Calculating The Shaded Area A Square Inscribed In A Circle
This article delves into a fascinating geometry problem involving a square inscribed within a circle. Specifically, we aim to calculate the area of the shaded region, which is the area of the circle minus the area of the square. The problem provides a crucial piece of information: the diameter of the circle is {sqrt{2}}$ millimeters. We will also leverage the properties of a 45°-45°-90° triangle to simplify our calculations. This is a classic problem that beautifully illustrates the interplay between different geometric shapes and the power of using known relationships to solve for unknowns. Understanding how shapes interact and how to calculate areas is a fundamental skill in mathematics, and this problem offers a great opportunity to practice and hone these skills. Mastering these concepts not only helps in academic settings but also provides a strong foundation for various real-world applications, from architecture and engineering to design and art. So, let's embark on this geometric journey and unravel the solution step by step. Our goal is to provide a clear and concise explanation, ensuring that the reasoning behind each step is easily understood. We'll break down the problem into smaller, manageable parts, making it accessible to learners of all levels. By the end of this article, you'll have a solid understanding of how to approach such problems and a newfound appreciation for the elegance of geometry.
To effectively tackle this geometry problem, a clear understanding of the given information and the desired outcome is paramount. We are presented with a scenario where a square is neatly inscribed within a circle. This means that all four vertices (corners) of the square lie precisely on the circumference of the circle. A critical piece of data is the diameter of the circle, which measures sqrt{2}}$ millimeters. Our primary objective is to determine the area of the shaded region. This shaded region is the area enclosed within the circle but lying outside the square. In other words, it's the difference between the circle's area and the square's area. To calculate these areas, we'll need to find the radius of the circle and the side length of the square. The problem also helpfully reminds us of a key property of 45°-45°-90° triangles$ units. This property will be instrumental in relating the diagonal of the square to the diameter of the circle. Visualizing the problem is often the first step towards solving it. Imagine a perfect square nestled inside a circle, its corners touching the circle's edge. The shaded region forms four segments around the square, each bounded by a side of the square and an arc of the circle. Understanding the relationships between the circle, the square, and the 45°-45°-90° triangle is crucial for finding the solution. The diameter of the circle, for instance, plays a dual role: it's the longest chord of the circle and also the diagonal of the inscribed square. By carefully considering these relationships, we can devise a plan to calculate the required area.
The first step in solving this geometric puzzle is to determine the radius of the circle. We are given that the diameter of the circle is {sqrt{2}}$ millimeters. Recall that the radius of a circle is simply half of its diameter. Therefore, to find the radius, we need to divide the diameter by 2. In this case, we have:
Radius = Diameter / 2
Radius = ({sqrt{2}}$) / 2
Radius = {sqrt{2}}$ millimeters
Now that we have the radius, we can proceed to calculate the area of the circle. The formula for the area of a circle is given by: Area = π * radius² Substituting the value of the radius we just calculated, we get:
Area of Circle = π * ({sqrt{2}}$)^2
Area of Circle = π * (36 * 2)
Area of Circle = 72Ï€ square millimeters
This represents the total area enclosed within the circle. We'll use this value later to find the shaded area. Calculating the radius is a fundamental step, as it forms the basis for calculating the circle's area. Understanding the relationship between diameter and radius is crucial in many geometry problems. The formula for the area of a circle is a cornerstone of geometry, and mastering its application is essential. With the area of the circle in hand, our next task is to determine the area of the inscribed square. This will involve leveraging the properties of the 45°-45°-90° triangle, as mentioned earlier in the problem statement.
Now that we know the radius of the circle, our next challenge is to find the side length of the inscribed square. This is where the provided information about the 45°-45°-90° triangle comes into play. The diagonal of the square is a crucial element here, as it coincides with the diameter of the circle. This is because the square's vertices touch the circle's circumference, making the diagonal a chord that passes through the circle's center. We already know that the diameter of the circle is sqrt{2}}$ millimeters. Therefore, the diagonal of the square is also {sqrt{2}}$ millimeters. Consider the square's diagonal as the hypotenuse of a right-angled triangle formed by two sides of the square. This triangle is a special case\sqrt2}$ = {sqrt{2}}$ To solve for x, we divide both sides of the equation by ${sqrt{2}}$$ = {sqrt{2}}$** is a direct application of the 45°-45°-90° triangle property. With the side length of the square determined, we are now in a position to calculate its area.
With the side length of the square now known to be 12 millimeters, we can easily calculate its area. The area of a square is found by simply squaring the length of one of its sides. Thus, the formula for the area of a square is:
Area = side²
In our case, the side length is 12 millimeters, so:
Area of Square = 12²
Area of Square = 144 square millimeters
This value represents the total area enclosed within the square. Now that we have both the area of the circle (72Ï€ square millimeters) and the area of the square (144 square millimeters), we are just one step away from finding the shaded area. The formula for the area of a square is a fundamental concept in geometry. Squaring the side length provides a straightforward way to calculate the area. Understanding the units is important: since the side length is in millimeters, the area is in square millimeters. With the areas of both the circle and the square calculated, we can now move on to the final step: finding the difference between these areas to determine the shaded region.
The final step in solving this problem is to calculate the area of the shaded region. As we established earlier, the shaded region is the area inside the circle but outside the square. To find this area, we simply subtract the area of the square from the area of the circle. We have already calculated these areas:
Area of Circle = 72Ï€ square millimeters
Area of Square = 144 square millimeters
Therefore, the area of the shaded region is:
Area of Shaded Region = Area of Circle - Area of Square
Area of Shaded Region = 72Ï€ - 144 square millimeters
This is the exact value of the shaded area. We can also approximate this value by using the value of π (approximately 3.14159):
Area of Shaded Region ≈ 72 * 3.14159 - 144
Area of Shaded Region ≈ 226.194 - 144
Area of Shaded Region ≈ 82.194 square millimeters
So, the area of the shaded region is approximately 82.194 square millimeters. Subtracting the area of the square from the area of the circle is the key to finding the shaded area. The expression 72π - 144 represents the exact value of the shaded area. Approximating the value of π allows us to get a numerical estimate of the shaded area. We have now successfully calculated the area of the shaded region, completing our solution to the problem.
In this article, we tackled a geometric problem involving a square inscribed within a circle and successfully calculated the area of the shaded region. We began by understanding the problem statement and identifying the key information: the diameter of the circle ({sqrt{2}}$ millimeters) and the relationship between 45°-45°-90° triangles. We then systematically broke down the problem into smaller, manageable steps. First, we found the radius of the circle by dividing the diameter by 2, which gave us a radius of {sqrt{2}}$ millimeters. This allowed us to calculate the area of the circle using the formula πr², resulting in an area of 72π square millimeters. Next, we determined the side length of the square by recognizing that the diagonal of the square is equal to the diameter of the circle. Using the properties of 45°-45°-90° triangles, we found the side length to be 12 millimeters. This enabled us to calculate the area of the square as 12², which equals 144 square millimeters. Finally, we calculated the area of the shaded region by subtracting the area of the square from the area of the circle: 72π - 144 square millimeters. We also approximated this value to be about 82.194 square millimeters. Solving this problem required a combination of geometric knowledge, algebraic manipulation, and careful attention to detail. The step-by-step approach we used demonstrates a powerful problem-solving strategy that can be applied to a wide range of mathematical challenges. Understanding the relationships between different geometric shapes, such as circles, squares, and triangles, is fundamental to success in geometry. This problem serves as a great example of how these relationships can be used to solve complex problems. By mastering these concepts and techniques, you'll be well-equipped to tackle future geometric challenges with confidence.
