GRE Geometry: Formulas, Figure Rules and 5 Practice Questions
Key Takeaways
- Geometry is one of four GRE Quant content areas: lines and angles, triangles, polygons, circles, 3D figures, area, perimeter, volume and the Pythagorean theorem.
- You never write a proof: ETS states that constructing proofs is not tested, and there is no trigonometry.
- ETS says geometric figures are not necessarily drawn to scale, so answer from what is given, not from how the figure looks.
- Coordinate systems and data graphs are drawn to scale, so you can read values off them.
- The 30-60-90 and 45-45-90 ratios, the Pythagorean theorem and the circle formulas solve most GRE geometry questions.
Table of Contents
- What geometry is tested on the GRE?
- How is the GRE Quant section set up?
- Which geometry formulas should you know?
- Can you trust the figures on the GRE?
- What are the common GRE geometry traps?
- GRE geometry practice questions with solutions
- Question 1 (multiple choice, select one)
- Question 2 (Quantitative Comparison)
- Question 3 (numeric entry)
- Question 4 (multiple choice, select one)
- Question 5 (multiple choice, select one or more)
- How should you practise GRE geometry?
Geometry is one of the four content areas of GRE Quantitative Reasoning. It covers lines and angles, triangles, quadrilaterals and other polygons, circles, three-dimensional figures, area, perimeter, volume and the Pythagorean theorem. You are never asked to write a proof. Most GRE geometry questions come down to a handful of formulas and one habit: never trust how a figure looks.
This guide lists what ETS says the geometry content covers, the formulas to know, the rules about figures, and five practice questions with full solutions.
The five practice questions are written by Galvanize in the official GRE formats. They are not ETS questions. For the whole Quant section, read our GRE Quant guide; for equations, slopes and graphs, see GRE algebra.
What geometry is tested on the GRE?
ETS lists the geometry topics on its Quantitative Reasoning content page:
- parallel and perpendicular lines
- circles
- triangles, including isosceles, equilateral and 30°-60°-90° triangles
- quadrilaterals and other polygons
- congruent and similar figures
- three-dimensional figures
- area, perimeter and volume
- the Pythagorean theorem
- angle measurement in degrees
ETS states that the ability to construct proofs is not tested, and that Quant does not include trigonometry. Coordinate geometry (slopes, intercepts and graphs of lines) sits under algebra in the ETS list, though it often appears alongside these topics.
How is the GRE Quant section set up?
The ETS test structure page sets out two Quant sections: 12 questions in 21 minutes, then 15 questions in 26 minutes, for 27 questions in 47 minutes. You get an on-screen calculator. ETS does not publish how many questions come from geometry.
Which geometry formulas should you know?
| Shape or idea | Formula or fact |
|---|---|
| Angles on a straight line | add up to 180° |
| Angles in a triangle | add up to 180° |
| Interior angles of any polygon | sum = (n − 2) × 180°, where n is the number of sides |
| Area of a triangle | ½ × base × height |
| Pythagorean theorem | a2 + b2 = c2 for a right triangle with hypotenuse c |
| Common right triangles | sides 3-4-5 and 5-12-13, and their multiples such as 6-8-10 |
| 45°-45°-90° triangle | sides in the ratio 1 : 1 : √2 |
| 30°-60°-90° triangle | sides in the ratio 1 : √3 : 2 |
| Rectangle | area = length × width; perimeter = 2(length + width) |
| Parallelogram and trapezoid | area = base × height; trapezoid area = ½(sum of parallel sides) × height |
| Circle | circumference = 2πr; area = πr2 |
| Arc and sector | a central angle of x° takes x/360 of the circumference and of the area |
| Rectangular box | volume = lwh; surface area = 2(lw + lh + wh); longest diagonal = √(l2 + w2 + h2) |
| Cylinder | volume = πr2h |
Two more facts come up often. In an isosceles triangle, the angles opposite the equal sides are equal. In similar figures, corresponding angles are equal and corresponding sides are in the same ratio, so if the sides scale by k, the area scales by k2 and the volume by k3.
Can you trust the figures on the GRE?
Not on their own. The mathematical conventions on the ETS Quantitative Reasoning page say geometric figures such as lines, circles, triangles and quadrilaterals are not necessarily drawn to scale, and that you should base your answers on geometric reasoning, not on estimating by sight. You can assume that lines shown as straight are straight and that points shown on a line lie on it, but an angle is only a right angle if it is marked or stated.
The exception is useful: ETS says coordinate systems, such as xy-planes and number lines, and data graphs are drawn to scale. So you can read values off a graph of a line, but not off a triangle.
This matters most in Quantitative Comparison. If a figure looks like an equilateral triangle but nothing says it is one, the honest answer may be that the relationship cannot be determined.
What are the common GRE geometry traps?
- Reading the figure instead of the facts. An angle that looks like 90° is not 90° unless the question says so.
- Mixing up radius and diameter. A circle with diameter 10 has radius 5 and area 25π, not 100π.
- Using the slanted side as the height. The height of a triangle or parallelogram is perpendicular to the base.
- Forgetting that area scales by the square. Doubling every side of a square multiplies its area by 4, not 2.
- Missing a hidden right triangle. Diagonals of rectangles, heights of equilateral triangles and radii drawn to a chord all create right triangles you can use.
GRE geometry practice questions with solutions
These five practice questions are written by Galvanize in the official GRE formats. Try each one before reading the solution.
Question 1 (multiple choice, select one)
A right triangle has legs of length 6 and 8. What is its perimeter?
(A) 20 (B) 22 (C) 24 (D) 28 (E) 48
Answer: (C) 24. The hypotenuse is √(62 + 82) = √100 = 10, a multiple of the 3-4-5 triangle. The perimeter is 6 + 8 + 10 = 24. Choice (E) is the two legs multiplied together, a common slip.
Question 2 (Quantitative Comparison)
A circle has radius 3.
Quantity A: the circumference of the circle Quantity B: the area of the circle
(A) Quantity A is greater. (B) Quantity B is greater. (C) The two quantities are equal. (D) The relationship cannot be determined from the information given.
Answer: (B). The circumference is 2π(3) = 6π and the area is π(3)2 = 9π. Comparing the numbers, 9π is greater than 6π. (The two are equal only when the radius is 2.)
Question 3 (numeric entry)
What is the measure, in degrees, of each interior angle of a regular hexagon?
Answer: 120. The interior angles of a six-sided polygon add up to (6 − 2) × 180° = 720°. A regular hexagon has six equal angles, so each one is 720° ÷ 6 = 120°.
Question 4 (multiple choice, select one)
A 30°-60°-90° triangle has a hypotenuse of length 10. What is its area?
(A) 25 (B) 25√3 / 2 (C) 25√3 (D) 50 (E) 50√3
Answer: (B) 25√3 / 2. The sides are in the ratio 1 : √3 : 2, so with a hypotenuse of 10 the short leg is 5 and the long leg is 5√3. The legs are perpendicular, so the area is ½ × 5 × 5√3 = 25√3 / 2, about 21.65. Choice (C) forgets the ½.
Question 5 (multiple choice, select one or more)
A rectangular box measures 2 by 3 by 4. Which of the following statements are true? Indicate all such statements.
(A) Its volume is 24. (B) Its surface area is 48. (C) Its longest diagonal is √29. (D) Its largest face has an area of 12.
Answer: (A), (C) and (D). Volume = 2 × 3 × 4 = 24, so (A) is true. Surface area = 2(2×3 + 2×4 + 3×4) = 2(6 + 8 + 12) = 52, so (B) is false. The longest diagonal is √(22 + 32 + 42) = √29, so (C) is true. The faces have areas 6, 8 and 12, so (D) is true. You must select all three to get the credit.
How should you practise GRE geometry?
- Learn the table above until you can write it from memory. The test gives you no formula sheet, so recall has to be instant.
- Redraw the figure. Sketch it on your scratch paper with the given facts labelled, and add any right triangles you can find.
- Write down what is given, not what it looks like. This one habit protects you from most figure traps.
- Practise under timing. At just over 1.7 minutes a question, knowing the 30°-60°-90° ratios cold saves real time.
To see how your geometry holds up across the whole section, take the free GRE mock test. For more practice in every content area, try our GRE sample questions with answers. To turn a practice score into a target, use the GRE score calculator.
If you are taking the GRE for a master’s abroad, our counsellors can help you plan your applications around your target score. Talk to an admissions counsellor.
Frequently Asked Questions
What geometry is on the GRE?
ETS lists parallel and perpendicular lines, circles, triangles, quadrilaterals and other polygons, congruent and similar figures, three-dimensional figures, area, perimeter, volume, the Pythagorean theorem and angle measurement in degrees.
Are GRE geometry figures drawn to scale?
Not necessarily. ETS says geometric figures such as lines, circles, triangles and quadrilaterals are not necessarily drawn to scale, so base your answer on the facts given. Coordinate systems and data graphs are drawn to scale.
Does the GRE give you geometry formulas?
No. You need to know the formulas for area, perimeter, volume, circles and special right triangles from memory.
Is trigonometry on the GRE?
No. ETS states that Quantitative Reasoning does not include trigonometry. Special right triangles such as 30-60-90 are tested through their side ratios.
Do I need to write proofs on the GRE?
No. ETS states that the ability to construct proofs is not tested.
What are the most useful GRE geometry formulas?
The Pythagorean theorem, the area of a triangle, the angle sum of a polygon, the circle formulas, and the side ratios of 30-60-90 and 45-45-90 triangles.
How many geometry questions are on the GRE?
ETS does not publish a count per content area. Geometry can also appear inside coordinate geometry and word problems.
How do I handle a geometry Quantitative Comparison?
Write down only what is stated, not what the figure suggests. If the facts allow more than one shape, test the extremes; if the result changes, the relationship cannot be determined.
What is the interior angle of a regular polygon?
The interior angles of an n-sided polygon add up to (n – 2) x 180 degrees. Divide by n for a regular polygon: each angle of a regular hexagon is 120 degrees.
How should I practise GRE geometry?
Learn the formulas until you can write them from memory, redraw each figure with the given facts labelled, look for hidden right triangles and practise under the section timing.
