Three congruent circles with a
radius of 43 centimeters are placed touching each other. Find the area of the part between
the circles giving the answer to the nearest square centimeter.
So we want to find the area of the
part between these circles. That’s this part here. We know that the circles are
congruent, that’s identical, each with the radius of 43 centimeters. If we join the centers of the three
circles, we have a triangle. Each side of this triangle consists
of two radii. And so, in fact, this is an
equilateral triangle with a side length of two times 43; that’s 86 centimeters. The area we’re looking to find is
inside this triangle. The other areas inside this
triangle, that’s these three areas here, are three congruent circular sectors. So our approach to finding the blue
area is going to be to find the area of the triangle and then subtract the area of
these three sectors.
The triangle, first of all. This is a nonright triangle, so we
need to use the formula a half 𝑎𝑏 sin 𝑐, where 𝑎 and 𝑏 represent two sides of
the triangle and 𝑐 represents the angle between them. Each of the side lengths of our
triangle are 86 centimeters. And as it’s an equilateral
triangle, all of the angles are 60 degrees. So we have a half multiplied by 86
multiplied by 86 multiplied by sin of 60 degrees. sin of 60 degrees is root three
over two, so the area of the triangle simplifies to 1849 root three, which we’ll
keep in its exact form for now.
Now, for the area of the sectors,
the area of each sector working in degrees is 𝜃 over 360 multiplied by 𝜋𝑟
squared, where 𝜃 is the central angle. But we have three of these
identical sectors, so we’ll multiply by three. That would be equivalent then to
the formula 𝜃 over 120 multiplied by 𝜋𝑟 squared, as three over 360 is one over
120. So substituting the angle of 60
degrees and the radius of the sector which, remember, is 43 centimeters, we have 60
over 120 multiplied by 𝜋 multiplied by 43 squared. 60 over 120 simplifies to a half
and 43 squared is 1849. So we have 1849𝜋 over two for the
area of the sectors.
We can now go ahead and evaluate
this using our calculators. Doing so gives 298.159. We’re required to give an answer to
the nearest square centimeter, so we’ll be rounding down. We found then that the area of the
part between the circles, which is the difference between the area of a nonright
triangle and three circular sectors, is 298 square centimeters.