the area of the square. It is given by the formula $A_{\text{shaded}} = A_{\text{circle}} - A_{\text{square}}$.

Q: Can you explain the concept of a $45{\circ}-45{\circ}-90^{\circ}$ triangle?

A: A $45{\circ}-45{\circ}-90^{\circ}$ triangle is a special right triangle with two legs of equal length and a hypotenuse that is $\sqrt{2}$ times the length of each leg. This property is crucial in solving the problem, as the square inscribed in the circle is formed by connecting the midpoints of the circle's diameter.

Q: How do you find the side length of the square using the properties of the $45{\circ}-45{\circ}-90^{\circ}$ triangle?

A: To find the side length of the square, you need to use the Pythagorean theorem. Let's denote the side length of the square as $s$. Since the diagonal of the square is $12 \sqrt{2}$ millimeters, we can use the Pythagorean theorem to find the side length of the square:

s2+s2=(122)2s^2 + s^2 = (12 \sqrt{2})^2

Simplifying the equation, we get:

2s2=2882s^2 = 288

Dividing both sides by 2, we get:

s2=144s^2 = 144

Taking the square root of both sides, we get:

s=12s = 12

Therefore, the side length of the square is $12$ millimeters.

Q: What is the significance of the radius of the circle?

A: The radius of the circle is half of the diameter, which is $6 \sqrt{2}$ millimeters. This value is used to find the area of the circle.

Q: Can you explain the concept of the shaded region?

A: The shaded region is the area between the circle and the square. It is the difference between the area of the circle and the area of the square.

Conclusion

In this article, we have answered some frequently asked questions related to the problem of finding the area of the shaded region when a square is inscribed in a circle. We hope that this article has provided you with a better understanding of the problem and its solution.

References

  • [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [2] "Mathematics for the Nonmathematician" by Morris Kline

Glossary

  • **$45{\circ}-45\circ}-90^{\circ}$ triangle** A special right triangle with two legs of equal length and a hypotenuse that is $\sqrt{2$ times the length of each leg.
  • Diameter: The distance across a circle passing through its center.
  • Radius: The distance from the center of a circle to its edge.
  • Square: A four-sided shape with equal sides and equal angles.
  • Shaded region: The area between the circle and the square.