Arc Lengths and Sector Areas

Arc Lengths (portions of perimeters) and Sectors (pie slices) seem more complicated than they really are. Both relate directly to the internal angle at the circle’s center, represented by θ in this diagram. Here are the equations:

Sector Area = A = θ/360 * πr²

Arc Length = L = θ/360 * 2πr

Literally, all we are doing is finding the fraction of the circle and applying it to either the area or circumference formula, respectively. Note there are only 2 variables. This is good to keep in mind for Data Sufficiency questions. It will be helpful to finding the solution to the following problem:

In the above diagram, a square is inscribed in a circle, which is inscribed in another square. Which is larger, the green region or the yellow region?

(1) The area of the larger square is 64.
(2) The area of the green region is 4.

(Standard Data Sufficiency answer choices apply)

Let’s start with Statement 1.

A(large square) = 64
s(large square) = 8
diameter = 8
r = 4

First, recognize that the figure is symmetrical. So while we may not explicitly be given an angle to find the sector area (not drawn) in which the yellow region resides, we do know its measure. The diagonals of a square intersect at a right angle, so we can deduce that the sector including the yellow region is 1/4 of the area of the circle. Since we know the radius…

A(large sector) = 1/4 * 16π = 4π

To find the yellow region itself, we must subtract the imaginary (not drawn) triangle from 4π. (Note that this imaginary triangle will be twice the green triangle.)

A(imaginary triangle) = 1/2 * r * r = 1/2 * 4 * 4 = 8

A(Yellow Region) = 4π – 8

The area of the green region can be found in two ways. Either we can see that it’s simply one-half of the 8 we just found, OR we can find both sides of the green triangle with the common 45-45-90 1:1:√2 formula. With a hypotenuse of 4, we derive 2√2 for each side, which yields an area of 4.

Which is greater,  4π – 8 or 4?

4π – 8. Sufficient. (We can save the calculations on the GMAT for DS questions, but it’s still good to go through it to practice for PS questions involving similar calculations.)

What about Statement 2?

If we know the area of the green triangle equals 4, and that it is an isosceles right triangle, then we can set up a simple equation to find its sides, which can be denoted x:

1/2 * x² = 4
x² = 8
x = 2√2

If x = 2√2, then the hypotenuse (r) = 2√2 * √2 = 4. From here, we follow the same logic as we did for Statement 1, and determine Statement 2 is sufficient. The answer choice is D.

One More Problem

A circular loop of wire is attached to the two straight wires of a dipole antenna at points X and Y. Point Z is the base of the antenna where the two straight wires meet. How far is point X from the center of the circular loop of wire?

(1) The perimeter of the circular loop of wire is 3 feet.
(2) The center of the circular loop of wire is 2 feet from point Z.

The key point to remember here is that all radii are equal. That is, the distance from the center of the circle (C, not drawn) to both X and Y is the same.

Statement 1: Before we mentioned any if you have any one piece of information about the circle, then you have it all. This is perfect example. If we know the circumference (perimeter) of the wire, we know the radius, regardless of the presence of Z in the diagram. Sufficient.

Statement 2: We can draw a line from Z to C, and mark that distance 2. Because both antenna are tangents to the circle, we know that angles CYZ and CXZ are both right angles, making 2 the hypotenuse. However, there is not additional information (angles or additional sides) that dictates the distance CX. You can visualize this lack of information; picture pinching point Z and dragging it “down” such that X and Y are still on the circle, but closer together. Since there is nothing restricting us from doing that, there is not enough information to accurately determine the length of either the two remaining sides of the right triangle. Insufficient. Answer Choice A.

Here are just a couple examples, but keep working hard. Givens in geometry provide a series of information, not just what’s stated. Make sure to keep that in mind for Data Sufficiency questions.

Good luck!

Read other articles in this series:

• Geometry Series Part 1: Circles Inscribed in Squares
• Geometry Series Part 2: Inscribed Triangles
• Geometry Series Part 3: Couple Common Circle Concepts

Q: What is a "safety school"?

A: A safety school is one where you are highly competitive and that is strong in your area of interest, but it not as highly regarded overall. Your goals are clearly supported by this school.

Q: How do you determine if a school supports your particular goals?

A: For admissions purposes, you can divide your goals into three categories: professional,  educational, and personal. To see if your safety school matches up with your goals, you'll need to answer these questions:

For professional goals: Does this program support what you plan to do after you graduate? Are graduates hired in good numbers for the positions you would like to obtain? Do companies that you'd be interested in working for recruit at this campus? Is the alumni network strong in your field?

For educational goals: Do the educational methods (case, lecture, project, or a mixture) appeal to you? Do the courses offered cover your fields of interest? Are there prominent professors in your area of interest?

For personal goals: Does this program meet your personal preferences? Will you be happy at this location and in this environment?

Q: Why would someone apply to a safety school rather than focus wholly on the more competitive schools?

A: If your application profile is less than perfect, you may not get into the competitive, Ivy League school of your dreams. If, however, you can find a school that matches your needs and that may be less discerning when it comes to GMAT scores or GPA, then applying to that safety school could protect you from an acceptance-less application season.

Q: Should anyone NOT apply to a safety school?

A: If you are an all-or-nothing sort of person, where it's HBS or bust, then it may not be worth your while to apply to a safety school. If you'd rather forego the MBA experience (this year at least) than go to a second-tier school, there's really no reason to apply to anything less than your first tier. In other words, if you would never consider attending a school, then there's no reason to apply to it.

Q: Should I apply to a safety school?

A: There's no clear-cut answer here. If you're worried about acceptance to your top choice schools and you've got the time and money to spend on yet another application, AND if your heart is set on going to business school next term, then applying to a safety school is probably a good idea. If, on the other hand, you'd rather wait out another year and reapply to your dream school (if you don't get in the first time around), then you'd have little reason to spend the time and money on filling out a safety school app.

For more information on safety schools, please see:

• Should You Apply to a Safety School
• MBA Admissions Tip: How Many Schools?
• Taming Grad School Application Jitters: What If I Don't I Get In?

What I found out was very interesting.

I started by obtaining the list of the 2009 top 20 ranked U.S. MBA programs as listed by U.S. News & World Report. I know that I’ve called into question the validity and/or usefulness of rankings in the past, but this provided a handy framework for my research. These are the schools:

#1 – Harvard
#2 – Stanford
#3 – Northwestern (Kellogg)
#3 – UPenn (Wharton)
#5 – MIT (Sloan)
#5 – Chicago (Booth)
#7 – UC Berkeley (Haas)
#8 – Dartmouth (Tuck)
#9 – Columbia
#10 – Yale
#11 – NYU (Stern)
#12 – Duke (Fuqua)
#13 – Michigan (Ross)
#14 – UCLA (Anderson)
#15 – Carnegie Mellon (Tepper)
#15 – UVA (Darden)
#17 – Cornell (Johnson)
#18 – UT Austin (McCombs)
#19 – Georgetown (McDonough)
#20 – UNC Chapel Hill (Kenan-Flagler)

Once I had my list in hand, I went to each of the schools’ websites, and pulled up the Class of 2011 entering class profiles for their full-time MBA programs. I printed each out, and spent an hour or so compiling the following data: undergraduate majors, median years of work experience, and number of undergraduate institutions represented. My goal was to do two things: (1) Find out which majors schools prefer, and (2) Determine whether a school’s penchant for a certain academic background could somehow trump the number of years of work experience preferred by a school. Except for a few notable exceptions (McDonough and Kenan-Flagler, who did not have their incoming class undergraduate majors available online), I was able to obtain the information I needed for this little admissions counseling experiment.

These were my results.

Most schools broke down their incoming class’ undergraduate major into three broad categories, which dealt with roughly the same academic backgrounds: Business/Business Administration/Economics, Engineering/Natural Sciences/Math/Technical Disciplines, and Humanities/Liberal Arts/Social Sciences. When averaged, this is what I found to be the percentage breakdown in accepted students for these schools’ incoming class of 2011:

Business/Business Administration/Economics                            34%
Engineering/Natural Sciences/Technical Disciplines/Math            33%
Humanities/Liberal Arts/Social Sciences                                    25%

Before we analyze these very interesting results, I would like to point out some also very interesting exceptions to these averages. Far and wide, most schools seemed to hover around these averaged percentages. However:

• Harvard Business School had Humanities/Social Sciences at 40% of their incoming class, and Business Administration at 26%.
• Stanford GSB also placed a heavy preference on Humanities/Social Sciences, with a whopping 47% of their 2011 incoming class majoring in those fields. Business got the short end of the stick with a measly 17% of the pie.
• Wharton also leans towards Humanities/Social Sciences, giving them 42% of their slots; Business only got 24%.
• MIT (Sloan) likes Engineering/Math (44%) more than Business (20%).
• NYU (Stern) was interesting, because they had a pretty even distribution across the board: Business (26%), Social Sciences (20%), Economics (20%), Engineering/Math/Science (18%), and Humanities/Arts (16%) all had similar numbers.
• Carnegie Mellon (Tepper) was a shocker in its preference of science and engineering majors—50% of their 2011 class favored these disciplines. And, if you think that’s a chunk, it’s nothing compared to their 2010 class (57%), or their 2009 class (60%).

So, what did this very brief, bird’s-eye view of undergraduate majors teach us? Probably something we already knew: That what you major in doesn’t really matter quite as much as how you do in that major, and how difficult your classes are. It doesn’t matter if you majored in Business Administration with a minor in Economics, or if you instead chose to double-major in Literature and Art History. The numbers tell a story that might make those English majors among you rejoice (“Hooray! No more downplaying my senior thesis on the works of Emily Brontë!”) and might make those Business majors among you curse (“You mean all those nights I spent slaving over numerical minutia don’t give me an edge?!”) However, there is a deeper layer to these results than just the percentages—and this layer exists because there are no overwhelming undergraduate major preferences. This lack of disparity amongst the numbers begs the question—if it’s not a major schools are looking for, then what are they looking for? And that’s where my other data (and a little bit of thinking) came in.

All schools had the majority of their incoming class as having at least 3 years of full-time work experience, and the vast majority of them had the median number of years at 5. UCLA (Anderson) even had a chart of the functional titles of their incoming class: 50% were Senior Managers, 33% were Middle Managers, 10% were Corporate Officers, and 7% were Professionals. Combine this information with our undergraduate major information and, after a brief bit of analysis, this is where I ended up: MBA programs aren’t looking for majors, or even specializations—honestly (although they won’t ever say this out loud) they probably don’t even care what your major was, as long as your GPA is decent and your GMAT score is good. What they’re looking for are qualities. This becomes even more apparent when you analyze the many and varied fields that MBA candidates come from: high tech, entertainment, financial services, real estate, education, military, biotechnology, and consulting are just a few of them, and the list goes on for pages. The preference, it turns out, is that there is no preference. Create a listing of qualities from this academic and professional information, and this is what you come up with:

Business majors will likely exhibit these qualities (among others): intellectual ability, organizational skills, good written and oral skills, quantitative ability, and the ability to work with others.

Engineering/Math majors will likely exhibit these qualities (among others): intellectual and analytical ability, organizational skills, self-motivation, a highly developed quantitative ability, an increased capacity to problem-solve, and the ability to get things done individually and as a group.

Humanities/Liberal Arts majors will likely exhibit these qualities (among others): intellectual ability, imagination, creativity, good written and oral skills, a sense of humor, and self-motivation.

Those with 3+ years of professional employment and success will likely exhibit these qualities (among others): intellectual and analytical ability, organizational skills, good written and oral skills, the ability to work well with others, problem-solving skills, self-confidence, maturity, a sense of humor, and managerial potential.

What do you immediately notice as you’re reading these lists? There are certain qualities (intellectual ability, organizational skills, problem-solving skills, a sense of humor, for example) that repeat themselves over and over again. This is what business schools are looking for, and not an academic pedigree. Now, don’t get me wrong: a good academic pedigree never hurt anyone’s chances during the application process. However, a business degree is not the end-all-be-all during MBA admissions, and an English degree isn’t an application-killer. Graduating from a certain school doesn’t guarantee admission (the number of schools represented in an incoming class varied from 144 to 207), and a degree from a second-tier school isn’t going to harm your chances. It boils down to your personal characteristics, professional experience, and still-untapped growth potential, and how you present these two aspects to the admissions committee. Take my little admissions counseling experiment to heart: focus on who and what you are, and not where you’ve come from. Don’t simply rely on pedigrees, be they undergraduate majors, school names, or corporate titles. The numbers tell a refreshingly simple truth: Getting into a top MBA program isn’t an old-boy network of academic specializations or Fortune 100 credentials. Shine a spotlight on your personal potential, and that will dazzle admissions committees more than a 5-page résumé and transcript ever could.

Okay, this sample problem is from our own archives. Set your timer for 2 minutes…. and… GO!

Both the average (arithmetic mean) and the median of a set of 7 numbers equal 20. If the smallest number in the set is 5 less than half the largest number, what is the largest possible number in the set?

(A) 40

(B) 38

(C) 33

(D) 32

(E) 30

You have an answer, right? Even if you don’t know what the answer is and you have to guess… you’ve still picked an answer, right? If you haven’t, go pick an answer before you keep reading!

As we discussed last week, the most important thing to notice here is the word “largest.” This one word is going to be the determining factor in how we set this problem up, right from the very beginning.

So, we have a set of 7 numbers. The average of those 7 numbers is 20; can we calculate anything from that? Yes – the sum! The basic formula for an average is A = S/n, where A is the average, S is the sum, and n is the number of items. We know A and n, so plug the two numbers in to get 20*7 = 140 for the sum.

The problem is asking us to maximize one figure: the last (and, of course, highest) number in the set. If all 7 numbers have to add up to 140, and we want to make one number as large as possible, then what do we have to do to the remaining six numbers?

We have to minimize all 6 of the remaining numbers – so, for the rest of the problem, we need to figure out how to make the other 6 numbers as small as possible.

Do we know anything about those 6 other numbers? We were told that the median is 20; what does that mean? Draw out some dashes on your scrap paper, one for each number in the set:

____   ____   ____   ____   ____   ____   ____

Now, how can we represent the fact that the median is 20? We have an odd number of terms. The median will be the middle term (the 4^th, in this case) and it will actually equal 20. So add that to your diagram, along with an “x” for the term we’re supposed to maximize:

____   ____   ____   _20_   ____   ____   __x__

The problem also gives us some info about the first term:

“the smallest number in the set is 5 less than half the largest number”

Hmm. We don’t know what the largest number is, of course – that’s what the problem asks us to maximize! But we’re calling that largest number “x” so let’s write the smallest term in terms of x: (½)x – 5. Add that to the diagram:

_(½)x – 5_   ____   ____   _20_   ____   ____   __x__

Okay, now what – what’s our goal again? Oh, right, we want to minimize everything that isn’t that last term, “x.” Okay, so what can we minimize? We can’t change the 1^st term; that’s going to be (½)x – 5 no matter what. And we can’t change the fourth term; that’s going to be 20 no matter what. What about the 2^nd, 3^rd, 5^th, and 6^th terms? What are the smallest possible values for each of those?

Let’s start with the rules for writing out a bunch of numbers in order to show a median. When you write a set of numbers to find the median, the requirement is to write the numbers from smallest to largest. Let’s say that we have to write these three numbers in order: 20, 14, 18. We would write them: 14, 18, 20. Moving to the right, each number is higher than the previous number. Moving to the left, each number is lower than the previous number.

Is that all? What if we had to write these three numbers in order: 20, 14, 20? Then, we would write: 14, 20, 20. Moving to the right, the second term, 20, is higher than the first term, 14, but the third term, 20, is equal to the second term, 20. So the full rule is not that the numbers have to increase as you move to the right or decrease as you move to the left. The rule is that, as you move to the right, the number has to be equal to or higher than the number to the left. Similarly, as you move to the left, the number has to be equal to or lower than the number to the right.

So, let’s get back to our problem. We’re trying to minimize the remaining slots. What is the smallest possible value for the 5^th term, keeping in mind that the number has to be equal to or higher than the 4^th term? The 4^th term is 20, so the smallest value for the 5^th term is also 20. For the same reason, the smallest value for the 6^th term is also 20.

_(½)x – 5_   ____   ____   _20_   _20_   _20_   __x__

What about the 2^nd and 3^rd terms? The 2^nd term has to be equal to or higher than the 1^st term, and the first term is (½)x – 5. Therefore, the smallest possible value for the 2^nd term is equal to (½)x – 5. For the same reason, the smallest value for the 3^rd term is also (½)x – 5.

_(½)x – 5_   _(½)x – 5_   _(½)x – 5_   _20_   _20_   _20_   __x__

Now we have representations for all seven terms: either real numbers or variable expressions. We know the seven terms add up to 140. Time to set up an equation and solve for x!

[(½)x – 5] + [(½)x – 5] + [(½)x – 5] + 20 + 20 + 20 + x = 140

(3/2)x – 15 + 60 + x = 140

(5/2)x = 95

x = 95(2/5)

x = 38

The correct answer is B.

Key Takeaways for Max/Min Problems:

1. figure out what variables are “in play” (what figures we can manipulate in the problem)
2. figure out whether each variable needs to be maximized or minimized in order to achieve the desired outcome (the thing the problem asks us to do)
3. do the work (carefully, as always!)

Note: the key takeaways are the same as last week, when we did some lower-level max/min problems. The basic process doesn't change; we just have a bit more we need to know and a bit more we need to do on the very hard problem we did this week.

What does it look like?

I've highlighted the places where the subjunctive differs from the "normal" indicative.  In the future tense, you see that I have "will/shall"; traditionally, shall is the simple first-person future form (I/we shall, but he/she/it/you/they will), though you are not likely to see it often in American English.  Shall can still be used to show certainty or obligation (You shall not pass!), and also appears in legal language.

Where will I see it?

There are some common places the subjunctive can appear in English; we will be covering all of these in this series:

• wishes (I wish that I were able to play the flute,  May the best man win)
• suppositions (If I were to go to the party, I wouldn't finish painting the house)
• demands and commands (She demanded that he leave her house immediately)
• suggestions and proposals (I suggest that the Board of Governors consider it a bit more)
• conditions contrary to fact (If I were master of the universe, college tuition would be free)
• statements of necessity (It's necessary that they be there for your safety and comfort)
• fearing with lest (I filled her car with gas lest she run out on her cross-country trip)
• idiomatic phrases  (As it were, be that as it may,  . . . need only . . .)

Wishes

Wishes are one of the two most common uses of the subjunctive in spoken English (suppositions the other, which we'll cover next time), when you use to wish, you use the past subjunctive were, even though the wish is not taking place in the past:

He wishes he were on vacation.
She wishes that the sun were out.

Note that you can use that, or not use it; the wish stays the same.

A GMAT-style question:

Despite the fantastic success he experienced in his first year of business, the owner still wishes that he is doing better.

A.  that he is doing better
B.  that he should be doing
C.  that he were doing better
D.  to be doing better
E.  to do better

Which one is correct in the original question?  Which one would be correct if we had a new question which replaced "wishes" with "wants"?

Answers next time in part 2: suppositions!

Being a great storyteller and a gifted writer can be a major advantage to the prospective b-school student. But be forewarned: A wonderful answer to a question not asked will not help you here. We can’t stress enough that you must answer the question.

Each school has its own set of questions. Although posed differently, all search for the same insights. Here’s one commonly asked question and what’s behind it.

Theme: Ethics

Describe an ethical issue you have faced in your professional life and how you dealt with it. What was the outcome?

Translation:

Do you even know what an ethical dilemma looks like? Are you tomorrow’s corporate miscreant? What kinds of decisions and judgments might you make in your future practices as a business leader?

The last few years have brought attention to the ethical issues of the business world and the failure of corporate self-governance. In the aftermath of the recent financial meltdown, b-schools don’t want to turn out graduates who are fast into their suspenders, fast into a deal, and fast to swindle their clients and shareholders.

The above question tests your judgment, integrity, and perspective. It’s most important to present a legitimate ethical dilemma here, one that has consequences. Applicants often write about the dilemma of not obeying a supervisor’s orders because they wanted to do things their way, a known better way. But this is not an ethics problem (unless the order was improper or illegal); it’s a management problem. Likewise, handing in a report to your boss that you know is full of errors is also not an ethical problem; it’s a trivial, single-impact, easy-to-fix problem.

If you were thinking of telling a story like one of those mentioned above, it may be because you wanted to play it safe. This is one of those uncomfortable, hot-seat essays after all. But playing it safe here would only make you appear clueless or morally bereft.

This essay requires you to roll up your ethical shirtsleeves and get down in the dirt. True ethical issues are neither clean nor pretty. Don’t shrink away from a discussion of failure here or present an overly optimistic, no-loose-ends solution.

It’s key that you write about an ethical dilemma in which there was no easy course—one that entailed costs either way. For example, let’s say you sold a product to a client and later discovered the product was faulty; your employer wanted you to keep mum. You’d built your sales relationships on trust and personal attention, so you wanted to be forthcoming. What did you do?

This essay must show that you can work through a complex ethical impasse, and it must highlight your sense of honor and conduct. This essay screams relevance. Make sure you shout back that you know right from wrong.

A square countertop has a square tile inlay in the center, leaving an untiled strip of uniform width around the tile. If the ratio of the tiled area to the untiled area is 25 to 39, which of the following could be the width, in inches, of the strip?

I. 1
II. 3
III. 4

a. I only
b. II only
c. I and II only
d. I and III only
e. I, II, and III

In case you haven't figured out the answer, the strip could be any width. Answer choice E is correct. How do we arrive to this answer?

The problem-solving version of this question is taken from p. 85 of the GMAT Quantitative Review, 1st Edition. The solution given is computational; the basic steps are:

Let = area of the square tile inlay
Let = area of the entire countertop

Then , so that .

Taking the square root of both sides, we see that t/s = 5/8, which means that the length of the side of the square tile inlay is 5/8 the length of the side of the entire square countertop. At this point, the solution in the QR proceeds to write an expression for the width of the strip, w, in terms of these two variables: w = (s - t)/2, and then substitute an expression for t in terms of s to obtain an expression for w in terms of one variable, s:

--> . Then .

Presumably, you need to be able to write w = ks for some constant k, in order to see that w can take on any positive value. This is a lot of computation, and it will surely put you beyond the the scant average of 2 minutes per question that you should allow. Is there a better way?

Imagine that this problem was posed as a data sufficiency question. It could look something like this:

A square countertop has a square tile inlay in the center, leaving an untiled strip of uniform width around the tile. What is the width, in inches, of the strip?

1. The ratio of the tiled area to the untiled area is 25 to 39.
2. The ratio of the tiled area to the total area of the countertop is 25 to 64.

Granted, this version of the question is slightly easier, because statement II gives you a clue to what your first step should be. But the same insight is necessary to solve this, and I maintain that it does not require computation. From either statement, you can reach the point reached in the solution given above: t = (5/8)s. Instead of plowing ahead with more computation, take a step back and use your imagination. The only constraint given by the problem is a proportion relating the length of a side of the tiled area to the length of a side of the table. FURTHERMORE, the question asks for width of the strip IN INCHES. This is slightly peculiar, given that the only other information about the table (a ratio of areas) does not mention ANY UNITS OF MEASUREMENT. What if the question had said "in feet." Or "in millimeters." Or "in miles." Any one of these is possible given the information in the statements. The fact is that there is no way to determine absolute length measurements in terms of a specific unit if only proportions are known. A simpler question might look like this:

If a red spherical balloon contains twice the volume of a blue spherical balloon, which of the following can be the surface area of the red balloon, in square meters?

1. 1
2. 2
3. 3

ANY of these numbers could be the surface area.

GMAT questions force the test-taker to make decisions about quantitative matters as quickly and as accurately as possible--very often without having to make calculations. The key behind these questions is to realize that computation is not necessary--a sense of what CAN be computed, however is indispensable.It makes sense that questions like these are on the GMAT. Very often in world of business, folks with an MBA are faced with the following decision: Should we spend all this time to do this computational task?

In the world of management, understanding when to ask this question can often be the difference between money wasted and money well-spent. On the GMAT, understanding this question will raise your score.

1. a² + b² = c², but only when a right triangle. If you don’t know it’s a right triangle, Pythagorean theorem does not apply!
2. Common special right triangles include 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples.)
3. 45-45-90 triangles are ALWAYS in the ratio 1:1:√2
4. 30-60-90 triangle are ALWAYS in the ratio 1:√3:2
5. Angles and opposite sides are in the same relative size order, but are NOT proportional.

Let’s continue with a standard diagram in which we have an equilateral triangle inscribed in a circle, which is inscribed in a square.

The center point of all three figures (triangle, circle, square) are all the same, but this is ONLY true if the triangle is equilateral. Therefore, if given ANY piece of information about the circle, square or triangle, we can derive the rest. We draw a perpendicular line from the center to the side of the triangle.

Note that the hypotenuses of the smaller triangles are equal to the radius of the circle. We also know that the smaller triangles are each 30-60-90 because you are taking the 120-degree internal angle from the circle’s center and cutting it in two. Here are your basic conversions:

r = ½d = ½s, where s is the side of the square.
The sides of the 30-60-90 triangles become ½r : (r√3)/2 : r respectively
The side of the equilateral triangle becomes 2*(r√3)/2 = r√3

If given the area of the square, we should be able to derive essentially any other information.

Area of an Equilateral Triangle

The area of an equilateral triangle equals (s²√3)/4. Memorize this. It will save you the time of drawing a 30-60-90 triangle, solving for the base, finding the height, multiplying and dividing by 2. That was long to write, imagine how long it takes to do!

If  the area of the square = 64 and we needed to find the area of the triangle, we just use the conversions above:

d = 8
r = 4
side of triangle = 4√3

Area of triangle = [(4√3)²√3]/4 = 16*3*√3 / 4 = 4*3*√3 = 12√3

Angle Relationships

Another important rule is that the interior angle created from of two radii extending to the outside of the circle is exactly twice the measure of any angle on the circle extending to those same points.  In the image above, 2b = a. This information is never explicitly stated on tests, but will come up on DS questions over and over.

Let’s take a look at this practice question:

In the figure above, a circle is inscribed in a square. What is the area of the shaded region?

(1) The perimeter of the square equals 32.
(2) x = 35

Statement 1 seems irrelevant to the question, but we can determine r by knowing that the length of the square’s side. If s = 32/4 = 8, then d = 8 and r = 4. This is insufficient, since we do not know the interior angle.

Statement 2 provides information about x, and from this, we know that the interior angle of the shaded region is 2(35) = 70. This is insufficient, since we do not know the size of the circle.

Together, we know both the size of the circle and the degree measure of the interior sector angle.

A(shaded) = (2x/360) * πr² = 70/360 * 16π, whatever the hell that comes out to. Remember, since it’s a Data Sufficiency question, we don’t actually need to calculate the number.

There are infinite variations of these concepts. Be flexible in your reasoning, and practice makes perfect!

Good luck!

Read Geometry Series Part 1: Circles Inscribed in Squares

Here is Dana’s analysis of The Princeton Review Cracking The GMAT book.

Overview

The Princeton Review Cracking the GMAT book is, like the Kaplan GMAT Premier Live Online book, a general GMAT strategy guide. However, apart from the natural similarities (such as separate chapters for the 5 types of problems you will see on the GMAT plus an essay section), the two books follow markedly different “paths to knowledge”.

As advertised from the beginning, The Princeton Review Cracking the GMAT is not a guide that emphasizes building subject-matter skills in the quantitative or verbal fields, but rather focuses on what the authors call “cracking the system” – tips and tricks to help you score higher with an obvious preference for process of elimination (POE) and picking numbers. Personally, I do not fully support such strategies for the GMAT (I consider them more as backup plans).  Here’s why: POE cannot and will never replace actually knowing the answer to a question, it will just improve your chances of getting it right. Picking numbers is also quite risky: I sometimes see this strategy discussed in online forums; people try to pick numbers, but particularly at an advanced level they pick the wrong set of numbers and everything falls apart.

The book is a pleasant read though, and I personally found the small info boxes on the sides of pages to be quite entertaining and interesting. The authors use “the average test taker” for examples of do’s and don’ts. However, I didn’t like the authors’ tone for describing the GMAT – I felt an overall negative vibe about the test, which was not the case when I read the PowerScore CR Bible for instance. An example would come from the Reading Comprehension chapter: “No one in her right mind would ever read one of these passages of her own free will. They are almost always boring.” I strongly disagree with this view, since I personally enjoyed most texts, particularly those dealing with the social sciences. My advice is to think of the GMAT as a sort of mind game you have to beat, an interesting little challenge. I assure you that feeling positive about the GMAT will make the experience a lot less painful.

Pros

• A good introduction to the GMAT, an easy read to make you understand what the test is about, how it’s scored, and what to expect on test day
• If your quantitative skills are not fresh, definitely give this book a chance.  It covers math basics pretty well. Again, the fact that it’s an easy read will probably help you reconnect with junior high and high school math
• Questions are structured in “bins” according to difficulty, which means you can focus on your particular level when you practice
• The book comes with a registration code you can use to gain online access to four CATs (computer-adaptive practice tests). However, I felt that these practice tests were a bit easier than the actual GMAT

Cons

• If you’re shooting for a higher score (above 600), this book won’t help much. You will need other resources to help you reach your goal
• At just over 280 questions, this book has fewer and easier practice questions as compared to other guides (like Kaplan Premier Live Online). Also the explanations could use some more work, particularly in verbal. I personally get really upset if the best explanation for why an answer is wrong is “because it’s out of scope”! This will definitely not help a test taker understand where her error in reasoning was!
• The focus on strategies such as POE and number picking means that you’ll have some potentially useful tools for G-day, but such strategies might backfire. Cracking the GMAT does not provide an in-depth analysis of question types and their respective approaches

Bottom Line

After going through The Princeton Review’s Cracking the GMAT book, all I can truly say is this: the book is good, but only for a certain level (that is, up to mid-600s). As a basic guide, it shines; as a tool for getting into that top 10 MBA, I do not think it delivers. If you do not already know your starting level, I advise you to do the following: go to MBA.com, download the free GMATPrep software, and take practice test #1 (save #2 for the last days of your prep!).  If you score significantly below 600, consider investing in this book. If you score higher than 600, you’d best search for a different resource since you’ll need a more rigorous program to get to a higher percentile score.

If you’re interested in purchasing The Princeton Review Cracking the GMAT book, click here.

Remember: exponents simply count the number of times something is multiplied by itself.

With exponents, the most important concept to take into the test is the concept of similar bases.  If we have similar bases, the number properties of exponents allow us to manipulate them.  For instance, and do not allow us to manipulate them.  However, if the bases are the same, we can simplify the terms.

“Ok. I got the same base.  Now what?”

For the test, memorize the times that exponents need to be added, multiplied, or subtracted.  Flipping the way we look at exponents will help you memorize what to do when the time comes.

• Add Exponents: We add the exponents together when we are multiplying terms with exponents with each other.  For example: 
• Multiple Exponents: We multiply the exponents together when we have a term with an exponent raised to another exponent.  For example: 
• Subtract Exponents: We subtract exponents when we are dividing terms with exponents by each other.  For example: 

What about the following statement? .  What can we do with the terms when we are adding them together?  Nothing.  You have to work out each term and add the result.

Our last point is what to do with negative exponents.  Negative exponents can be simplified by dividing 1 by the term.  For example: is simply: .

Continue to work on your number properties questions.  These are the source of some frustrating questions.  However, a few simple principles will get you through it.