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How to Find the Area of a Shaded Region

Finding the area of a shaded region is a common GMAT geometry question type, and one that students tend to struggle with for whatever reason. In our article “GMAT Geometry – Area of Shaded Regions,” we explained that you want to think about the shaded region of a geometric shape as “leftovers.” The key, in other words, is to subtract the smaller geometric figure from the larger geometric figure, and whatever is “left over” is the shaded region (depending on how the figure is drawn).

Here is another useful tip for you in terms of how to find the area of the shaded region: Start with the end in mind, and write out exactly what you’re looking for in terms of geometric formula components before actually diving in to try to solve anything. To see what I mean, watch the video below as I explain how to apply this useful GMAT tip to solve this sample GMAT problem solving question:

Find the area of the shaded region – GMAT problem solving

Q: Viewed from the outside inward, the figure above depicts a square-circle-square-circle, each enclosed within the other. If the area of square ABCD is 2 square units, then which of the following expresses the area of the darkened corners of square EFGH?*

(A) 2 – ¼π
(B) 2 – ½π
(C) 1 – ¼π
(D) ½ – ⅛π
(E) 1 – ½π

* Excerpted from Brandon Royal’s Game Plan for the GMAT

For your convenience, here is a full transcript of the video above:

Hello. This is Brett Ethridge, founder of Dominate the GMAT. In this lesson
we’re going to take a look at the topic of how to find the area of a shaded
region.

Now, these GMAT geometry questions can be difficult, because to find the
area of the shaded region usually requires us to do several different
geometric steps to get to the final answer. Before I dive in, I’m going to
let you press pause and take a stab at this question first. Then we’ll work
through it together. Go ahead and press pause.

All right, how did you do? Before I give away the answer, let’s work
through this together. Ultimately to find the area of a shaded region we
want to start with the end in mind. There are always two steps. If you
struggle with how to move forward on a question, be very crystal clear
about what information you ultimately need to find. Do something.

You see that it’s really going to be two things, because in this shaded
region there’s no clear formula for how to find the area of the shaded
region. It’s a bizarre shape. However, what do you notice? You want to
think of the shaded region as leftovers. If we could take the area of this
smaller square, EFGH, and subtract the area of this smaller circle, C, what
is going to be leftover, the shaded region.

That’s ultimately how we want to do it. So the answer is going to be the
area of the smaller square, EFGH, minus the area of that smaller circle,
circle C. If we can find those two things separately and subtract them,
that will give us the answer. Actually, in fact, that makes sense, because
when we look at the answer choices that’s what all the answer choices are.
They’re all comprised of two parts, something minus something else. Well,
the something is this, the area of EFGH, minus something else. Well the
something else is the area of the small circle, C.

More specifically, how do we find the area of a square? Well, that’s just
base times height. Of course, the base and the height are the same because
it’s a square. So you can think of it as side squared. We want this side
and we want this side. If we figure out that side, then we know the area
of EFGH, minus, what’s the area of a circle? The formula for the area of a
circle is Pi r squared. If you have to review our lessons on GMAT circles,
or GMAT geometry, you can certainly find that information on our website.

But, how do we do that, because a medium difficulty question would stop
right there. They would only give you a circle inside a square. Here’s the
problem. This is a difficult question on how to find the area of a shaded
region, because it’s not only a circle inside a square, but that whole
thing is inside another circle, inside another square, and they’ve only
given us information about that big square.What do they tell us? They tell
us that the are of the big square, which of course is base times height,
equals two.

Well, what do we do with that? We’ve got to work backwards, because that’s
all the information we have. So, what then is the side? What times what is
going to equal two? By definition, that’s something squared. The side
squared is going to equal two, then that means the side itself is simply
the square root of two. That’s how you would solve that. The square root of
two times the square root of two is going to equal two.

Here’s what we know. We know that the side of the large square, ABCD, is
the square root of two. The side is the square root of two. How does that
help us? We have to be creative, because ultimately we need to figure this
out, and we’re ultimately looking for this side, and this radius.

What are we going to do? Let’s look. We know that side DC is the square
root of two, but guys, if we slide that up, do you see that’s also the
radius of the larger circle? The radius of the larger circle is going to be
the same length, the square root of two, as the larger side. How does that
help us? It doesn’t help us find the diameter of the smaller circle. I
mean, we could guess. We could maybe guess, but we don’t know exactly what
that is. What can we do? We need to be clever. See, this is a 700 point
GMAT geometry question, because it recognizes that we need to do what? What
if we swivel it up? Is the diameter of the circle, EG, the same as the
diameter as we previously drew? Absolutely. There we go.

So, EG is also the square root of two. That’s the same length. Now is that
helpful? Maybe. Can we figure out now the sides of the smaller square?

You can if you recognize that we have just created 45-45-90 right
triangles. Again, you may need to brush up on your triangle rules, but if
you cut a square in half diagonal-wise, you’ve created two 45-45-90 right
triangles, because you’ve bisected a 90 degree angle.

We know the rules of a 45-45-90 right triangle. In fact, the template that
we use is one one root two. They made it easy for us. The hypotenuse is
already root two, we don’t have to do any fancy math. We know for sure that
the sides of the 45-45-90 triangle then are just one.

Again, if you need to brush up on right triangle rules, go ahead and do so,
but that’s it. That helps us find the area of EFG. In fact, it’s base times
height, which is just one times one, which is one. So it’s going to be one
minus something else.

Guys, if you couldn’t go any further, would this be helpful in eliminating
some wrong answer choices? Absolutely. Only the answer choices C and E
start that way. I’m going to get rid of A. I’m going to get rid of B. I’m
not going to get rid of E, because E starts with one minus. I’d probably
get rid of D as well, although it is possible that we would reduce the
whole thing, so D could still be in the running; but if you were forced to
guess now we’re guessing between C and E.

Let’s take it a step further. Is that going to be helpful in finding the
area of circle C? How can we figure that out? Well, that’s easy, isn’t it?
If the whole diagonal here is the square root of two, then what do we do?
That tells us that the side is one. That tells us that the side here is
one, and just as we saw when we were looking at the large square, this side
one, when we shifted down, do you see that is also the same as the radius
of the smaller square? The big shaded diameter here is going to be the same
length, one, as the side of the square, EF.

Now we can find the radius, because the radius is simply half the diameter,
which is just one half. So we have one half as the radius, and when we’re
looking at the formula for the area of the circle as Pi r squared, well
that’s Pi times one-half squared. Of course, when you square a fraction it
gets smaller, so that’s one-fourth Pi. Ultimately, we have one minus one-
fourth Pi. Look at that. That happens to be one of our answers, answer
choice C. There you go. Check it and move on.

It’s complicated, but I hope you see how a 700 point, difficult GMAT
geometry question, that asks you to find the area of the shaded region, can
become much simpler if you start with the end in mind, and you ultimately
write out what you’re looking for, one area minus the other area, and what
is leftover becomes the area of the shaded region. There are a lot of
moving parts here. Make sure you know your rectangle rules. Make sure you
know your square rules. Make sure you know your circle rules and formulas.
Make sure you know your triangles, especially your right triangles, here we
worked with a 45-45-90 right triangle, and apply it.

For more tips, if you want any of these lessons, visit DominateTheGMAT.com.
For other tips like this as well you can check out our blog. I encourage
you to visit DominateTheGMAT.com.

I thank you very much. I hope you’ve learned something in this lesson, and
go out and dominate the GMAT.

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