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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 [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]\(2x\)[/tex] units.

A. [tex]\((150 \sqrt{3} - 75 \pi) \, \text{ft}^2\)[/tex]
B. [tex]\((300 - 75 \pi) \, \text{ft}^2\)[/tex]
C. [tex]\((150 \sqrt{3} - 25 \pi) \, \text{ft}^2\)[/tex]
D. [tex]\((300 - 25 \pi) \, \text{ft}^2\)[/tex]



Answer :

GinnyAnswer
To find the area of the shaded region, we need to calculate the area of the hexagon and subtract the area of the inscribed circle from it.

### Step 1: Calculate the Area of the Hexagon

A regular hexagon can be divided into 6 equilateral triangles. The area [tex]\( A \)[/tex] of an equilateral triangle with side length [tex]\( a \)[/tex] is given by:

[tex]\[ A = \frac{\sqrt{3}}{4} a^2 \][/tex]

Since the hexagon is made up of 6 such triangles, the area of the hexagon [tex]\( A_{\text{hexagon}} \)[/tex] is:

[tex]\[ A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4} a^2 \][/tex]

Given [tex]\( a = 10 \)[/tex] feet, we have:

[tex]\[ A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4} \times 10^2 = 6 \times \frac{\sqrt{3}}{4} \times 100 = 150\sqrt{3} \text{ square feet} \][/tex]

### Step 2: Calculate the Area of the Inscribed Circle

The radius [tex]\( r \)[/tex] of the inscribed circle in a regular hexagon is equal to the height of one of the equilateral triangles, which is given by:

[tex]\[ r = \frac{\sqrt{3}}{2} a \][/tex]

Given [tex]\( a = 10 \)[/tex] feet, we have:

[tex]\[ r = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \text{ feet} \][/tex]

The area [tex]\( A_{\text{circle}} \)[/tex] of a circle is given by:

[tex]\[ A_{\text{circle}} = \pi r^2 \][/tex]

Substituting [tex]\( r = 5\sqrt{3} \)[/tex]:

[tex]\[ A_{\text{circle}} = \pi (5\sqrt{3})^2 = \pi \times 75 = 75\pi \text{ square feet} \][/tex]

### Step 3: Calculate the Shaded Area

The shaded area is the area of the hexagon minus the area of the inscribed circle:

[tex]\[ A_{\text{shaded}} = A_{\text{hexagon}} - A_{\text{circle}} = 150\sqrt{3} - 75\pi \][/tex]

Therefore, the area of the shaded region is:

[tex]\[ \boxed{150 \sqrt{3} - 75 \pi \text{ square feet}} \][/tex]

This matches the first provided option in the problem statement.

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