We want to find the area
of the shaded region or the area shaded in blue.
We first want to devise a strategy
in order to find this area.
I think the best strategy is going to be
to find the area of these four half circles
or four semi-circles, meaning the area here,
here,
and finally here.
Once we find this combined area,
notice how if we subtract out the area
bounded by the circle and the square,
meaning this area,
this area,
and finally this area,
we'd be left with the area of the shaded region.
We'll be using the area formula for a circle
and the area formula for a square.
So the area of a circle is equal to
Pi r squared, where r is the radius.
And the area of a square is equal to
s squared, where s is the length of one side.
Let's begin by determining the area
of the four semi-circles.
When we combine the area of four semi-circles,
that would be equal to the area of two full circles.
So the area of the four semi-circles
is going to be equal to two times Pi r squared.
Now looking at one of the semi-circles,
notice how the diameter is equal to four units
and because the radius is equal to half the diameter,
or this length here, the radius is equal to two.
Which means the area of the four semi-circles
is equal to two times Pi times two squared.
Two squared is equal to four.
And therefore the area of the four semi-circles
is equal to eight Pi.
Let's go ahead and label this four semi-circles.
So we'll be using this area again later.
Now let's work on determining the area
bounded by the circle and the square.
And this is going to take a little bit more work.
In order to find the area of the circle,
we'll have to determine the radius.
Before we find the radius though,
we'll need to determine the diameter
and notice how this would be the diameter of the circle.
But it's also the diagonal of the square.
So we'll first find this length
and then find half of it to determine
the radius of the circle.
Notice how the diagonal of the square
forms two right triangles, one here and one here.
And we can use this to determine the length
of the diagonal which is also the hypotenuse
of one of the right triangles.
There's a couple ways of doing this.
Looking at one of the right triangles,
because these two acute angles are congruent,
they both measure 45 degrees.
Whenever we have a 45, 45, 90 right triangle,
the lengths of the three sides
will always be multiples of one, one and square root two.
The three sides of a 45, 45, 90 right triangle
always hold this relationship.
Notice how for our right triangle,
the two legs have a length of four,
so we multiply all three sides by four,
we'd have four times one here,
four times one here and four times square root two,
which means the length of this diagonal
is equal to four square root two units.
Another way to find this length
would be to use the Pythagorean Theorem.
Let's also show that.
Pythagorean Theorem states that for any right triangle,
a squared plus b squared equals c squared,
where a and b are the lengths of the two legs
and c is the length of the hypotenuse.
But you notice in this case, both a and b are four,
so we'd have four squared plus four squared
equals c squared.
Four squared is equal to 16.
16 plus 16 is equal to 32.
So we have 32 equals c squared.
Taking the square root of both sides.
The square root of c squared is c.
And because 32 is equal to 16 times two
and the square root of 16 is equal to four,
we have c equals four, square root two.
So again, this tell us that the diameter
of the circle is equal to four square root two,
the radius is equal to half the diameter
or 1/2 times four square root two,
which is equal to two, square root two.
Now we have all the information we need in order to find
the area bounded by the circle and the square.
The area bounded by the circle and the square
is going to be equal to the area of the circle,
which is Pi r squared minus the area
of the square which is s squared.
So substituting in the values,
we'd have area equals Pi times r squared,
where the radius is equal to two, square root two
minus s squared where s is the length of
one side of the square, which is four,
so minus four squared.
Let's show some work on two, square root two squared.
Two square root two squared is equal to
two square root two, times two square root two.
Two times two is equal to four.
Square root two times square root two
is equal to the square root of four which is two.
So two square root two, squared is equal to eight.
So the area of the boundary region
is equal to Pi times eight or eight Pi
minus four squared or minus 16.
Again this is the area of the region
bounded by the circle and the square.
So now we'll use this area
and the area of the four semi-circles
to determine the area of the shaded region.
The area we're looking for is equal to
the area of the four semi-circles, which is eight Pi,
minus the area bounded by the circle and the square
which we now know is eight Pi minus 16.
So to help simplify here, if we want we can think of
distributing a negative one because of the subtraction.
So we'd have eight Pi and then minus eight Pi
and then plus 16 and notice how
eight Pi minus eight Pi is zero.
So the area of the shaded region
or the region in blue is exactly 16 square units.
I hope you found this helpful.