In the figure, 𝑃𝑆𝑅, 𝑅𝑇𝑄, and
𝑃𝐴𝑄 a three semicircles of diameters 10 centimeters, three centimeters, and seven
centimeters, respectively. Find the perimeter of the shaded
region. Take 𝜋 is equal to 3.14.
For this question, we’ve been given
a shape and we’ve been asked to find its perimeter. The perimeter of the shape that
we’ve been given is comprised of three semicircular arcs. If we define the perimeter as 𝑥,
we can say that 𝑥 is the sum of these three arcs. The arcs are 𝑃𝑆𝑅, 𝑅𝑇𝑄, and
Now, the question tells us the
diameter of the semicircles which form these arcs. This information can be used to
answer the question in the following way. We first recall that the perimeter
— also known as the circumference 𝑐 of a circle — is given by two 𝜋𝑟, where 𝑟 is
the radius of the circle.
Since the semicircle is a circle
that’s been cut in half, the arc length of a semicircle will be half that of the
circumference of the original circle; that is, 𝑐 divided by two. Dividing both sides of our equation
by two, we see that this is equal to two 𝜋𝑟 over two. Cancelling the two on the top and
bottom half of our fraction, we see that this is equal to 𝜋 times 𝑟.
Now, for this question, we haven’t
been given the radius of the circles, which have formed our semicircular arcs rather
we’ve been given the diameters. We can now recall that the radius
of a circle is half of its diameter. We can now put these two factors
together by replacing the 𝑟 in our original formula by 𝑑 over two.
When we do so, we find that the
length of a semicircular arc, here given by 𝑐 over two, is equal to 𝜋 times the
diameter divided by two. This formula will allow us to move
forward with our question. And we’ll put it to one side to
make room for some working.
Looking back at the equation that
we have for 𝑥, which we defined as the perimeter of the shaded region, we can now
replace the length 𝑃𝑆𝑅 with 𝜋 times the diameter of the semicircle which formed
𝑃𝑆𝑅 divided by two. We can also do the same thing for
the semicircles 𝑅𝑇𝑄 and 𝑃𝐴𝑄.
The question has given us that the
diameter of 𝑃𝑆𝑅 is 10 centimeters. And so we can rewrite our first
term as 𝜋 times 10 over two. The diameter of 𝑅𝑇𝑄 is three
centimeters. And so our second term becomes 𝜋
times three over two. Finally, the diameter of 𝑃𝐴𝑄 is
seven centimeters. And our last term becomes 𝜋 times
seven over two.
We can now use the fact that all
three of our terms have a factor of 𝜋. We can take out this factor to
simplify our next line of working, which then becomes 𝜋 times 10 over two plus
three over two plus seven over two. 10 plus three plus seven is equal
to 20. And so our bracket simplify to 20
over two. 20 divided by two is 10. And so we can replace this in our
We’re now left with the fact that
𝑥, the perimeter of our shape, is equal to 𝜋 times 10. Here, we know that the question has
given us an approximate value of 𝜋 to use, which is 3.14. We now substitute this value into
our equation. 3.14 times 10 is equal to 31.4. And we also remember to add back in
the units of length, which here are centimeters.
This is now our final answer for
the question. And we have found that based on the
semicircular diameters which we’ve been given the perimeter of the shaded region is