Given the two squares and a circle,
what is the circle's area?
Well, we know the area of a circle is equal to pie R
squared, where R is the radius.
So lets see if we can analyze the given diagram
and determine the radius of the circle.
And there's actually several ways of doing this.
We take a look at the inner square
and construct the diagonal,
which would be this segment here.
Notice how the diagonal of the square
is also the diameter of the circle.
So if we can determine the diameter,
we know the radius is going to be half the diameter,
which we can then use to find the area of the circle.
Also, notice how when we form the diagonal of the square,
it cuts the square into two right triangles.
We have one on the left and one on the right.
We also know all four sides of the square are congruent,
and therefore, both of these are isosceles right triangles,
which means that two acute angles
of the right triangles measure 45 degrees.
This is 45 degrees and this is 45 degrees,
and the same in the other right triangle.
Forty-five degrees here and forty-five degrees here.
So we can determine the hypotenuse of these right triangles,
which is also the diagonal of the square and the diameter
of the circle using the relationship among the three sides
of the 45-45-90 right triangle,
or we can use the Pythagorean theorem.
And let's show both.
For any 45-45-90 right triangle,
a triangle is going to be proportional
to the right triangle where the two legs
have the length of one and the hypotenuse
has a length of square root two.
Notice how our 45-45-90 right triangles
have legs that measure six units.
So if we multiply each side of this triangle by six,
we can determine the length of the hypotenuse.
So if we multiply each side by six,
notice how the hypotenuse is six square root two units,
which is the length of the hypotenuse here.
So this measures six square root two units.
But let's also show how we can do this using
the Pythagorean theorem.
The Pythagorean theorem states if 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.
Notice in our right triangles,
A and B are both six, which would give us the equation
six squared plus six squared equals C squared.
Six squared is equal to 36.
Thirty-six plus 36 is equal to 72.
So that's 72 equals C squared.
Now taking the square root of both sides of the equation,
the squared of C squared is C.
The squared of 72 is equal to the squared of 36 times two.
And since the squared of 36 is six, we get the same value.
The hypotenuse C is equal to six square root two.
So again, this length here, which we now know is six
square root two, gives us the diameter of the circle.
So we know the diameter is equal to six square root two,
and the radius, which is half the diameter,
which would be from the center of the circle
to the edge of the circle is half the diameter.
So we have the radius R is equal
to one half times six square root two,
which is equal to three square root of two.
Now that we have the radius,
we can determine the area of the circle.
The area is equal to pie times R squared,
or pie times three square root two squared.
Now let's show some work on three square root two squared.
That would be three square root two
times three square root two.
Three times three is nine.
Square root two times square root two is equal to
the square root of four, which simplifies perfectly to two.
So three square root two squared is equal to 18,
and therefore, the area of the circle
is equal to pie times 18, or 18 pie.
So we know the area is 18 pie square units.
So we now have the area of the circle,
but I do want to show how to find
the radius using one additional method.
Let's go ahead and clear off all the information we have
here except for the special relationship among
the three sides of any 45-45-90 right triangle.
Again, any 45-45-90 right triangle will be proportional
to a right triangle where the legs measure one and one
and the hypotenuse measures square root two.
Now this time, let's focus on this
right triangle here formed by the intersection
of the inner square and the outer square.
The points where the inner square intersects
the outer square is the mid point of each side
of the larger square, and therefore,
again, focusing on this right triangle,
the two legs here are congruent,
and therefore, we have another 45-45-90 right triangle.
And therefore, this triangle, once again,
is proportional to this triangle here on the right.
And also, notice how the length of this leg here
would be the same as the radius of the circle
because it would start here on the edge of the circle
and go to the center of the circle.
So by determining this leg of this right triangle,
we can determine the radius of the circle,
which we already know is three square root two.
But here, notice how we only have one side
of the right triangle, so we'll have to determine
the length of the leg by setting up a proportion.
Setting up a proportion using corresponding sides
to this triangle and this triangle,
the ratio of six to square root two,
or the ratio of the two hypotenuse
must be equal to the ratio of the legs.
So if we label this leg R,
the second ratio would be R to one.
Again, the ratio of the hypotenuse must be equal to the
ratio of the legs because these two triangles are similar.
Now we can cross multiply and solve for R.
So square root two times R must equal six times one.
So square root two R must equal six.
Dividing both sides by square root two,
we would have R equals six divided by square root two,
and now if we rationalize the denominator by multiplying
by the squared of two over the squared of two,
we would have R equals six square root two
divided by square root two times square root two,
which equals two, giving us the same
radius of three square root two.
So I just wanted to show this as well because
there are several ways that we can determine
the radius of the circle and then find the area,
which we already know is equal to 18 pie square units.
I hope you found this helpful.