Circle & Square: Find Shaded Area!

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Circle & Square: Find Shaded Area!

Hey guys! Let's dive into a cool geometry problem where we're figuring out the area of a shaded region. This involves a square chilling inside a circle, and it's all about finding the sweet spot in between. We've got a circle with a diameter of $12 \sqrt{2}$ millimeters, and a square perfectly nestled inside it. Our mission, should we choose to accept it, is to calculate the area of the space that's inside the circle but outside the square. Buckle up, because we're about to break this down step by step!

Understanding the Problem

Okay, first things first. Let's really understand what we're dealing with here. We have a circle, right? And inside this circle, there's a square. Imagine drawing a square inside a circle so that all four corners (or vertices, if we're feeling fancy) of the square are touching the edge of the circle. That's what we mean by "inscribed." The shaded region is basically the leftover space – the parts of the circle that aren't covered by the square. To find this, we're going to need to do a little math magic involving areas.

Now, the key piece of information we're given is the diameter of the circle: $12 \sqrt{2}$ millimeters. Remember, the diameter is the distance across the circle, passing through the very center. This is super important because it's directly related to the radius, which we'll need to calculate the circle's area. Also, the diameter of the circle is the diagonal of the square. This will help us find the side length of the square and then its area. We know that in a $45{\circ}-45{\circ}-90^{\circ}$ triangle, if the legs each measure $x$ units, then the hypotenuse measures $x\sqrt{2}$ units. This relationship is crucial because it connects the side length of the square to its diagonal (which is also the diameter of the circle). This is like a secret code that unlocks the problem for us!

Breaking Down the Steps

  1. Find the Radius: Remember, the radius is half the diameter. So, we'll divide the given diameter by 2.
  2. Calculate the Circle's Area: We'll use the classic formula for the area of a circle: $A = \pi r^2$, where A is the area and r is the radius.
  3. Determine the Side Length of the Square: Here's where that $45{\circ}-45{\circ}-90^{\circ}$ triangle rule comes into play. The diagonal of the square is the diameter of the circle, so we can use the relationship to find the side length.
  4. Calculate the Square's Area: The area of a square is simply side length times side length (or side length squared): $A = s^2$, where A is the area and s is the side length.
  5. Find the Shaded Area: Finally, we'll subtract the area of the square from the area of the circle. This will give us the area of the shaded region.

Step-by-Step Solution

Alright, let's get our hands dirty and crunch some numbers!

1. Find the Radius

The diameter of the circle is $12 \sqrt{2}$ millimeters. To find the radius (r), we divide the diameter by 2:

r = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2}$ millimeters So, the radius of our circle is $6 \sqrt{2}$ millimeters. ### 2. Calculate the Circle's Area Now that we have the radius, we can use the formula for the area of a circle: $A = \pi r^2$. Plugging in our value for *r*: $A = \pi (6 \sqrt{2})^2 = \pi (36 \cdot 2) = 72 \pi$ square millimeters Therefore, the area of the circle is $72 \pi$ square millimeters. ### 3. Determine the Side Length of the Square This is where the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle comes in handy. The diagonal of the square is the diameter of the circle, which is $12 \sqrt{2}$ millimeters. Let's call the side length of the square *s*. In a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, the hypotenuse (which is the diagonal of our square) is $s\sqrt{2}$. So we have: $s \sqrt{2} = 12 \sqrt{2}

To find s, we divide both sides by $\sqrt{2}$:

s = \frac{12 \sqrt{2}}{\sqrt{2}} = 12$ millimeters Thus, the side length of the square is 12 millimeters. ### 4. Calculate the Square's Area The area of a square is side length squared: $A = s^2$. Using our side length of 12 millimeters: $A = 12^2 = 144$ square millimeters So, the area of the square is 144 square millimeters. ### 5. Find the Shaded Area Finally, to find the shaded area, we subtract the area of the square from the area of the circle: $Shaded Area = Circle Area - Square Area = 72 \pi - 144$ square millimeters We can factor out a 72 to simplify this a bit: $Shaded Area = 72(\pi - 2)$ square millimeters ## Final Answer And there we have it! The area of the shaded region is $72(\pi - 2)$ square millimeters. That's the space inside the circle but outside the square. *Awesome*, right? ## Key Takeaways This problem is a fantastic example of how different geometric concepts can come together. We used the relationship between the diameter and radius of a circle, the formula for the area of a circle, the properties of a square, and the special $45^{\circ}-45^{\circ}-90^{\circ}$ triangle ratio. Here are some key things to remember: * **Radius and Diameter:** The radius is half the diameter. * **Area of a Circle:** $A = \pi r^2

  • Area of a Square: $A = s^2$
  • 45-45-90 Triangles: The sides are in the ratio $x:x:x\sqrt{2}$

By mastering these concepts, you'll be well-equipped to tackle similar geometry challenges.

Practice Makes Perfect

Geometry is all about practice, guys! The more problems you solve, the better you'll become at recognizing patterns and applying the right formulas. Try working through some similar problems with different dimensions or shapes. You could even try inscribing other shapes inside circles or squares. The possibilities are endless!

So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!