Square In A Circle: Diagonal & Perimeter With Area 38 M²

Let's dive into a classic geometry problem where we explore the relationship between a square inscribed in a circle. We're given that the area of the circle is 38 m², and our mission is to find two key properties of the square: the exact length of its diagonal and its exact perimeter. This is a fantastic exercise in connecting geometric concepts like area, diagonals, and perimeters, and it's sure to sharpen your problem-solving skills. So, grab your thinking caps, and let's get started!

a) Determining the Exact Length of the Square's Diagonal

To find the exact length of the diagonal of the square, we need to connect the information about the circle's area to the square's dimensions. The key insight here is that when a square is perfectly inscribed within a circle, the diagonal of the square is equal to the diameter of the circle. This is a fundamental geometric relationship that allows us to bridge the gap between the circle and the square.

Our starting point is the circle's area, which is given as 38 m². We know that the area of a circle is calculated using the formula A = πr², where A represents the area and r represents the radius of the circle. Therefore, we can set up the equation 38 = πr² and solve for r. To isolate r², we divide both sides of the equation by π, giving us r² = 38/π. Taking the square root of both sides, we find that r = √(38/π).

Now that we have the radius, we can easily find the diameter, which is simply twice the radius. So, the diameter d = 2r = 2√(38/π). As we established earlier, the diameter of the circle is equal to the diagonal of the inscribed square. Therefore, the exact length of the diagonal of the square is 2√(38/π) meters. This is our answer for part (a). We've successfully used the relationship between the circle's area and its radius to determine the diagonal of the inscribed square. Remember, visualizing these geometric relationships is crucial for solving such problems. Understanding that the diagonal of the square is the diameter of the circle was the key to unlocking this part of the problem.

b) Calculating the Exact Perimeter of the Square

Now that we know the exact length of the diagonal of the square, we can move on to finding its perimeter. To do this, we'll need to determine the side length of the square first. The diagonal of a square divides it into two right-angled triangles, where the diagonal acts as the hypotenuse and the sides of the square are the legs. This brings the Pythagorean theorem into play, a fundamental concept in geometry that relates the sides of a right-angled triangle.

Let's denote the side length of the square as s. According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (sides of the square). So, we have the equation s² + s² = (2√(38/π))². Simplifying this, we get 2s² = 4(38/π). Dividing both sides by 2, we have s² = 2(38/π) = 76/π. Taking the square root of both sides, we find the side length s = √(76/π) meters.

With the side length in hand, calculating the perimeter is straightforward. The perimeter of a square is simply four times the length of one of its sides. Therefore, the perimeter P of the square is P = 4s = 4√(76/π) meters. This is the exact perimeter of the square. We've now successfully found both the diagonal length and the perimeter of the square, using the information about the circle's area and applying geometric principles like the Pythagorean theorem. This problem highlights the interconnectedness of different geometric concepts and the power of using known formulas and theorems to solve problems.

Conclusion

In this problem, we've successfully navigated the relationship between a circle and an inscribed square. We started with the area of the circle and used it to determine the diagonal length and perimeter of the square. The key takeaways here are the understanding that the diagonal of the inscribed square is equal to the diameter of the circle, and the application of the Pythagorean theorem to relate the diagonal and side length of the square. These are valuable tools in your geometric problem-solving arsenal.

Geometry problems often require us to visualize shapes, understand their properties, and connect different concepts. By breaking down complex problems into smaller, manageable steps and applying the right formulas and theorems, we can arrive at elegant solutions. Keep practicing, and you'll become a geometry whiz in no time!

For further exploration of geometry and related concepts, consider visiting Khan Academy's Geometry section, a fantastic resource for learning and practicing mathematical skills.