Answer:
The area of the shaded portion of the figure is [tex]9.1\ cm^2[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
we know that
The shaded area is equal to the area of the square less the area not shaded.
There are 4 "not shaded" regions.
step 1
Find the area of square ABCD
The area of square is equal to
[tex]A=b^2[/tex]
where
b is the length side of the square
we have
[tex]b=4\ cm[/tex]
substitute
[tex]A=4^2=16\ cm^2[/tex]
step 2
We can find the area of 2 "not shaded" regions by calculating the area of the square less two semi-circles (one circle):
The area of circle is equal to
[tex]A=\pi r^{2}[/tex]
The diameter of the circle is equal to the length side of the square
so
[tex]r=\frac{b}{2}=\frac{4}{2}=2\ cm[/tex] ---> radius is half the diameter
substitute
[tex]A=\pi (2)^{2}[/tex]
[tex]A=4\pi\ cm^2[/tex]
Therefore, the area of 2 "not-shaded" regions is:
[tex]A=(16-4\pi) \ cm^2[/tex]
and the area of 4 "not-shaded" regions is:
[tex]A=2(16-4\pi)=(32-8\pi)\ cm^2[/tex]
step 3
Find the area of the shaded region
Remember that the area of the shaded region is the area of the square less 4 "not shaded" regions:
so
[tex]A=16-(32-8\pi)=(8\pi-16)\ cm^2[/tex]
---> exact value
assume
[tex]\pi =3.14[/tex]
substitute
[tex]A=(8(3.14)-16)=9.1\ cm^2[/tex]
