A Circle Is Inscribed In A Regular Hexagon With A Side Length Of 10 Feet. What Is The Area Of The Shaded Region?Recall That In A $30^{\circ}-60^{\circ}-90^{\circ}$ Triangle, If The Shortest Leg Measures $x$ Units, Then The Longer
Introduction
In geometry, a regular hexagon is a six-sided polygon with all sides and angles equal. When a circle is inscribed in a regular hexagon, it touches the midpoints of each side of the hexagon. In this article, we will explore the problem of finding the area of the shaded region in a regular hexagon with a side length of 10 feet, where a circle is inscribed within it.
Understanding the Problem
To find the area of the shaded region, we need to understand the properties of the regular hexagon and the inscribed circle. A regular hexagon can be divided into six equilateral triangles, each with a side length of 10 feet. The inscribed circle touches the midpoints of each side of the hexagon, creating six congruent 30°-60°-90° triangles.
Properties of a 30°-60°-90° Triangle
In a 30°-60°-90° triangle, the shortest leg measures x units, and the longer leg measures x√3 units. The hypotenuse measures 2x units. In this case, the shortest leg of each 30°-60°-90° triangle is half the side length of the hexagon, which is 5 feet.
Finding the Radius of the Inscribed Circle
To find the radius of the inscribed circle, we need to find the length of the longer leg of each 30°-60°-90° triangle. Since the shortest leg is 5 feet, the longer leg is 5√3 feet. This is also the radius of the inscribed circle.
Finding the Area of the Shaded Region
The area of the shaded region is equal to the area of the regular hexagon minus the area of the inscribed circle. To find the area of the regular hexagon, we need to find the area of one of the equilateral triangles and multiply it by 6.
Finding the Area of an Equilateral Triangle
The area of an equilateral triangle with side length s is given by the formula:
A = (√3/4)s^2
In this case, the side length of the equilateral triangle is 10 feet. Plugging in the value, we get:
A = (√3/4)(10)^2 A = 25√3
Finding the Area of the Regular Hexagon
Since the regular hexagon can be divided into six equilateral triangles, the area of the regular hexagon is six times the area of one equilateral triangle:
A = 6(25√3) A = 150√3
Finding the Area of the Inscribed Circle
The area of a circle with radius r is given by the formula:
A = πr^2
In this case, the radius of the inscribed circle is 5√3 feet. Plugging in the value, we get:
A = π(5√3)^2 A = 25π(3) A = 75π
Finding the Area of the Shaded Region
The area of the shaded region is equal to the area of the regular hexagon minus the area of the inscribed circle:
A = 150√3 - 75π
Conclusion
In this article, we explored the problem of finding the area of the shaded region in a regular hexagon with a side length of 10 feet, where a circle is inscribed within it. We used the properties of the regular hexagon and the inscribed circle to find the area of the shaded region. The final answer is:
A = 150√3 - 75π
Final Answer
Introduction
In our previous article, we explored the problem of finding the area of the shaded region in a regular hexagon with a side length of 10 feet, where a circle is inscribed within it. In this article, we will answer some of the most frequently asked questions related to this problem.
Q: What is the relationship between the side length of the regular hexagon and the radius of the inscribed circle?
A: The radius of the inscribed circle is equal to the length of the longer leg of each 30°-60°-90° triangle, which is half the side length of the hexagon multiplied by √3. In this case, the radius of the inscribed circle is 5√3 feet.
Q: How do you find the area of the shaded region?
A: The area of the shaded region is equal to the area of the regular hexagon minus the area of the inscribed circle. To find the area of the regular hexagon, we need to find the area of one of the equilateral triangles and multiply it by 6. The area of the inscribed circle is found using the formula A = πr^2, where r is the radius of the circle.
Q: What is the formula for finding the area of an equilateral triangle?
A: The area of an equilateral triangle with side length s is given by the formula:
A = (√3/4)s^2
Q: How do you find the area of the regular hexagon?
A: Since the regular hexagon can be divided into six equilateral triangles, the area of the regular hexagon is six times the area of one equilateral triangle:
A = 6(25√3) A = 150√3
Q: What is the relationship between the area of the regular hexagon and the area of the inscribed circle?
A: The area of the shaded region is equal to the area of the regular hexagon minus the area of the inscribed circle:
A = 150√3 - 75π
Q: Can you provide a step-by-step solution to finding the area of the shaded region?
A: Here is a step-by-step solution to finding the area of the shaded region:
- Find the area of one equilateral triangle using the formula A = (√3/4)s^2, where s is the side length of the triangle.
- Multiply the area of one equilateral triangle by 6 to find the area of the regular hexagon.
- Find the area of the inscribed circle using the formula A = πr^2, where r is the radius of the circle.
- Subtract the area of the inscribed circle from the area of the regular hexagon to find the area of the shaded region.
Q: What is the final answer to the problem?
A: The final answer is:
A = 150√3 - 75π
Conclusion
In this article, we answered some of the most frequently asked questions related to the problem of finding the area of the shaded region in a regular hexagon with a side length of 10 feet, where a circle is inscribed within it. We hope that this article has provided a clear understanding of the problem and its solution.