Answer:
19.58 unit²
Step-by-step explanation:
To find the area of the shaded region, we use the formula:
[tex]A_{shaded}=A_{circle} - A_{hexagon}\\\\[/tex]
As we know that,
[tex]A_{circle} = \pi r^2[/tex] = [tex]\pi 6^2[/tex]
=36πunit²
Next is to split up the hexagon into 6 regular triangles.
[tex]A_{hexagon} = 6. A_{triangle}[/tex]
=6 x [tex]\frac{1}{2} bh[/tex]
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
[tex]A_{hexagon}= 6.\frac{1}{2}.6.3\sqrt{3}[/tex] => 54√3unit²
[tex]A_{shaded}=A_{circle} - A_{hexagon}\\\\[/tex]
=36π-54√3
= 19.58 unit²
