Question

Recall that in a 30 – 60 – 90 triangle, if the shortest leg measures x units, then the longer leg measures xStartRoot 3 EndRoot units and the hypotenuse measures 2x units. (150StartRoot 3 EndRoot – 75π) ft2 (300 – 75π) ft2 (150StartRoot 3 EndRoot – 25π) ft2 (300 – 25π) ft2.

Answer 1

The area of the shaded region is  $\rm (150\sqrt{3} \ - 75\pi ) \ feet^2$ option first is correct.

It is given that a circle is inscribed in a regular hexagon with sides of 10 feet.

It is required to find the shaded area (missing data is attached shown in the picture).

What is a circle?

It is defined as the combination of points that and every point has an equal distance from a fixed point ( called the center of a circle).

We have a hexagon with a side length of 10 feet.

We know the area of the hexagon is given by:

$\rm A = \frac{3\sqrt{3} }{2} a^2$  where a is the side length.

$\rm A = \frac{3\sqrt{3} }{2} 10^2$ ⇒ $150\sqrt{3}$ $\rm feet^2$

We have the shortest length = x feet and from the figure:

2x = 10

x = 5 feet

The radius of the circle r = longer leg

$\rm r = x\sqrt{3} \Rightarrow 5\sqrt{3}$ feet

The area of the circle a = $\pi r^2$  ⇒ $\pi (5\sqrt{3} )^2 \Rightarrow 75\pi \ \rm feet^2$  

The area of the shaded region = A - a

$\rm =(150\sqrt{3} \ - 75\pi ) \ feet^2$

Thus, the area of the shaded region is  $\rm (150\sqrt{3} \ - 75\pi ) \ feet^2$

Learn more about circle here:

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