### Video Transcript

Three semicircles each of diameter
three centimeters, a circle of diameter 4.5 centimeters, and a semicircle of radius
4.5 centimeters are drawn in the given figure. Find the area of the shaded
region.

Looking at our figure, we see the
three semicircles of diameter three centimeters. And we can label the area of one of
those semicircles 𝐴 sub three. We also can see the circle with a
diameter of 4.5 centimeters highlighted in green. We’ll name this area 𝐴 sub
two. And finally, there’s the larger
semicircle of radius 4.5 centimeters. We’ll call this area 𝐴 sub
one. And given all this, we want to
solve for the area of the shaded region in the figure. We’ll refer to this area as 𝐴 sub
sh.

And by looking at the figure, we
can see that that area is equal to 𝐴 sub one, the area of the larger semicircle,
minus 𝐴 sub two, the green circle, minus two times 𝐴 sub three, the area of the
smaller semicircle, plus 𝐴 sub three. We add this final 𝐴 sub three term
onto this expression because of the extension of the shaded region below the line of
the larger semicircle. Looking at this expression, we see
we can combine the two 𝐴 sub three terms. This simplifies our expression to
𝐴 sub one minus 𝐴 sub two minus 𝐴 sub three. That’s the area of the shaded
region.

We’ll now use the information given
about each to these areas to solve for 𝐴 sub sh. As we do, we’ll use the fact that
the area of a circle is equal to 𝜋 times its radius squared. And that two times the radius of a
circle is equal to its diameter. We can now write out the area 𝐴
one in terms of this relationship. That area is equal to 𝜋 times the
radius, 4.5 centimeters, squared all divided by two, because 𝐴 one is a semicircle
rather than a full circle.

Next, calculating the area 𝐴 two,
this is equal to 𝜋 times the diameter of the circle divided by two. That gives us the radius all
squared. And finally, the area 𝐴 three is
equal to 𝜋 times the diameter of three centimeters divided by two all squared. And all that divided by two because
𝐴 three is a semicircle rather than a full circle. We can now begin to simplify this
expression for 𝐴 sub sh. And we’ll start doing it with the
factors of two that appear in the denominators of each term.

In the expression for 𝐴 one, that
factor of two is already separated out. In the expression for 𝐴 two, one
over two quantity squared can be rewritten as one over four. And in the expression for 𝐴 three,
we have one over two squared, which will give us one over four. And that itself divided by two,
which will give us overall one over eight.

As we consider once more this
entire expression, we see that a factor of 𝜋 over two appears in each of the three
terms. When we factor 𝜋 over two out from
each term, we’re left with the resulting expression in our brackets. We see that this expression
involves 4.5 centimeters quantity squared minus 4.5 centimeters quantity squared
over two. These two terms combine to equal
4.5 centimeters quantity squared over two, leaving us with just two terms in
brackets. We see that now we can factor out
one over two from each of the terms. We can now start to calculate each
of these remaining terms.

We’ll start with three centimeters
quantity squared over two. This is equal to nine centimeters
squared over two or 4.5 square centimeters. When we calculate the area of the
first term, 4.5 centimeters quantity squared, multiplying 4.5 times itself, we find
a result of 20.25. 20.25 square centimeters minus 4.5
square centimeters is equal to 15.75 square centimeters. If we want to calculate an exact
answer for this solution, we can notice that 15.75 can be rewritten as a
fraction. 63 divided by four is equal to
15.75. So when we substitute this exact
fraction in for our decimal and multiply through by 𝜋 over four, we find that the
area of the shaded region is exactly 63𝜋 over 16 square centimeters.

If we want to calculate a decimal
approximation of this area though, we can assume a specific truncated value for
𝜋. We can assume that 𝜋 is exactly
equal to 3.14. To calculate then this approximate
area of the shaded region, we’ll multiply 63 by this value for 𝜋. The result of multiplying these
values together is 197.82. So now, as a final step, we’ll
divide this result by 16.

Setting up this long division, we
find that 16 goes into 19 one time. 16 divides into 37 two times. It divides into 58 three times. It divides into 102 six times. And it divides into 60 three
times. Since we approximated 𝜋 to two
decimal places, we’ll only keep two decimal places in this result of our long
division. This tells us that 𝐴 sub sh is
approximately 12.36 centimeters squared. That’s the area of the shaded
region in the figure.