### Video Transcript

In the figure, a square 𝑂𝐴𝐵𝐶 is
inscribed in a quadrant 𝑂𝑃𝐵𝑄 of a circle. If 𝑂𝐴 equals 20 centimeters, find
the area of the shaded region. Use 𝜋 equals 3.14.

The area of the shaded region is
equal to the area of quadrant 𝑂𝑃𝐵𝑄 minus the area of the square 𝑂𝐴𝐵𝐶. A quadrant is one-fourth of a
circle. If we find the area of a circle by
multiplying 𝜋 times the radius squared, the area of the quadrant is one-fourth of
that. So we would say 𝜋𝑟 squared
divided by four. And of course, the area of a square
equals one of its sides squared.

Our question tells us that 𝑂𝐴,
the side of the square, measures 20 centimeters. We can plug in 20 for the side of
the square. We’re missing one piece of
information we need to find the area of the quadrant. We need to know what the radius is
of this circle. We remember that a radius is a
straight line from the center to the circumference of a circle. Our circle is centered at point
𝑂. It has a radius at 𝑂𝑄 and
𝑂𝑃. We also know a third radius for
this quadrant.

The line segment 𝑂𝐵 is a straight
line from the center of this circle to its outside. This diagonal across the square
turns it into two congruent triangles. Because we have a square, we know
that side length 𝐴𝐵 is also equal to 20 centimeters. If we find the length of 𝑂𝐵,
we’ll find the radius of this quadrant. We could of course use the
Pythagorean theorem and say 20 squared plus 20 squared equals the hypotenuse
squared.

But a simpler way is to recognize
that this is a 45-45-90 degree triangle. And that means its side lengths
will be in the ratio one to one to the square root of two. Our two smaller sides measure 20
centimeters. And that means our larger side will
be 20 centimeters times the square root of two. The radius of this triangle is 20
times the square root of two.

To solve, we plug in 20 times the
square root of two in for our radius. 20 times the square root of two
squared equals 800. 800𝜋 over four minus 20 squared
which equals 400. In the next step, we can divide 800
by four. And we’ll be left with 200𝜋 minus
400. This is really the furthest we can
go without using our approximation 3.14 for 𝜋. We substitute 3.14 for 𝜋.

At this point, I would probably do
something like this. Instead of leaving 200, I would say
100 times two times 3.14. Two times 3.14
equals 6.28. Multiply that by 100. Move the decimal point two places
to the right. And we get 628. Don’t forget our subtract 400
that’s there.

628 minus 400 is the area of our
shaded region. And it equals 228. We’re dealing with area. And all of our units were measured
in centimeters. So the final answer should be
written as 228 centimeters squared.