Question

A regular hexagon is circumscribed about a circle with a radius of 6. Find the area of the shaded region shown. Give the exact answer. (Do not approximate or any square root.)

Answer 1

Answer:

19.58 unit²

Step-by-step explanation:

To find the area of the shaded region, we use the formula:

$A_{shaded}=A_{circle} - A_{hexagon}$

As we know that,

$A_{circle} = \pi r^2$ = $\pi 6^2$

           =36πunit²

   

Next is to split up the hexagon into 6 regular triangles.

$A_{hexagon} = 6. A_{triangle}$

               =6 x $\frac{1}{2} bh$

the base is equal to the radius i.e 6 as the triangles are regular.The height can be represented by taking one of the triangles and drawing a line down the middle.

So, the newly formed triangle is a 30°-60°90° right triangle.

(see figure 1 in attachment)

Here, a= 6/2=> 3

h= a√3 => 3√3

Substituting the required values in the formula of area of hexagon, we get

$A_{hexagon}= 6.\frac{1}{2}.6.3\sqrt{3}$ => 54√3unit²

$A_{shaded}=A_{circle} - A_{hexagon}$

           =36π-54√3

            = 19.58 unit²