What Is The Area Of The Shaded Region When A Circle Is Inscribed In A Regular Hexagon With A Side Length Of 10 Feet?
In the realm of geometry, shapes and their relationships often present intriguing challenges. One such challenge involves calculating the area of a shaded region formed when a circle is inscribed within a regular hexagon. This problem combines the properties of regular hexagons, circles, and the application of trigonometric principles. Let's delve into the step-by-step solution to this geometric puzzle.
Understanding the Problem: A Circle Inside a Hexagon
The problem states that we have a regular hexagon with a side length of 10 feet. Inside this hexagon, a circle is perfectly inscribed, meaning it touches each side of the hexagon at exactly one point. Our goal is to determine the area of the shaded region, which is the area within the hexagon that is not occupied by the circle. This means we need to calculate the area of the hexagon, calculate the area of the circle, and then subtract the circle's area from the hexagon's area. Let's break down the process.
1. Deconstructing the Hexagon: Finding the Area
A regular hexagon can be divided into six congruent equilateral triangles. This is a crucial observation because we know the side length of the hexagon, which is also the side length of each equilateral triangle. The side length is given as 10 feet. To find the area of the hexagon, we first need to find the area of one equilateral triangle and then multiply it by six.
The formula for the area of an equilateral triangle with side s is given by (√3 / 4) * s². In our case, s = 10 feet. Substituting this value into the formula, we get:
Area of one equilateral triangle = (√3 / 4) * (10²) = (√3 / 4) * 100 = 25√3 square feet.
Since the hexagon is made up of six such triangles, the total area of the hexagon is:
Area of hexagon = 6 * 25√3 = 150√3 square feet. This is a key value we'll use later.
2. The Inscribed Circle: Determining the Radius and Area
Now, let's focus on the circle inscribed within the hexagon. The radius of this circle is the perpendicular distance from the center of the hexagon to any of its sides. This distance is also the height (or altitude) of one of the equilateral triangles we discussed earlier. To find this height, we can use the properties of a 30-60-90 triangle.
When we draw an altitude in an equilateral triangle, it bisects the base and creates two 30-60-90 right triangles. In our case, the hypotenuse of this right triangle is 10 feet (the side of the equilateral triangle), the shorter leg is half the base (5 feet), and the longer leg is the altitude we're trying to find. In a 30-60-90 triangle, the ratio of the sides opposite the 30°, 60°, and 90° angles is x : x√3 : 2x. Here, the shorter leg (opposite the 30° angle) is x = 5 feet. The longer leg (opposite the 60° angle), which is the radius of the inscribed circle, is therefore x√3 = 5√3 feet.
Now that we have the radius of the circle, we can calculate its area using the formula for the area of a circle, which is πr², where r is the radius. Substituting r = 5√3 feet, we get:
Area of circle = π * (5√3)² = π * (25 * 3) = 75π square feet. This is the area of the circle inscribed within the hexagon.
3. Finding the Shaded Area: Subtraction is Key
Finally, to find the area of the shaded region, we subtract the area of the circle from the area of the hexagon:
Area of shaded region = Area of hexagon - Area of circle = 150√3 - 75π square feet.
We can approximate this value by using the values √3 ≈ 1.732 and π ≈ 3.14159:
Area of shaded region ≈ 150 * 1.732 - 75 * 3.14159 ≈ 259.8 - 235.61925 ≈ 24.18 square feet (rounded to two decimal places).
Therefore, the area of the shaded region is approximately 24.18 square feet. This is the area inside the hexagon but outside the inscribed circle.
Breaking Down the Geometry: Hexagons, Circles, and 30-60-90 Triangles
To solve the problem of the shaded area within a regular hexagon housing an inscribed circle, a multifaceted approach intertwining geometric principles is essential. This geometric challenge requires a deep dive into understanding the properties of hexagons, circles, and the special relationships within 30-60-90 triangles. Mastering these concepts is crucial for accurately calculating the desired shaded area. Let's dissect the problem step-by-step, ensuring clarity and precision in our solution.
Understanding Regular Hexagons: Symmetry and Subdivision
A regular hexagon, a six-sided polygon with equal sides and equal angles, exhibits a high degree of symmetry. This symmetry is a cornerstone in solving geometric problems related to hexagons. One of the most insightful ways to leverage this symmetry is by visualizing the hexagon as a composite of six congruent equilateral triangles. Each triangle shares a common vertex at the center of the hexagon, and their bases form the sides of the hexagon. This decomposition is pivotal because it transforms the complex shape of a hexagon into simpler, more manageable equilateral triangles.
The area of an equilateral triangle with side length s is given by the formula (√3 / 4) * s². Since the hexagon is composed of six such triangles, the total area of the hexagon can be calculated by multiplying the area of one equilateral triangle by six. This approach streamlines the calculation, allowing us to focus on determining the side length of the equilateral triangles, which is equivalent to the side length of the hexagon.
The Inscribed Circle: Tangency and Radius
An inscribed circle within a regular hexagon is a circle that perfectly fits inside the hexagon, touching each of its sides at exactly one point. These points of contact are called points of tangency. The center of the inscribed circle coincides with the center of the hexagon, and the radius of the circle is the perpendicular distance from the center to any side of the hexagon. This radius is a critical parameter, as it directly influences the area of the circle, which we need to subtract from the hexagon's area to find the shaded region.
Finding the radius of the inscribed circle involves leveraging the decomposition of the hexagon into equilateral triangles. The radius is equal to the height (or altitude) of one of these equilateral triangles. To determine this height, we can utilize the properties of special right triangles, specifically the 30-60-90 triangle, which naturally arises when we draw the altitude in an equilateral triangle.
30-60-90 Triangles: Unlocking the Radius
When an altitude is drawn in an equilateral triangle, it bisects the base and forms two congruent 30-60-90 right triangles. These triangles have a unique side ratio that allows us to easily calculate unknown side lengths if one side is known. The side ratio in a 30-60-90 triangle is x : x√3 : 2x, where x is the length of the side opposite the 30° angle, x√3 is the length of the side opposite the 60° angle, and 2x is the length of the hypotenuse (opposite the 90° angle).
In the context of our hexagon problem, the hypotenuse of the 30-60-90 triangle is the side length of the equilateral triangle (and the hexagon), the shorter leg (x) is half the side length of the hexagon, and the longer leg (x√3) is the altitude of the equilateral triangle, which is also the radius of the inscribed circle. By identifying these relationships and applying the 30-60-90 triangle side ratio, we can accurately determine the radius of the inscribed circle.
Calculating Areas: Hexagon, Circle, and the Shaded Region
With the foundational geometric concepts in place, the calculation of areas becomes straightforward. The area of the hexagon is six times the area of one equilateral triangle, the area of the circle is πr², where r is the radius, and the area of the shaded region is the difference between the hexagon's area and the circle's area. The accuracy of the final result hinges on the precise determination of the hexagon's side length, the radius of the inscribed circle, and the correct application of the area formulas.
The problem emphasizes the elegance and interconnectedness of geometry. By dissecting the hexagon, understanding the properties of the inscribed circle, and leveraging the special relationships within 30-60-90 triangles, we can methodically solve for the shaded area. This approach not only yields the solution but also reinforces a deeper appreciation for the beauty and logic inherent in geometric problem-solving.
Step-by-Step Solution: A Practical Guide
In summary, we methodically determine the shaded area formed by a circle inscribed within a regular hexagon by following a step-by-step approach. This problem exemplifies the practical application of geometric principles and their interconnectedness. By dissecting the hexagon into equilateral triangles, calculating the radius of the inscribed circle using 30-60-90 triangle properties, and applying area formulas, we can accurately solve for the desired shaded area. This step-by-step guide provides a clear and concise pathway to understanding the solution process.
Step 1: Hexagon Area Calculation
The initial step involves calculating the area of the regular hexagon. Recognizing that a regular hexagon comprises six congruent equilateral triangles is crucial. Given the side length of the hexagon (10 feet), we can calculate the area of each equilateral triangle using the formula: Area = (√3 / 4) * s², where s represents the side length.
Substituting the given side length (10 feet) into the formula yields: Area = (√3 / 4) * 10² = 25√3 square feet. Since there are six equilateral triangles within the hexagon, we multiply this result by six to obtain the total area of the hexagon: Hexagon Area = 6 * 25√3 = 150√3 square feet. This step is foundational, as it provides one of the two key area values needed for the final calculation.
Step 2: Circle Radius Determination
The next step involves finding the radius of the inscribed circle. This radius is equivalent to the height (or altitude) of one of the equilateral triangles that compose the hexagon. To determine this height, we utilize the properties of 30-60-90 triangles. When an altitude is drawn in an equilateral triangle, it bisects the base and creates two 30-60-90 right triangles.
In our scenario, the hypotenuse of this 30-60-90 triangle is the side length of the equilateral triangle (10 feet), the shorter leg is half the side length (5 feet), and the longer leg, which we seek, is the altitude and also the radius of the circle. Applying the 30-60-90 triangle side ratio (x : x√3 : 2x), we deduce that the longer leg (radius) is 5√3 feet. This critical value allows us to calculate the circle's area.
Step 3: Circle Area Calculation
With the radius of the inscribed circle determined, we proceed to calculate its area. The formula for the area of a circle is Area = πr², where r is the radius. Substituting the calculated radius (5√3 feet) into the formula yields: Circle Area = π * (5√3)² = 75π square feet. This step provides the second key area value needed for the final calculation.
Step 4: Shaded Area Calculation
Finally, we calculate the shaded area by subtracting the area of the circle from the area of the hexagon. This represents the area within the hexagon that is not occupied by the circle. Using the values calculated in the previous steps, we get: Shaded Area = Hexagon Area - Circle Area = 150√3 - 75π square feet.
To obtain an approximate numerical value, we substitute the values √3 ≈ 1.732 and π ≈ 3.14159: Shaded Area ≈ 150 * 1.732 - 75 * 3.14159 ≈ 24.18 square feet (rounded to two decimal places). Therefore, the area of the shaded region is approximately 24.18 square feet. This final step completes the solution, providing the desired area of the region inside the hexagon but outside the inscribed circle.
By systematically following these steps, we can accurately solve the problem of the shaded area formed by a circle inscribed within a regular hexagon. This approach highlights the power of breaking down complex geometric problems into simpler, manageable steps.
Conclusion: Geometry in Action
In conclusion, the problem of finding the shaded area of a circle inscribed in a regular hexagon elegantly demonstrates the practical application of geometric principles. By understanding the properties of hexagons, circles, and 30-60-90 triangles, we can systematically solve this type of geometric challenge. The solution involves deconstructing the hexagon into equilateral triangles, determining the radius of the inscribed circle, calculating the areas of both the hexagon and the circle, and finally, subtracting the circle's area from the hexagon's area to find the shaded region. This process showcases the interconnectedness of geometric concepts and the power of step-by-step problem-solving.
This type of problem not only enhances our understanding of geometry but also reinforces the importance of visual reasoning and analytical skills. The ability to break down complex shapes into simpler components and apply relevant formulas is a valuable skill in various fields, including engineering, architecture, and design. The problem serves as a reminder that geometry is not just an abstract mathematical concept but a practical tool that can be used to solve real-world problems. The final result, approximately 24.18 square feet, represents the area within the hexagon that is not covered by the inscribed circle, highlighting the precision and elegance of geometric calculations.