A circle is inscribed in a regular hexagon with side length 10 feet. What is the area of the shaded region?
Recall that in a [tex]$30^{\circ}-60^{\circ}-90^{\circ}$[/tex] triangle, if the shortest leg measures [tex]$x$[/tex] units, then the longer leg measures [tex]$x \sqrt{3}$[/tex] units and the hypotenuse measures [tex]$2 x$[/tex] units.
A. [tex]$(150 \sqrt{3}-75 \pi) ft ^2$[/tex]
B. [tex]$(300-75 \pi) ft ^2$[/tex]
C. [tex]$(150 \sqrt{3}-25 \pi) ft ^2$[/tex]
D. [tex]$(300-25 \pi) ft ^2$[/tex]
Answer (2)
Calculate the area of the regular hexagon by dividing it into six equilateral triangles and using the formula for the area of an equilateral triangle: 150 3 square feet.
Determine the radius of the inscribed circle, which is equal to the apothem of the hexagon: 5 3 feet.
Calculate the area of the inscribed circle using the formula A = π r 2 : 75 π square feet.
Subtract the area of the circle from the area of the hexagon to find the area of the shaded region: 150 3 − 75 π ft 2 .
Explanation
Problem Analysis We are given a regular hexagon with side length 10 feet, and a circle is inscribed within it. Our goal is to find the area of the shaded region, which is the area inside the hexagon but outside the circle.
Calculate Hexagon Area First, we need to calculate the area of the regular hexagon. A regular hexagon can be divided into six equilateral triangles. The area of an equilateral triangle with side length s is given by the formula 4 s 2 3 . Since the side length of the hexagon is 10 feet, the area of one equilateral triangle is 4 1 0 2 3 = 4 100 3 = 25 3 Since there are six such triangles in the hexagon, the total area of the hexagon is 6 × 25 3 = 150 3 square feet
Determine Circle Radius Next, we need to find the radius of the inscribed circle. The radius of the inscribed circle in a regular hexagon is equal to the apothem of the hexagon. The apothem is the distance from the center of the hexagon to the midpoint of a side. In the 3 0 ∘ − 6 0 ∘ − 9 0 ∘ triangle formed by the center, the midpoint of a side, and a vertex, the apothem is the longer leg. The shorter leg is half the side length, which is 5 feet. Using the given property of 3 0 ∘ − 6 0 ∘ − 9 0 ∘ triangles, the longer leg (apothem or radius of the inscribed circle) is 5 3 feet.
So, the radius of the inscribed circle is r = 5 3 feet.
Calculate Circle Area Now we can calculate the area of the inscribed circle using the formula A = π r 2 . Substituting r = 5 3 , we get A = π ( 5 3 ) 2 = π ( 25 × 3 ) = 75 π square feet
Calculate Shaded Area Finally, we subtract the area of the circle from the area of the hexagon to find the area of the shaded region: A re a s ha d e d = A re a h e x a g o n − A re a c i rc l e = 150 3 − 75 π square feet
Final Answer Therefore, the area of the shaded region is 150 3 − 75 π square feet.
Examples
Hexagons and circles frequently appear in architecture and design. For example, consider a hexagonal gazebo with a circular seating area inside. Calculating the shaded area (the area of the gazebo floor not covered by the seating) is a practical application of this geometric problem. This helps in determining the amount of flooring needed or the open space available for other purposes, blending mathematical problem-solving with real-world design considerations.
The area of the shaded region between a regular hexagon and an inscribed circle is given by the formula 150 3 − 75 π square feet. This calculation involves determining the area of the hexagon and the circle and subtracting the two areas. The correct answer from the choices provided is option A.
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