The given figure shows two half circles
and two-quarters of another circle. Find the perimeter of the shaded region,
taking 3.14 as an approximate value for 𝜋.
In this question then, we have two
different sizes of circle. Tracing our finger or pen all the way
around the edge of the figure, we see that it is composed firstly of the arc of a
semicircle. Then the arc of a different-sized quarter
circle. Then the arc of a semicircle which is
congruent to the first. And finally the arc of a quarter circle
which is congruent to the other quarter circle.
In total then, what we have is the full
circumference of the smaller circle — that’s the orange one — and half the circumference of
the larger circle — that’s the pink one. We know that the circumference of a
circle can be found using the formula 𝜋𝑑, where 𝑑 is the diameter of the circle. So we just need to determine the diameter
of each of these circles.
From the figure, we can see that the
smaller circle has a diameter of 39 centimeters. So its circumference and its contribution
to the perimeter of the full figure is 39𝜋. When we think about the pink circle,
however, this measurement of 39 centimeters is the radius of this circle. So the diameter is twice that. It’s 78 centimeters. The circumference of the full pink circle
then would be 78𝜋. But remember, we only have half the
circumference. So the length of the semicircular arc is
78𝜋 over two.
In fact, we find then that the two values
are the same. Both the circumference of the full orange
circle and the length of the semicircular arc for the pink circle are 39𝜋. In total then, we have an exact perimeter
of 78𝜋. But looking back at the information given
in the question, we’re asked to use 3.14 as an approximate value for 𝜋.
We can use a column method to work out
314 multiplied by 78, giving 24492. And we then need to divide this value by
100 to give the answer to the decimal calculation 3.14 multiplied by 78. This gives a value of 244.92. And the units for this perimeter are the
same as the units given for the individual lengths in the question. They are centimeters.