A. 12 + 12π
B. 24 + 12π
C. 24 + 24π
D. 48 + 12π
E. 48 + 24π
Here’s How to Crack It
Make sure you write everything you need on the figure, including the relevant square and circle formulas.
Since you are dealing with shapes within shapes, now is also a good time to figure out what the link is,
what these two (or three, in this case) shapes have in common. Don’t worry about digging up a formula
for the perimeter of that hourglass-shaped thing. Stick with the shapes you know.
For this problem, the links occur on the right and left sides of the figure: The left and right sides of the
square are the same as the diameters of each of the semi-circles. We can therefore say that if all sides of a
square are equal, each of these sides is 12. Since the diameter of each semicircle is equivalent to a side
of the square, then the diameters of both semicircles must also be 12.
Let’s use this information to find the perimeters of each of these semicircles. Remember, when you’re
dealing with circles, the perimeter is called the circumference, which can be found with the formula C =
πd. Since you’re dealing with a semi-circle here, you’ll need to divide its circumference in half. Let’s
find the circumference of the semicircle with center E.
The semicircle with center F will have the same circumference because it has the same diameter. Now all
we need to do is add up our perimeters to find our answer. You know the sides of the square will each be
12 and the circumference of each semicircle will be 6π, so the total perimeter of the shaded region will
be 12 + 12 + 6π + 6π = 24 + 12π, as in (B).
So let’s hear it once more:
Don’t do more work than you have to on Geometry problems by trying to remember every weird
formula you’ve ever learned. Stick to the formulas you know, and work with the Basic Approach.
