Complete Guide to GRE Geometry with Examples

You’ve taken countless practice tests, vacuumed up over 1000 vocabulary words, and committed the most important GRE geometry formulas to memory. You walk into the test center, knock out the first verbal section, and then you see the first quant problem: a square inscribed in a circle that is inscribed in a larger square. You freeze up. GRE geometry formulas pinball around your brain but nothing rings a bell. The clock ticks imperiously as you’re forced to move on to the next question.

Many a test taker can relate to this experience because GRE geometry questions can oftentimes be difficult to master. In this post, I will break down what you need to know for the GRE geometry section. I’ll also provide you with strategies to help boost your performance while discussing techniques that are often used but that might be hurting you.

I’ll also discuss how GRE geometry formulas are essential but are the final step in a problem-solving process, especially for the more difficult problems.

Finally, to help you put the concepts in this post into practice, I’ll include GRE geometry questions with solutions. First, though, let’s examine exactly what geometry topics you need to master for test day.

Here are the topics we’ll cover:

• Which Geometry Topics Are Tested on the GRE?
  1. Triangles
  2. Quadrilaterals
  3. Circles
  4. Lines and angles
  5. Coordinate geometry
  6. Three-dimensional shapes
  7. Regular polygons
• Must-Know GRE Geometry Formulas
  1. Triangles
  2. Quadrilaterals
  3. Circles
  4. Lines and angles
  5. Coordinate geometry
  6. Three-dimensional shapes
  7. Regular polygons
• The Key to the Geometry Question Type – Pattern Recognition
  • Example Problem: Perimeter Region
  • Example Problem: Isosceles Right Triangle
• Strategies for Solving GRE Geometry Problems
  1. Label Everything
  2. Identify Trigger Clues in Questions
  3. Focus on Relationships, Not Numbers
  4. Create a Pattern Cheat Sheet
  5. Review Incorrect Answers Thoughtfully
• Practice Questions: Test Your Geometry Skills
  • GRE Geometry Practice Question Solutions
• Avoid These Common Geometry Mistakes
  • Misleading Diagrams
  • The Little Things
  • Focusing on Formulas, Not Patterns
• Conclusion: Conquer GRE Geometry with Confidence
• FAQs
  • How many geometry questions are on the GRE?
  • What geometry topics appear most frequently on the GRE?
  • How does GRE geometry differ from high school geometry?
• What’s Next?

Which Geometry Topics Are Tested on the GRE?

The main geometry topics tested on the GRE are listed below. You can think of this list as your checklist for studying geometry. So, if you need to see which geometry concepts you still need to work on, use the following as your guide. Note that the GRE does not test trigonometric functions (e.g., sine, cosine, etc.)

1. Triangles

• 45-45-90 triangles (isosceles right triangles)
• 30-60-90 triangles
• Equilateral triangles
• Similar triangles
• Pythagorean triples: 3:4:5, 5:12:13, 7:24:25

2. Quadrilaterals

• Squares and rectangles (perimeter and area)
• Parallelograms
• Rhombuses
• Perimeter and area of these figures
• Properties of quadrilaterals

3. Circles

• Radius, diameter, circumference, area
• Arc length, sector area
• Inscribed figures (triangles or quadrilaterals, typically)

4. Lines and angles

• Parallel and perpendicular lines, intersecting lines
• Exterior angles
• Supplementary and complementary angles

5. Coordinate geometry

• Slope, slope formula
• x- and y- intercepts
• Intersecting lines
• Parabolas (typically only on harder sections)
• Geometric shapes displayed on the coordinate plane

6. Three-dimensional shapes

• Surface area and volume
• Cubes and prisms
• Right circular cylinders
• Spheres (rare and typically only on harder sections)

7. Regular polygons

• Pentagon
• Hexagon
• Properties of regular polygons
• Total degree measure

TTP PRO TIP:

Know the 7 major geometry topics tested on the GRE and their subtopics.

Must-Know GRE Geometry Formulas

Below are GRE geometry formulas. Some are basic formulas, and others are more advanced. But all fall into the must-know category. I’ve aligned them to pair with the seven topic categories above.

1. Triangles

45-45-90 Triangle:

• Sides are in ratio x : x : x√2
• If leg is x, hypotenuse is x√2
• Area = side² / 2

30-60-90 Triangle:

• Sides have ratio x : x√3 : 2x
• If shortest side is x, then hypotenuse is 2x, and third side is x√3
• Area = (b ✕ h)/2, where b = the base and h = the perpendicular height

Equilateral Triangle:

• Height (h) = (side length × √3)/2
• Area = (side length² × √3)/4

2. Quadrilaterals

Rectangle:

• Area = length × width
• Perimeter = 2(length + width)

Square:

• Area = side²
• Perimeter = 4 × side

Trapezoid:

• Area = h(a + b)/2, where h is height and a and b are parallel sides

3. Circles

Basics:

• Diameter = 2r, where r is the radius
• Circumference = 2 π r
• Area = π r²
• π ≈ 3.14

Sectors:

• Arc length = (θ/360°) × 2 π r, where θ is the central angle in degrees
• Sector area = (θ/360°) × π r²

4. Lines and angles

• For parallel lines, m₁ = m₂, where m₁ and m₂ are the slopes of the two lines
• For perpendicular lines, m₁ ✕ m₂ = -1
• Adjacent angles on a line = 180°
• Complete circle = 360°

5. Coordinate geometry

• Slope = (y₂ – y₁)/(x₂ – x₁)
• Distance = √[(x₂ – x₁)² + (y₂ – y₁)²]
• Midpoint of line segment = ((x₁ + x₂)/2, (y₁ + y₂)/2)

6. Three-dimensional shapes

Rectangular Solid/Box:

• Volume = length × width × height
• Surface Area = 2(lw + lh + wh)

Cube (special case):

• Volume = side³
• Surface Area = 6 × side²

Cylinder:

• Volume = π r² h
• Surface Area = 2π h + 2π r²

Sphere:

• Volume = (4/3)π r³
• Surface Area = 4π r²

7. Regular polygons

• Total degree measure = [(n-2)/n] * 180, where n = number of sides

TTP PRO TIP:

Commit the geometry formulas to memory.

Let’s now consider a useful approach to decoding geometry problems: pattern recognition.

The Key to the Geometry Question Type – Pattern Recognition

Think of the GRE geometry question as a mystery that has to be solved. The first step to solving the mystery is getting a key, which unlocks a door. Behind that door is the answer. The key is the formula. The mystery is a pattern that you must recognize. Unless you recognize that pattern, the formula is of no use.

Example Problem: Perimeter Region

What is the perimeter of the region shown above? (Note: figure not to scale)

Solution:

Notice there are two sides whose lengths are unknown. First, there’s the horizontal side right below the ‘6’. Next, there’s the vertical side at the extreme right of the figure. Knowing the perimeter formula inside and out won’t help at this point.

How do you unlock the mystery; how do you answer the question correctly?

A problem like this is an example of the type of skill the GRE wants to test. It requires you to do some critical thinking before you use the formula. You can’t just plug and chug because it appears that you don’t have enough information.

Above, the pattern to discern is that the shape can be made into a rectangle without altering the lengths of the existing sides. To see how, let’s take a look at the figure below.

It’s now easy to see that the two unknown lengths are of no consequence. The figure is now seen to be a rectangle, with length 14 and width 10. The perimeter is calculated by adding the four side lengths: 14 + 10 + 14 + 10 = 48.

Example Problem: Isosceles Right Triangle

In triangle ABC, angle B equals 90° and AB equals BC. If the hypotenuse, AC, has a length of 4 units, find the area of triangle ABC.

• 2
• 2√2
• 4
• 4√2
• 8

Solution:

Knowing that the formula for the area of any right triangle is (b ✕ h)/2 doesn’t actually help you solve the mystery. On GRE triangle questions, you’ll need to recognize that a right triangle with two equal sides will always be a 45:45:90 triangle, also called an isosceles right triangle. (See below.) We know that the sides of a 45:45:90 triangle have a ratio of x : x : x√2.

Because side AC is not one of the equal sides, it must be the hypotenuse. This corresponds to the x√2 in the diagram below.

Therefore, x√2 = 4. We need to solve for x, so we divide both sides by √2, obtaining x = 4/√2. We must rationalize the denominator, so we multiply the fraction by √2/√2, obtaining x = 4√2 / 2, which simplifies to x = 2√2. Now we’ve solved the mystery by finding the base and height (the key to the problem). To unlock the door, we can plug in the base and height to calculate the area.

Base = 2√2

Height = 2√2

Area = (2√2 x 2√22) / 2

Area = 4

Answer: C

The problem we have just solved is typical for a GRE geometry question. The 45:45:90 triangle is the most common triangle presented on the GRE. The pattern recognition usually involves noticing that the two equal sides have a 90-degree angle between them. From there, you can easily solve for the hypotenuse. Or, if the hypotenuse length is given, as happened in this problem, then you use the fact that x√2 is the formula for the hypotenuse, where x is the side length of either side of the triangle.

Another common way to test you on a 45-45-90 triangle is to present you with a square and ask you to calculate the length of the square’s diagonal. If you recall the pattern that a square has four equal sides with a 90-degree angle between each of them, then you can see that the square’s diagonal splits the square into two identical 45-45-90 triangles. Thus, the diagonal of the square is the hypotenuse of each of the triangles.

TTP PRO TIP:

Be familiar with the 45-45-90 triangle and its many variations.

Strategies for Solving GRE Geometry Problems

How do you get better at GRE geometry? The strategies below will unlock the answer.

1. Label Everything

The moment you see a diagram, start labeling the information provided: side lengths, angles, etc.. For example:

• If two sides are equal, mark them as “x.”
• If the diagram shows a perpendicular intersection, add a 90° angle mark.
• For inscribed shapes, identify radii, diameters, or shared points.

This organization often makes patterns pop out. Even if the problem doesn’t require all the labels, the process helps clarify your thinking.

2. Identify Trigger Clues in Questions

Certain phrases in GRE geometry problems serve as hints for specific patterns. Train yourself to recognize these clues:

• “Isosceles right triangle” → Think 45:45:90
• “Inscribed shapes” → Relate the radius/diameter of the circle to the inscribed figure.
• “Parallel lines” → Look for corresponding or supplementary angles.

The more familiar you are with these triggers, the faster you’ll recognize what the problem is really asking.

3. Focus on Relationships, Not Numbers

GRE geometry problems are less about crunching numbers and more about understanding relationships. For instance:

• If dealing with inscribed shapes, focus on how the circle’s radius relates to the square’s side length.
• For intersecting lines, consider angles and slopes over exact measurements.

By thinking conceptually, you’ll spend less time bogged down by calculations.

4. Create a Pattern Cheat Sheet

As you practice, keep a running list of common patterns/relationships. Here are some examples::

• “Right triangles inscribed in circles always have the hypotenuse as the diameter.”
• “The diagonals of a rectangle bisect each other and are equal in length.”
• “If the product of two lines’ slopes is -1, then the lines are perpendicular.”

Review your GRE geometry cheat sheet regularly to internalize these patterns.

5. Review Incorrect Answers Thoughtfully

When you miss a question, ask yourself:

• “What pattern did I fail to see?”
• “Could I have solved this faster by simplifying the diagram?”

This reflective practice builds the intuition needed to spot similar patterns on future questions.

TTP PRO TIP:

When you’re studying geometry, use a variety of strategies to discern patterns and help you remember approaches to solving problems.

Practice Questions: Test Your Geometry Skills

Now it’s your turn! Below are five GRE geometry practice questions that cover the range of concepts we’ve discussed. The questions are roughly in order of difficulty from easiest to hardest.

Question 1

Triangle ABC is positioned in the coordinate plane such that one vertex lies at the origin (0, 0), and another vertex is located at (5, 12). What is the area of triangle ABC?

• 24
• 30
• 48
• 65
• Cannot be determined by the information provided.

Question 2

In the figures above, quadrilateral ABCD is a square, and quadrilateral EFGH is a rectangle where EF > FG. If the perimeters of both quadrilaterals are equal, which of the following statements must be true?

• AB > EH
• 2AB > EF
• The area of ABCD > EFGH

• I only
• III only
• I and II
• I and III
• I, II, and III

Question 4

The perimeter of semicircle C above is 5π + 10. What is the area of the semicircle?

• 5π
• 10π
• 25π/2
• 25π
• 50π2

GRE Geometry Practice Question Solutions

1. The pattern here is recognizing that triangle ABC is a right triangle. Line BC is perpendicular to the x-axis, forming a right angle at BCA because points B and C both have the x-coordinate 5. A triangle’s base and height always meet at a right angle. So, AC (base) = 5; BC (height) = 12, and Area = (12×5)/2 = 30. Answer: B)
2. The first condition is true. The sides of the square must be longer than the short side of the rectangle because the perimeters of the two shapes are equal. The second condition can be disproved using AB = 2 and EF = 3. The last condition requires pattern recognition: the maximum area of a quadrilateral given a fixed perimeter is a square. Play with different numbers to spot this! Answer: D)
3. Both of these triangles are right triangles because their sides are Pythagorean ratios. Triangle ABC corresponds to a 3:4:5 and DEF a 5:12:13. Therefore, both contain a 90-degree angle, which is the largest angle in both triangles. Answer: C).
4. The perimeter of the semicircle equals half the circumference + twice the radius, giving us: π r + 2r, or r π + 2r. This aligns with 5π + 10, with r corresponding to 5. Given the radius is 5, the area of the semicircle is π r² /2 = 25π/2 Answer: C).
5. Hard GRE geometry questions like this take lots of practice. Think of each vertex as a possible option when creating a triangle. With an octagon, you have 8 vertices from which to choose the three vertices. Therefore, we can use the following combination formula:
   8! / (3! 5!)
   With a pentagon, there are 5 vertices to choose from a total of 8:
   8! / (5! 3!)
   Notice that the denominator for each is the same, with only the ordering of 3! and 5! different. Answer: C)

TTP PRO TIP:

Use the detailed solutions to learn the patterns and to gain insight into solving GRE geometry problems.

Avoid These Common Geometry Mistakes

Now that you know the strategies and are memorizing the must-know formulas, there are still common geometry mistakes that can trip you up on test day.

Misleading Diagrams

Things are not what they seem–at least when it comes to geometry diagrams.

When you see a diagram, put on your skeptic’s cap! A shape might look like a square but unless you can prove it, you can’t assume the angles equal 90 degrees. Oh, and you can’t even assume the shape is a square just because all the sides appear the same length.

Unless you want to fall for the trap, always back up your assumptions with information in the question and facts provided in the diagram.

The Little Things

Trying to decipher the pattern to these questions takes a lot of brain power. So, it’s easy to make careless mistakes. Luckily, for many students, these mistakes fall into several predictable categories. So look out for them!

• Confusing radius and diameter
• Confusing circumference and area
• Confusing area with perimeter
• Forgetting to divide base x height by 2 when calculating the area of a triangle.

Focusing on Formulas, Not Patterns

I mentioned this earlier but it’s super important: don’t get too caught up with GRE geometry formulas. They only help you at the final step of what may be a long journey. For instance, in question #5 above, you might conclude that you need to memorize the combination formula when figuring out how many smaller shapes you can make from a larger shape.

Instead, understand the logic behind the formulas. This will help you with similar questions that are different enough from each other that replicating what you did to solve one question won’t quite work on the other.

TTP PRO TIP:

Being aware of the common types of geometry mistakes, you can recognize when you make them and then learn from them so you don’t make them again.

Conclusion: Conquer GRE Geometry with Confidence

At the end of the day, you will still need to know geometry facts and formulas for the GRE. But remember, they are part of a much bigger conceptual picture.

So, start analyzing shapes—circles, trapezoids, triangles-–see how they are put together. Understanding the various properties of shapes will help you to be able to better identify patterns in the questions.

Remember: recognizing patterns in GRE geometry isn’t just a skill—it’s the key to unlocking even the toughest questions. Start practicing these strategies, keep a lookout for common clues, and focus on the relationships behind the numbers. The more you train your brain to spot patterns, the less intimidating geometry will feel—and the faster you’ll solve problems on test day!

FAQs

How many geometry questions are on the GRE?

You can expect to see 3-5 geometry questions per quantitative section on the GRE, making up roughly 20% of all quant questions. On a typical test with two scored quantitative sections, this means you’ll encounter 6-10 geometry questions total. However, the exact number can vary between tests. These questions are usually distributed across different difficulty levels, with more challenging geometry problems appearing in the second section if you performed well in the first.

What geometry topics appear most frequently on the GRE?

The distribution of geometry topics on the GRE follows a fairly consistent pattern:

• Triangles (30-35% of geometry questions)
  • Special emphasis on right triangles, especially 30-60-90 and 45-45-90 triangles
  • Triangle properties and area calculations
• Circles (25-30% of questions)
  • Circle properties, area, and circumference
  • Inscribed figures and arc measures
• Coordinate Geometry (20-25% of questions)
  • Distance and midpoint calculations
  • Slope calculation and linear equations
• Quadrilaterals (15-20% of questions)
  • Rectangles, squares, and their properties
  • Area and perimeter calculations
• Other topics (5-10% of questions)
  • Three-dimensional figures
  • Regular polygons
  • Compound shapes

How does GRE geometry differ from high school geometry?

The GRE takes a distinctly different approach to testing geometry compared to high school math classes. Understanding these differences is crucial for effective preparation:

Key differences:

1. Focus on Problem-Solving vs. Proofs
   • High School: Emphasis on formal proofs and theoretical understanding
   • GRE: Emphasis on quick, practical problem-solving and pattern recognition
   • Why it matters: You need to develop rapid solution strategies rather than detailed proofs
2. Limited Formula Set
   • High School: Covers extensive formulas and theorems
   • GRE: Tests a focused set of fundamental formulas
   • Why it matters: You can concentrate on mastering a smaller set of essential tools
3. Question Format
   • High School: Straightforward questions testing single concepts
   • GRE: Questions combining multiple concepts in novel ways
   • Why it matters: Practice recognizing how different concepts interact

What’s Next?

Now that you’ve gotten a strong grasp on the fundamentals of GRE geometry, let’s outline your next steps. That way, you’ll be fully prepared for test day.

Immediate Next Steps:

1. Take a Diagnostic Test
   • Use the practice questions above to assess your current level
   • Identify which geometry topics need the most work
   • Time yourself to establish your baseline speed
2. Create Your Formula Sheet
   • Download or write out all the formulas covered in this guide. This GRE math formulas cheat sheet can help too.
   • Focus first on memorizing the high-frequency formulas
   • Create flashcards for quick daily review
3. Start Pattern Recognition Practice
   • Begin with easier questions to build confidence
   • Gradually increase the difficulty as you recognize common patterns
   • Keep a log of question types that consistently challenge you

If you’re still feeling overwhelmed or uncertain about GRE geometry, consider signing up for our guided GRE prep program where you’ll get tons of geometry practice and more.

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