If The Area Of A Circle Is 89.42 Square Inches, What Is The Length Of A Side Of A Regular Hexagon Inscribed In The Circle?
Introduction: Exploring the Geometry of Circles and Hexagons
In the realm of geometry, circles and hexagons hold a special allure, their symmetrical forms captivating mathematicians and enthusiasts alike. This article delves into an intriguing problem that intertwines these two shapes: determining the side length of a regular hexagon inscribed within a circle of a given area. We will embark on a step-by-step exploration, unraveling the geometric principles and calculations required to arrive at the solution. This problem not only tests our understanding of geometric relationships but also highlights the practical application of mathematical concepts in real-world scenarios. Before we dive into the specifics of Problem 19, let's lay a solid foundation by revisiting the fundamental properties of circles and hexagons. A circle, defined as the locus of points equidistant from a central point, possesses a rich set of characteristics, including its radius, diameter, circumference, and area. The area of a circle, the two-dimensional space it occupies, is calculated using the well-known formula A = πr², where A represents the area and r denotes the radius. This formula serves as a cornerstone for our problem-solving journey, enabling us to connect the given area of the circle to its radius, a crucial step in determining the hexagon's side length. Now, let's shift our focus to the regular hexagon, a six-sided polygon with all sides and angles equal. A remarkable property of a regular hexagon is its ability to be divided into six congruent equilateral triangles, each with its vertices at the center of the hexagon and at two adjacent vertices of the hexagon. This characteristic proves invaluable when we inscribe the hexagon within a circle, as the radius of the circle becomes the side length of these equilateral triangles. This connection between the circle's radius and the hexagon's side length forms the crux of our solution strategy. By understanding these fundamental properties and relationships, we equip ourselves with the necessary tools to tackle Problem 19 effectively. The interplay between the circle's area and the inscribed hexagon's geometry provides a fascinating example of how geometric principles can be applied to solve practical problems. As we proceed, we will meticulously dissect the problem, applying these concepts to arrive at the precise side length of the hexagon.
Problem Statement: Decoding the Geometric Puzzle
Problem 19 presents a classic geometric challenge: Given a circle with an area of 89.42 square inches, determine the length of the side of a regular hexagon inscribed within that circle. This problem encapsulates the essence of geometric problem-solving, requiring us to connect seemingly disparate pieces of information to arrive at a solution. The keywords here are area of the circle, regular hexagon, and inscribed. Understanding these terms and their implications is crucial for successful problem-solving. The area of the circle, provided as 89.42 square inches, serves as our starting point. It's the known quantity that we will use to unlock other geometric properties of the circle, specifically its radius. The radius, the distance from the center of the circle to any point on its circumference, is a fundamental parameter that links the circle to the inscribed hexagon. A regular hexagon, as we discussed earlier, possesses six equal sides and six equal angles. Its symmetrical nature allows us to divide it into six congruent equilateral triangles, a key characteristic that simplifies our calculations. The term inscribed signifies that the hexagon is drawn inside the circle such that all its vertices lie on the circle's circumference. This geometric constraint establishes a direct relationship between the circle's radius and the hexagon's side length. The problem essentially asks us to decipher this relationship and use it to calculate the unknown side length. To solve this problem, we will employ a combination of geometric principles and algebraic manipulation. We will first use the given area of the circle to calculate its radius. Then, we will leverage the relationship between the circle's radius and the side length of the inscribed regular hexagon. This relationship stems from the fact that the radius of the circle is equal to the side length of the equilateral triangles that make up the hexagon. By carefully applying these concepts, we can systematically unravel the geometric puzzle and arrive at the solution. The problem-solving process will not only yield the answer but also deepen our understanding of the interplay between circles and hexagons. It will reinforce our ability to translate geometric concepts into mathematical equations and use them to solve practical problems. As we move forward, we will meticulously execute each step, ensuring clarity and precision in our calculations.
Solution: A Step-by-Step Geometric Journey
To find the length of the side of the regular hexagon inscribed in the circle, we embark on a step-by-step solution, utilizing geometric principles and algebraic manipulations. First, we recall the formula for the area of a circle: A = πr², where A is the area and r is the radius. We are given that the area of the circle is 89.42 square inches. So, we can set up the equation: 89.42 = πr². To find the radius r, we divide both sides of the equation by π (approximately 3.14159): r² = 89.42 / π ≈ 28.46. Next, we take the square root of both sides to solve for r: r ≈ √28.46 ≈ 5.33 inches. Now that we have determined the radius of the circle, we can leverage the relationship between the radius and the side length of the inscribed regular hexagon. A regular hexagon can be divided into six congruent equilateral triangles, each with its vertices at the center of the hexagon and at two adjacent vertices of the hexagon. The side length of each equilateral triangle is equal to the radius of the circle. Therefore, the side length of the regular hexagon is equal to the radius of the circle, which we found to be approximately 5.33 inches. Thus, the length of the side of the regular hexagon inscribed in the circle is approximately 5.33 inches. This solution demonstrates the power of geometric reasoning and algebraic manipulation in solving problems involving circles and polygons. By understanding the relationships between different geometric properties, we can systematically unravel complex problems and arrive at accurate solutions. The key to this problem lies in recognizing the connection between the circle's radius and the hexagon's side length, a connection that stems from the hexagon's unique geometric structure. The division of the hexagon into equilateral triangles provides a direct link between these two parameters, allowing us to calculate the unknown side length based on the known area of the circle. As we conclude this step-by-step solution, it's important to reflect on the process we undertook. We began with the given information, the area of the circle, and strategically applied geometric principles and algebraic techniques to arrive at the desired result, the side length of the inscribed hexagon. This approach exemplifies the problem-solving process in mathematics, where careful analysis, strategic planning, and precise execution are essential for success.
Answer and Conclusion: Confirming the Solution and Reflecting on the Process
Based on our step-by-step solution, the length of the side of the regular hexagon inscribed in a circle with an area of 89.42 square inches is approximately 5.33 inches. Therefore, the correct answer is (c) 5.33. This result aligns with our understanding of the geometric relationships between circles and hexagons, further solidifying the accuracy of our solution. To ensure the validity of our answer, it's always prudent to perform a quick check. We can visualize the hexagon inscribed within the circle and mentally assess whether a side length of 5.33 inches seems reasonable given the circle's area. This intuitive check provides an additional layer of confidence in our solution. In conclusion, Problem 19 serves as an excellent example of how geometric principles and algebraic techniques can be combined to solve practical problems. The problem-solving process involved several key steps: understanding the problem statement, identifying the relevant geometric concepts, formulating a solution strategy, executing the calculations, and verifying the result. Each step played a crucial role in our journey towards the correct answer. The core concept underlying the solution is the relationship between the circle's radius and the side length of the inscribed regular hexagon. By recognizing that the hexagon can be divided into six congruent equilateral triangles, we established a direct link between these two parameters. This connection allowed us to calculate the unknown side length based on the known area of the circle. The problem also highlights the importance of accurate calculations and attention to detail. A small error in any step could lead to an incorrect result. Therefore, it's essential to perform each calculation carefully and double-check our work. Furthermore, the problem emphasizes the value of visualization in geometry. By visualizing the hexagon inscribed within the circle, we can gain a better understanding of the geometric relationships and develop a more intuitive sense of the solution. This visual approach can often help us identify potential errors and ensure the reasonableness of our answer. In summary, Problem 19 not only provides a specific solution but also offers valuable insights into the problem-solving process in mathematics. It reinforces the importance of understanding geometric principles, applying algebraic techniques, performing accurate calculations, and visualizing geometric relationships. These skills are essential for success in mathematics and beyond.
Keywords
Area of a circle, regular hexagon, inscribed hexagon, side length, radius, geometry, equilateral triangles, problem-solving, mathematical calculations.
Problem 19 Rewrite
If a circle has an area of 89.42 square inches, what is the length of one side of a regular hexagon that is inscribed inside the circle? The options are: a) 6.12, b) 4.22, c) 5.33, and d) 5.89.