Find the area of the shaded part of
the quadrant in the diagram in terms of 𝜋.
So we’re looking to find the blue
area in the diagram. We’re told that we have a
quadrant. That’s a quarter of a circle, which
means that this is a sector whose central angle is a right angle. Now, looking carefully at the
diagram, we can see that this blue area is made up of a quadrant where the radius of
25 centimeters from which a quadrant of a smaller circle which has a radius of 17
centimeters has been removed. The area we’re looking for then
will be the difference between the areas of these two quadrants. Now the question asks us to give
our answer in terms of 𝜋, we might be asked to do this for two reasons, either
because an exact answer is required or because we’re expected to answer this problem
without a calculator.
We haven’t been told whether to
work in degrees or radians, so let’s choose to work in degrees. We know that the area of a sector
whose central angle is 𝜃 and whose radius is 𝑟 is 𝜃 over 360 multiplied by 𝜋𝑟
squared. In this problem, the central angle
is 90 degrees, so we have 90 over 360 multiplied by 𝜋𝑟 squared, which simplifies
to one-quarter multiplied by 𝜋𝑟 squared or 𝜋𝑟 squared over four. This, of course, makes sense
because we know that a quadrant is quarter of a circle, so the area of a quadrant
will be quarter of the full circle’s area. For the area of the larger quadrant
first of all then, we have one-quarter multiplied by 𝜋 multiplied by 25
squared. And for the smaller quadrant, it’s
one-quarter multiplied by 𝜋 multiplied by 17 squared.
If we want, we can factor by 𝜋
over four giving 𝜋 over four multiplied by 25 squared minus 17 squared. 25 squared is 625 and 17 squared is
289. 625 minus 289 is 336. So we have 336𝜋 over four. And finally, we can cancel a factor
of four from the numerator and denominator to give 84𝜋. The units for the radius were
centimeters, and so the units for the area will be square centimeters. So we found that the area of the
shaded part of the quadrant in terms of 𝜋 is 84𝜋 square centimeters.