Question

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 ( 30^{circ}-60^{circ}-90^{circ} ) triangle, if the shortest leg measures ( x ) units, then the longer leg measures ( x sqrt{3} ) units and the hypotenuse measures ( 2x ) units. ( (150 sqrt{3}-75 pi) mathrm{ft}^{2} ) ( (300-75 pi) mathrm{ft}^{2} ) ( (150 sqrt{3}-25 pi) mathrm{ft}^{2} ) ( (300-25 pi) mathrm{ft}^{2} )

Explanation:

Step1: Calculate the area of the regular hexagon

A regular hexagon can be divided into 6 equilateral - triangles. The side - length of the hexagon \(a = 10\) ft.
The formula for the area of a regular hexagon \(A_{hexagon}=6\times\frac{1}{2}\times a\times h\), where \(h\) is the apothem.
In a \(30 - 60-90\) triangle (formed by splitting one of the six equilateral triangles of the hexagon), if the side of the hexagon (hypotenuse of the \(30 - 60 - 90\) triangle) is \(a = 10\) ft, the shorter leg (half of the side of the hexagon) \(x = 5\) ft, and the apothem (longer leg) \(h = 5\sqrt{3}\) ft.
The area of the hexagon \(A_{hexagon}=6\times\frac{1}{2}\times10\times5\sqrt{3}=150\sqrt{3}\) \(ft^{2}\).

Step2: Calculate the area of the inscribed circle

The radius of the inscribed circle \(r\) is equal to the apothem of the hexagon. So \(r = 5\sqrt{3}\) ft.
The formula for the area of a circle is \(A_{circle}=\pi r^{2}\). Substituting \(r = 5\sqrt{3}\) into the formula, we get \(A_{circle}=\pi\times(5\sqrt{3})^{2}=75\pi\) \(ft^{2}\).

Step3: Calculate the area of the shaded region

The area of the shaded region \(A = A_{hexagon}-A_{circle}\).
\(A=(150\sqrt{3}-75\pi)\) \(ft^{2}\).

Answer:

\((150\sqrt{3}-75\pi)\text{ ft}^2\) (the first option)