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The name “triangle” means, sensibly enough, “three angles” and indeed every triangle has three angles and three sides:

This much is hopefully familiar. But just as with circles, there are many special terms people have devised for triangles. When it comes to the sides of a triangle, we can have:

Equilateral Triangles: All sides have the same length.

Isosceles Triangles: Two sides have the same length as each other; the third has a different length.

Note that in an isosceles triangle:

Equal Sides Have Equal Angles: In an isosceles triangle, the angles opposite from the sides of equal length must have equal degree.

Thus, in the below diagram:

angles \alpha and \theta are equal.

Scalene Triangles: All of the sides are of different lengths.

We can also classify triangles according to their angles:

Acute Triangles: All angles are less than 90 degrees.

Right Triangles: One angle is exactly 90 degrees.

We use a square to mark a 90 degree angle.

Right triangles have lots of special further features which we will talk about here.

Obtuse Triangles: One angle is more than 90 degrees.

Now, we can also compare two triangles to each other. We can say that they are:

Similar: Two triangles are similar if their angles have the same values.

So in the above diagram, even though one of the triangles is obviously bigger than the other, we can say that they are “similar” (in the technical, mathematical sense defined above) because their angles have the same values. And the definition makes sense since, after all, those two triangles do look pretty similar (in our ordinary, day-to-day sense)!

These triangles are also similar to one another; it doesn’t matter if you rotate or flip the second triangle. All that matters is whether the angle values for the two figures are the same.

The reason why we care about the similarity of triangles is:

Similar Triangles Share Proportions: If two triangles are similar, then there is some constant ratio you can multiply the sides of one triangle by in order to get the sides of the other triangle.

For example, if the following two triangles are similar:

then there is some constant, call it r, such that:

    \[ar = A\]

    \[br = B\]

    \[cr = C\]

This can be very helpful in trying to find certain side lengths or the area of a triangle as we shall see.

Congruent: Two triangles are congruent if their angles are the same and their sides are the same length.

You can think of congruence as signaling that those two triangles are effectively the same triangle. The lines are the same length, the angles are the same, the area is the same, and so on.

And again, it doesn’t matter if you rotate or flip one of the triangles. It is still true that their angles and side lengths are the same, since rotating or flipping a shape won’t change any of that.

Now, when it comes to finding the area of a triangle, you need to know the base of the triangle and its height. The base can be any side of the triangle, but the height needs to extend perpendicular (at a right angle to) the base, and reach the highest point of the triangle, away from the base, like so:

Traditionally, we use b to denote the base and h to denote the height. Thus, we get:

Triangle Area Formula: \frac{1}{2}bh

Finally, we note two fundamental facts about triangles:

Angles of a Triangle Sum to 180: For any triangle, the sum of its interior angles is 180 degrees.

Triangle Inequality: The sum of the length of any two sides of a triangle is greater than the length of its third side.

Practice Problems:

1. Find the value of \theta.



2. Find the value of \theta.


3. A triangle has legs that are 4 and 7 units long. What integer values are possible for the third side?


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No doubt we’re all familiar with circles:

Less familiar, though, may be the many technical terms for certain parts of the circle — the kinds of terms that most people leave behind in high school. But knowing what those terms mean can help you on the GRE by giving you a vocabulary that helps you to understand the formulas and the problems (it’s easier to learn how to find the area of a circle’s sector, rather than “the bit of the circle left over when you cut off the circular bit”).

Circle: A 2D shape whose points are always the same distance from some central point.

Center: The center of a circle is the point within the circle from which every point of the circle is the same distance. We often label the center and use it to identify the circle (e.g. “the circle whose center is at A”).

Radius: Any line from the center of a circle to any point on the circle. We often use the variable r for the length of the radius.

The radius of a circle is often very important because it allows us to find the area of that circle:

Area of a Circle: The area of a circle with radius r is \pi r^2.

Diameter: Any line that connects two points on the circle and the center of the circle. We often use the variable d for the length of the diameter.

And of course, using the diameter, we can always find the radius of a circle:

Diameter Formula: For a circle with radius r, the diameter d = 2r.

Circumference: The distance around the circle.

And we can find the circumference of a circle using the diameter:

Circumference Formula: The circumference of a circle with a radius r is 2\pi r.

Thus, the area of a circle, its radius, its diameter, and its circumference are all related. Using any one of these values, we can find the other three!

Chord: Any line segment that connects two points on a circle.

Note also that:

Diameter is Longest Chord: The diameter of a circle is also a chord of the circle and, in fact, it is the longest chord on a circle.


Arc: Any portion of a circle’s circumference located between two points on the circle.

Sector: A portion of the circle enclosed by two radii and an arc

Some GRE problems will ask you to find the area of some sector of the circle. We will talk about how to handle those problems later (%TODO insert hyperlink here).

Tangent: A straight line that touches a curve or circle at a single point

Point of Tangency: The point at which a tangent line touches a curve. Note that the tangent line (in red below) will be perpendicular to any line connecting the center and the point of tangency.

Finally, circles can be drawn within or around other shapes:

Inscribed: Circles can be inscribed within other shapes. An inscribed circle is the largest possible circle that can be drawn wholly within another shape. For example:


Circumscribed: Circles can also be drawn around other shapes. A circumscribed circle is the smallest possible circle that can be drawn wholly outside of another shape. For example:

And circles can share a center: 

Concentric Circles: Two circles are concentric if they share the same center.

Practice Problems:

1. Draw a circle inscribed within a square


3. The radius of the below circle is 1. What is the area of the shaded region?


4. Which is longer?
AC                  BD
A. <u>AC is longer.
B. <u>BD is longer.
C. They are equally long.
D. It cannot be determined.


5. The radius of the following inscribed circle is 5, and the radius drawn connects to a point of tangency. What is the area of the shaded region?


6. The shaded region has an area of 5. What is the area of the circle?



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Many GRE math problems involve 2D shapes, which include triangles, squares, and in general, the kind of shapes you might remember seeing in first grade. It turns out these shapes actually have a number of interesting properties, knowledge of which matters for doing well on the GRE. These shapes are often defined by the number of sides they have. Now, there is no 2D shape that is made of just 1 line:

And similarly, there is no 2D shape with just 2 lines:

But starting with three lines, we get:

No doubt you know that this is called a triangle. With four lines we get the general category of quadrilaterals (or four-sided shapes):

This category includes trapezoids, squares, rectangles, parallelograms, and rhombuses:

(The arrows above indicate that those two lines are parallel to one another).

And we can also have a 2D shape defined by a constant distance from a central point:

and we call this a circle.

Now, of courses there are other 2D shapes; we can have pentagons (5 sides), hexagons (6 sides), and more. (A chiliagon has 1000 sides). But for the GRE, we will generally need just the above.

Finally, we also need the idea of a regular polygon.

Regular Polygon: Regular polygons have sides of equal length and angles of equal degree.

So, for example, the regular 3-sided polygon is the equilateral triangle:

which, as the name equi-lateral suggests, has three sides of equal length. And its angles are all 60 degrees.

The regular polygon for four sides is the square:

where each angle is 90 degrees.

You may have noticed a pattern here: 60 degrees for 3 sides; 90 degrees for 4 sides. And this pattern holds for any number of sides:

Angles of a Regular Polygon
If we have a regular polygon of n sides, each angle is 30(n-2) degrees.

This formula can be important when trying to find certain angle values; for example:

Example 1
You have a seven-sided regular polygon. Give the degree of an angle of the polygon.


Example 2
You have a regular octagon. What is the sum of all of the angles of the octagon?


And the fact that regular polygons have sides of equal length can help in problems involving the perimeter of a regular polygon:

Example 3
A regular 29-sided polygon has a perimeter of 116. What is the length of its sides?


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Geometry, unlike the other mathematical subjects tested on the GRE, has special rules. Basically, these rules prohibit test-takers from just eye-balling the answers based off of the diagrams. So for example, in the following problem:

Example 1
Find the value of x:

You cannot just look at the figure and guess that the mystery leg is the same length as the other leg of the triangle. For if we used the Pythagorean Theorem to calculate the solution:


So our first rule is:

Rule 1: Shapes and figures are not necessarily drawn to scale.

So in the following problem:

Example 2
Is side B longer than side A?

The correct answer is that it cannot be determined which side is longer.

However, this does not mean that the figures drawn are totally misleading. For we also have:

Rule 2: Shapes and figures do show the relative position of different objects.

What does this mean? It means that if you see something like:

Then you can conclude that the triangle is to the left of the square. Or if you see:

You can conclude that those two points are on that line, and that point a is to the left of point b.

Finally, the rules are different for co-ordinate systems and the number line:

Rule 3: Co-ordinate systems and the number line are to scale.

The main co-ordinate system you will see on the GRE is the xy-plane:

and the number line is just the following diagram:

On such systems, the objects are to scale. So if you see:

you can conclude that the value for b is twice as large as the value for a. Or if you see:

you can conclude that the area of the smaller square is 1/4 of the area of the larger square.

How do you conclude that?

So in short, remember that the proportions of shapes cannot be trusted, but the relative position of shapes can be trusted, and if there is co-ordinate system or a number line, then even the proportions can be trusted.

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This guide covers all the geometry you need for the GRE math section. We will try to make the content fairly intuitive, combining formulas with some explanation of why the formula makes sense. Doing well on geometry problems can make a big difference for your GRE math score since geometry questions make up about 15% of all GRE math questions.

Most geometry questions will ask you to do one of four things:

  1. Find the line length
    Example 1
    In the following figure, find the value of x
  2. Find the angle
    Example 2
    Find the value of \theta:
  3. Find the area (of a 2D shape) or surface area (of a 3D shape), or
    Example 3
    Find the area of the shaded region:
  4. Find the volume (of a 3D shape)
    Example 4
    Find the volume of the rectangular prism below:

But these problems can be difficult because they give you seemingly odd bits of information. For example, they might ask:

Example 5
The area of the following triangle is 20. Find c:

and you will have to figure out a way to use what you know (the area formula for triangles, the Pythagorean Theorem) to figure out the answer:


There are many little factoids which matter in geometry, and these will be found throughout our guide. Now, the sheer quantity of formulae can get overwhelming, so we will distinguish between must-know formulas like:

Pythagorean Theorem: For any right triangle where a, b are the legs of the triangle and c is the hypotenuse, a^2 + b^2 = c^2.

and the less critical formulas like:

30-60-90 Right Triangle: For any right triangle with angles of 30, 60, and 90 degrees, the side lengths have the following ratio:

The must-know formulas are, well, a must-know if you want to do well on geometry. But if you are okay with missing a few geometry questions on the exam, you can afford to ignore some of the less critical formulas. Each such formula comes up maybe once per test.

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As an LSAT tutor, one of the question types my students most struggle with is “resolve the paradox.” As a law school applicant, the paradox that most nettled me was a paradox centered on law school itself: given that law school is so arduous, why are application figures so robust? Does law school simply attract masochists (or whiners)? Is the well-trod pathway to wealth and political power what tempts people to accept such a hideous fate?

But these questions assume a basic premise, which is that law school is actually hard. Is this true?

Is Law School Hard?

Sorry to disappoint, but the answer to this question is an emphatic “yes”! Of course law school is hard! Have you never talked with a law student? Observe any 1L during finals, or any bedraggled OCI participant, and you will witness the rigors of law school wrought on the human body. Before law school I was youthful and energetic; by November of my 1L year I looked like the “Before” picture in an eye cream ad, and that was on my good days.

Making the decision to attend law school requires accepting that it will likely be difficult.

Why Is Law School (Usually) Hard?

Some people—veterans, parents, interns for Meryl Streep’s character in “The Devil Wears Prada”—don’t find law school particularly onerous because they have survived tougher gauntlets. But most law students are mostly untested in the ways that law schools challenge their students. Looking at the profile of the typical 1L, this is hardly mystifying: a fairly young, relatively recent humanities grad possessed of an abbreviated work history and a roster of academic successes in a context where they are somewhat easy to come by. Despite the maxim that past outcomes do not guarantee future results, most people enter law school either overconfident, underprepared, or both.

A Tale of Two Law School Experiences

One characterization of law school has it like this: you’ll be thrown into a group of dozens of strangers in a pseudo-professional, contentious setting. You will leave behind a lifetime of continuous and easily-won academic validation for a system in which you are evaluated anonymously, for the first and only time, by a professor whose primary interactions with you will be to point out the weaknesses in your reasoning and comprehension in front of the 50+ snickering strangers whose respect you most covet. Unlike the forgiving grading curves of undergrad, which stretched from lowly B+ to unremarkable A+, desirable grades in law school are in limited supply, and you are competing for them against the people with whom you spend most of your waking hours. Remember also that, like you, these people have probably selected into this profession because of a yen for confrontation and an ability to work hard.

But another characterization has it like this: you have to go to class for between 12 and 17 hours per week, with no other responsibilities. To prepare, you will have to read approximately 25-50 pages per class. In the one-in-twenty chance that you get cold-called and can’t remember every single detail of the case, you can just access one of the many overzealous outlines floating around your school and grow comfortable with CONTROL+F and bullshitting—skills that, incidentally, are useful for any attorney.

So Which Experience Will I Have?

The answer to this question is that it’s largely up to you. If you are a welter of insecurities who regards law school success as the paramount test of intelligence and worth, then you might have the former experience. To ensure that the former perspective doesn’t overtake the latter reality, try to keep the following in mind:

  • Take law school seriously: it is a professional school, and how well you do will likely shape your career prospects. But that doesn’t mean that you can’t screw up...
  • ... And don’t be afraid to screw up: the first time a student in your section gets cold-called and doesn’t know the answer, visceral group mortification sets in. Every time after that point is pretty uneventful, because half the class is on Facebook or shopping online.
  • Treat yourself kindly: go to the gym, eat well, avoid forming bad habits, get lots of sleep, have fun, and relax. I learned this lesson the hard way: as a 1L, I took to pouring myself a fluorescent yellow glass of Mountain Dew each night and setting it beside my bed. Why? Because I knew I would be too exhausted to go make coffee in the morning, and I needed a kick to get myself out of bed. I thought, “I can rest when I am a 2L!” until a mentor encouraged me to sleep more regularly and more often, eat better, and exercise. I was doubtful at first, but another paradox that I was happy to resolve was that my comprehension and attitude improved with more leisure and relaxation time and better self-care.
  • Maintain your values: decide why you are in law school and stick to it. This doesn’t mean you can’t grow or change your mind, but resist the urge to jump through hoops just because your peers are doing so. Law students slobber over honors like clerkships and law review, but these might not be right for you, might not matter for your career, and might just make you miserable. There are things in life worth suffering and striving for, but make sure they’re important to you before you commit to them.
  • Your classmates are going to be your future colleagues, so get to know them through study groups, clubs, and other extracurricular activities.

If you’ve been through law school, what other tips or suggestions do you have? Let us know in the comment section. If you’ve got questions, let us know too.

  • Conor Ahern

    Conor Ahern


    Conor works as a civil rights attorney for the City of New York, and has been moonlighting as an LSAT tutor for two years. Immediately following law school, he worked as a Ford Fellow at the ACLU’s Women’s Rights Project. He enjoys reading fiction and making bad puns. He is a graduate of the University of Virginia and of Harvard Law School.

  • Featured image: Markus Spiske

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    As part of a new series here in the 7Sage blog, we've asked our community leaders (Mentors, who were selected from among their peers for their outstanding contributions and character, and Sages, who are community leaders who scored above 170 on the LSAT) to answer a series of questions and provide us with their LSAT wisdom.

    This series is just a sampling of the kind of wisdom ready at hand to anyone in our Discussion Forums.

    What's the biggest myth about the LSAT?

    Mentor Sam “That 2-3 months is plenty of time for a 170+ score. It's not, unless you're a genius or have somehow mastered logic prior to ever being exposed to the LSAT.”


    Sage Alex “In my opinion, the biggest myth about studying for the LSAT is that you should only expect to improve your score by 10 points.  This claim is not only false, but very detrimental to future law students. Relaying to people that 10 points is all you should expect to gain from studying deters test-takers from devoting enough time to study for the LSAT until they reach their goal score.”


    Mentor Brett “In my mind the 2 biggest myths about studying for the LSAT are that you only have to study for 3 months and that you won’t see improvement in any one section. This is a long test, and the skills that it tests aren’t things that you can truly learn and master in 3 months for most people. It’s one where it may take you 3-5 months just to get through the curriculum and then from there you take another 3-6 in taking tests. But the process is worth it in the end. “Also, the assumption that ‘You can’t improve on RC, LG, or LR’ is completely false.  Everyone taking this test is different but all of the sections are exactly the same at their base; the questions all test a set of skills and are created by professionals who can exploit the psychology of test takers. All you have to do is learn the skills, avoid the pitfalls, and be able to be able to do this efficiently and confidently.”


    Sage Allison “One myth about this test is that it's just an input, output equation with how many hours you put in leading to a particular score. That is part of the equation, but cranking through PTs without good reflection is bad practice. You absolutely must scrutinize your comprehension, question your thought processes, and accept that you have erroneous ways of reasoning that need to be corrected in order to excel on this test. Do your homework, but reflect on your methods and spend time to diagnose your weaknesses.”


    Mentor Daniel “That high scores are rare and achievable only among those who are great at tests, geniuses, or savants destined for Harvard. I was once told by an instructor of a test-prep company that I could reasonably expect a score five points higher than my diagnostic. I'm now sitting at an average score increase more than four times that, and I firmly believe at least most should be aiming for a score fifteen-to-twenty points higher than their diagnostic. It takes time, but a score that much higher is doable.”

    Featured image: theilr

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    Ok. LSAT Logical Reasoning—you got this! You're logical. You're reasonable. You destroy (or repair) arguments all the time on Twitter or Tumblr. You've even done some debate in high school or college. How hard can it be?

    And then you take your first PT after completing the Core Curriculum.

    First few questions are a little wordier than you'd like, but you feel like you got this. You get to question 4 and ...

    ... Pff ... You know you got this! Just had to get up to speed, that's all. Things are going fine until ...

    ... You come to a Necessary Assumption question with a really unattractive answer choice that just nags at you. Why did they even bother putting that one in there? And then there's this other answer choice that sounds like EXACTLY what the argument needs ... But is it the right answer choice? And then ...

    ... A Most Strongly Supported question with an answer choice that seems to be just soft enough, just specific enough, just irrefutable enough to fit the bill for the right answer choice. It's got all the hallmarks of a right answer choice for MSS. So ... You ... Slowly ... Circle ... the AC ...

    ... And run smack dab into a Parallel Flaw question that takes up the entire left hand side of the page. So you find the flaw in the stimulus ... And then you try to remember if you're supposed to map out the logic in the Answer Choices ... Or is that for the other Parallel question type? You thought you HAD this ...

    ... And even though you're on FIRE with the next 3 questions, finding those main points, honing in on those flaws in the support structure, naming those assumptions, you're still thinking about that question 2 pages back.

    You finish the PT and you question your whole existence for a good 10 minutes.

    Then you remember ...

    ... You've got dreams to grab ahold of.

    So you pick yourself up and get ready for some Blind Review.

    And you think to yourself ...

    And maybe it would.

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    You worked hard in undergrad and now you're getting ready to start applying.  But do you know how hard is it to get into law school?  Or more importantly, how do you get into a good law school?

    The one thing you need to know about how to get into law school

    The answer can be summed up in four letters. LSAT.  You need to demolish the LSAT.  That's the one thing you need to know.

    In the topsy-turvy world of law school applications, LSAT is king.

    Isn’t GPA / Personal Statement / Recommendations / Whatever More Important for Getting into Law School?

    What about GPA?  First, your GPA is pretty much set.  Even if you still have another year of grades before you send in your applications, the A in GPA will ensure that the impact of your best efforts won't have much of an impact.  Secondly, even though most people agree that GPA is the second most important admissions criteria, it is not nearly as important as the LSAT.  A rule of thumb many students use is +1 LSAT point = +0.1 GPA.  It's reasonably common for students to improve 10 points on the LSAT with 4 months of studying.  Good luck bringing your GPA from 3.3 to 4.3 with 4 months of studying :D.

    What about Personal Statements, Recommendations, Extracurriculars, Job Experience and Interviews? They make a difference, but not that much.  If you have a lame-duck recommendation or a douchey personal statement, it can tank you.  If you were the President of your home country it'll really help.

    Most of the time these aren't going to make a big difference.  At least not compared to the LSAT.  Most of the time, you should put effort into making these shine only after you've taken the LSAT.

    WTF?  Why do law schools care so much about the LSAT?

    There are some obvious reasons, and at least one non-obvious one.

    Among the obvious reasons is that the LSAT isn't subject to grade inflation/deflation and competitiveness of different colleges.  For example, are a 4.33 GPA from Greendale Community College and a 4.33 GPA from MIT equally impressive?  Probably not, at least not academically :)  The LSAT acts as an equalizer.

    Another reason is that the LSAT tests abstract logic and reasoning, as well as time pressured reading comprehension skills.  Both of these are extremely important in law school when you grind through endless readings and try to pull out the arguments and implications.

    Here's one non-obvious reason: US News & World Report Rankings.

    They rank 200 or so US law schools using a bunch of metrics.  One of these metrics is the LSAT.  The better the median LSAT score of a school's students, the better the school's rankings.  The better the school's rankings, the higher the prestige.  More prestige lets the school attract better law students (prestige is like crack for law students), and get a higher median LSAT.  The circle of LSAT continues.

    What now?

    If you want to go to a good law school, now you know how to get in. You study your ass off for the LSAT.  Sign up for a free trial to get started on LSAT prep.  Or jump right in and prep for LSAT with a full 7Sage Course (there is a 14 day money-back guarantee in case you change your mind).

    Featured image: How to Get Into Law School - LSAT (photo attribution David Ortez)


    If you're taking practice LSAT PrepTests, then you need to simulate the test environment.  It is crucial so that you are ready for the real thing.

    It’s really important to experience testing with a simulated proctor so you’re not thrown off on the test day by a person announcing a five minute warning or by the lack of time between the first three sections. In a test that’s as psychological as the LSAT, practicing dealing with those things is critical. - Robyn B.

    If you're using our video proctor with real LSAT instructions, that's a great start!  But setting up an LSAC approved timer, and finding a place with just the right amount of ambient noise can be a bit of a pain...

    So we made an LSAT Proctor App for iPhone/iPad/iPod just for you ;)  We designed it to be easy to use, while including all the features we knew were important from teaching thousands of LSAT students.  Simulate the LSAT test environment perfectly, anytime, anywhere.

    This includes procedurally generated distraction noises, turbo mode, realistic virtual timer, five minute warnings, real instructions, and more.  If you like it, give us a great rating!  It'll encourage us to keep making great LSAT tools available for free :D

    Get Proctor App


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