GRE Geometry: Formulas and Sample Questions

In addition to arithmetic, algebra, and data analysis, geometry is one of the four major math courses tested on the GRE. Geometry problems on the GRE come in various forms. Some questions are multiple-choice, numerical entry, or quantitative comparison questions. Other test questions are unscaled graphs. Geometry covers 15% of the test. Practicing terms, solutions, and test samples are essential to obtaining a higher GRE score. This article covers formulas, concepts, and practice questions for the GRE Geometry test.

Are geometry topics included in the GRE quantitative reasoning?

Yes, GRE Quantitative Reasoning includes geometry topics that are not limited to lines and angles, triangles, area, perimeter, and other geometry-related topics.

What geometry concepts are included in the GRE?

Everything that is listed below includes the following geometry concepts that are included in the GRE:

  • Degrees.
  • Lines.
  • Angles.
  • Squares and Rectangles.
  • Triangles.
  • Pythagorean Theorem.
  • Circles.
  • Coordinate System.
  • Solid Geometry.

The following topics could be some of the questions that will be asked during the GRE Geometry examination. These concepts are either asked multiple times, or just a few times wherein the examinees will be providing their answers coming from the possible answers listed.

What are the GRE rules and formulas to remember in Geometry?

The GRE rules and formulas to remember in Geometry are as follows:

  • Angles and Lines

Note:

Straight: Any line that does not bend or curve

Horizontal: A line that runs parallel to something's top or bottom.

Vertex: Two lines that intersect

  • All Triangles

Area=12×bh

If a GRE geometry question asks for the area of the triangle, it will provide a way to find at least one base and the corresponding height.

  • Isosceles Triangles

One geometry rule related to this last fact concerns one elite category of triangles: the isosceles triangles. These are triangles with two equal sides.

  • Right Triangles

There are two sides touching at the right angle, and these are called legs. On the other hand, the longest side is called the hypotenuse.

Note:

There are two very special triangles that you have to understand for GRE geometry.

  1. 30-60-90 triangle
  2. 45-45-90 triangle (isosceles right triangle)
  • Rectangles

Area = length x breadth

Perimeter = 2 (l + b)

  • Rectangular Solids

Area = length x breadth

Perimeter = 2 (l + b)

  • Squares

All sides are equal. Each angle equals 90 degrees. Diagonals bisect each other at right angles.

Area = side * side

Perimeter = 4 * side

  • Trapezoids

A = 1/2 [L + L2] *

Perimeter = sum of the lengths of all the sides = L + L1 + L2 +L3

  • Circles

A circle’s perimeter is called the ‘circumference of the circle’.

Radius: The radius of a circle, the line segment joining the center to any point on the circle.

Chord: The chord is the line segment joining any two points on the circle.

Diameter: The diameter of a circle is the line segment joining any two points of the circle passing through the center.

Arc: A part of circumference is the arc of the circle.

Sector: An area bounded by the circumference (arc) at an angle at the center (as the vertex) is called sector.

SECANT is a line passing through the circle intersecting at 2 points.

TANGENT is a line passing through the circle intersecting at only 1 point.

Circumference of the circle is the measurement of the boundary edge of the circle.

C = 2 π r = π d, where π = 222/7 or 3.14 approximately

Area of the circle

A = π r2, where π = 222/7 or 3.14 approximately

‘r’ can be replaced by ‘d/2’

So A = πd2/4

  • Quadrilaterals

A quadrilateral has 4 straight lines and a close figure. The total sum of all interior angles of a quadrilateral is 360 degrees.

Square, rectangle, parallelogram, trapezium, and rhombus are all quadrilateral.

  • Parallelograms

Area = length x breadth

Perimeter = 2 (l + b)

Opposite sides, and angles are equal. Diagonals will bisect each other but it is not at right angles.

  • Special Parallelograms

Rhombus

Area of the Rhombus = (a * b)/ 2

Perimeter = 4 *length of side.

  • Three-Dimensional Shapes

V=lwh

SA=2lw+2wh+2lh

The surface area is simply the sum of the areas of the six rectangular faces.

  • Coordinate Geometry

The geometry rules concerning slope are very important to remember.

To find the y-intercept, set x equal to 0 and solve for y.

To find the x-intercept, set y equal to 0 and solve for x.

  • Pythagorean Theorem

It states that the sum of individual squares of base and height is equal to the square of hypotenuse of a right angled triangle. (1)

How important is geometry in the GRE as a whole?

GRE Geometry is essential because it is part of the most tested math concepts in GRE Quantitative Reasoning, which covers approximately 25–30% of the exam. Additionally, geometry is one of the four chapters in the GRE Math Subject Test Examination.

How to Approach GRE Geometry on Test Day

On your scheduled test for GRE Geometry, being able to think with a direct yet calculated approach is important to get along with the questions being asked about different numbers. These approaches can help you with it:

  • Assess what diagrams you can assume. Geometry diagrams on the GRE are not drawn to scale. Because of this, one technique that you can use is to estimate values, but also remember that it should not always be done, especially for angles, right angles, parallel lines, or anything that requires calculations. However, you can also use estimations, especially for lines that are assumed to be straight, circles, and any other shapes that follow the rules for basic geometry.
  • Visualize the shape to arrive at the best answer. Some of the GRE geometry problems do not supply diagrams. For this approach, it is better to draw the shapes for yourself in order to visualize what is being asked in the question. This could help you determine the formula that you will be using.
  • Utilize the numbers as a sample. Often, quantitative comparisons give a geometry problem that does not offer actual numbers or information to be used to plug in and solve the problem. If this happens during your examination, just try to plug in made-up numbers and see what happens, then verify it with a solution.
  • Approach the inscribed shapes as though they were two distinct shapes. Some combinations can be seen during your GRE examination. When this happens, the first approach that is truly helpful to you is to independently analyze each shape, then use the formula for each shape to solve for what is being asked.

All of these approaches are useful on your test day. Knowing that geometry and other quant exams are tricky, this could be very helpful for you to follow or do when given these types of questions.

List of GRE Geometry Terms

Before you take an examination for your scheduled GRE Geometry test, one of the helpful ways to get you through it is by familiarizing common geometry terms that you might encounter on your examination. Here are the following geometry terms present in GRE:

  • Acute angle. Any angle that is smaller than 90 degrees.
  • Arc. These are all points between two points on the edge of a circle.
  • Area. The amount of space enclosed by a 2D shape.
  • Axis. The line segment of a right circular cylinder joins the center points of the circular bases.
  • Central angle. The vertex at the center of a circle is created by the intersection of two radii.
  • Circumference: The length around a circle.
  • Congruent. Equal to or the same as in shape and size.
  • Cube. A type of rectangular solid with six square faces that are all the same length and width (and share equal areas).
  • Diameter. The length of a straight line cutting a circle in half and passing through the center; equal to double the radius.
  • Equilateral triangle. A triangle with three sides of the same length and three 60-degree angles.
  • Isosceles triangle. A triangle with two sides of the same length and two equal angles.
  • Obtuse angle. Any angle larger than 90 degrees and smaller than 180 degrees
  • Parallel lines. Lines that never intersect and have the same slope
  • Perimeter. The total length around a shape.
  • Perpendicular lines. lines that intersect to form right angles.
  • Polygon. Any 2D shape created by straight lines, including triangles, squares, rectangles, parallelograms, and trapezoids.
  • Quadrilateral. Any four-sided polygon, such as squares, rectangles, parallelograms, and trapezoids.
  • Radius (radii). The length of a straight line connects the center of a circle to any point on the edge of the circle equals half the diameter.
  • Rectangular solid. A 3-D object with six faces.
  • Right angle. a 90-degree angle, usually designated by a small square.
  • Right Circular cylinder. A 3-D object with two congruent circular bases and an axis perpendicular to the centers of its bases.
  • Right triangle. A triangle with a 90-degree angle, two legs (shorter sides), and a hypotenuse (the longest side).
  • Sector. Any (shaded) region of a circle enclosed by an arc and two radii.
  • Similar. Two shapes with the exact same shape (and therefore the same angles) but different sizes; similar shapes share equal ratios of their sides and angles.
  • Surface area. The total exterior area of a 3-D object.
  • Tangent. A line or shape intersecting a shape at exactly one point.
  • Vertex (vertices). The point at which two lines meet to form an angle.
  • Volume. The space occupied by a 3-D object.

These are the common terms that you might encounter in the questions written on your examination sheet. Familiarizing yourself with them can provide you with insight into what formula to use to obtain an answer from the questions being asked. (2)

What is the GRE Geometry Syllabus?

The geometry syllabus can be found below.

GRE GEOMETRY

DEFINITION

Parallel and perpendicular lines

Parallel lines never intersect. Perpendicular lines intersect at 90 degrees. Moreover, one must keep in mind that if two lines intersect, the sum of the consequently formed four angles is 360°.

Circles

In a plane, there is a point O, and a positive number ‘r’. The total set of points in that plane that are a distance of ‘r’ units from O is called the circle. That point ‘O’ is called the center of the circle, and ‘r’ is known as the radius. The diameter of a circle is always twice the radius.

Triangles

Every triangle of GRE geometry has three sides and interior angles. The interior angles of a triangle add up to 180° degrees. Scalene triangles have no similar angles, isosceles are where two angles are similar and all angles in an equilateral triangle are similar. There are right angles that are 90 degrees. There are special triangles, namely, the equilateral triangle, and isosceles triangle.

Quadrilaterals and other polygons

A parallelogram is a quadrilateral with two similar sides. All the parallelograms have four points that always go hand-in-hand.

opposites sides are parallel:

AD // BC and AB // CD

opposites sides are equal:

AD = BC and AB = CD

opposite angles are equal:

angle BAD = angle BCD and angle ABC = angle CDA

diagonals bisect each other:

AE = EC and BE = ED

Congruent similar figures

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length.

Three-dimensional figures, area, perimeter, and volume

A rectangular solid comprises 6 sides with all of them being rectangles, and all the faces connect at right angles. Suppose, it has a height ‘h’, length ‘l’, and width ‘w’. Then the volume and surface area would result in:

V = lwh; SA = 2lw + 2wh + 2lh

Pythagorean theorem

It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

Angle measurement in degrees

The angles opposite each other have to be equal. Because these angles meet only at a vertex, they are known as the “vertical” angles.

This syllabus is the overall content of the possible coverage that will be asked on the GRE Geometry examination. This basically has the objective of assessing the candidate’s ability to understand and use basic mathematical skills, such as formulas and concepts. (3)

How to make your GRE geometry preparation effective?

If you want to ace your GRE Geometry examination, one of the best ways to make it happen is to prepare for it. This preparation will help you increase your confidence while taking the exam. Here are some of the tips on how to make your preparation effective:

  • Important Concepts Should be Memorized. Familiarizing yourself with some of the major concepts in geometry will help you perform well during your examination. Try your best to remember and memorize the formulas necessary for the field of geometry. Do not go overboard on what is irrelevant, as it will take much of your time and, thus, it will not be as useful as you had expected.
  • Utilize flashcards. Flashcards will allow you to retain key concepts, especially when you use them for geometry or other subjects. This will help you to remember just the main points of the basic and major concepts without having to memorize word for word.
  • Practice answering realistic practice questions. Try answering questions that are similar to what is being tested during the exam. This will allow you to adjust to your own pace and give you the time to train yourself to answer all the questions within the time given.

Remember that in every exam, preparation is one of the essential things that we should do to maintain composure during the time we will be taking the test. Once we allow ourselves to practice, the more we have the chance to attain the highest score that we want to achieve.

How can I learn GRE geometry easily?

One of the best ways to learn GRE Geometry easily is to use a learning material that mainly focuses on the area of this subject. Reading various publications without familiarizing yourself with the material necessary to pass the examination can divert your attention away from the truly important topics that an examinee must understand. Stick with a book that covers basic formulas of geometry and contains pre-test and post-test assessment questions. Furthermore, make time to go over everything from the fundamentals to the more complex topics to give you a range and idea of what to expect. (4)

What are some examples of GRE geometry questions?

Answering practice questions is an effective way to help a test-taker achieve their score goal. Here are some practice questions:

Example 1

O is the center of the circle above.

The length of AB is 14.

Quantity A: The area of the circle.

Quantity B: 49π

Which of the following is true?

Possible Answers:

  • The relationship cannot be determined.
  • The two quantities are equal.
  • Quantity B is greater.
  • Quantity A is greater.

Correct answer:

Quantity A is greater.

Explanation:

O is the center of the circle above.

The length of AB is 14.

Quantity A: The area of the circle.

Quantity B: 49π

Do not be tricked by this question. It is true that AB can be split into halves, each of which are 7 in length. These halves are not, however, radii to the circle. Since this does not go through the center of the circle, its length is shorter than the diameter. This means that the radius of the circle must be greater than 7. Now, if it were 7, the area would be 49π. Since it is larger than 7, the area must be larger than 49π. Quantity A is larger than quantity B.

Example 2

O is the center of the circle above.

The circumference of the circle above is 30π.

Quantity A: The length of AB.

Quantity B: 30

Which of the following is true?

Possible Answers:

  • Quantity A is larger.
  • The two quantities are equal.
  • Quantity B is larger.
  • The relationship cannot be determined.

Correct answer:

Quantity B is larger.

Explanation:

Now, we know that the circumference of a circle is:

C=2πr or C=πd

This means that the diameter of our circle must be 30. Given this, we know that the AB must be shorter than 30, for the diameter is longer than any chord that does not pass through the center of the circle. Quantity B is larger than quantity A.

Example 3

What is the circumference of a circle with an area of 36π?

Possible Answers:

  • 32
  • 12π
  • 15π
  • None of the other answers

Correct answer:

12π

Explanation:

We know that the area of a circle can be expressed: a = πr2

If we know that the area is 36π, we can substitute this into said equation and get: 36π = πr2

Solving for r, we get: 36 = r2; (after taking the square root of both sides:) 6 = r

Now, we know that the circumference of a circle is expressed: c = πd. Since we know that d = 2r (two radii, placed one after the other, make a diameter), we can rewrite the circumference equation to be: c = 2πr

Since we have r, we can rewrite this to be: c = 2π*6 = 12π

Example 4

Which is greater: the circumference of a circle with an area of 25π in2, or the perimeter of a square with side length 7 inches?

Possible Answers:

  • The circumference of the circle is greater.
  • The relationship cannot be determined from the information given.
  • The two quantities are equal.
  • The perimeter of the square is greater.

Correct answer:

The circumference of the circle is greater.

Explanation:

Starting with the circle, we need to find the radius in order to get the circumference. Find by plugging our given area into the equation for the area of a circle:

A=πr2

25π=πr2

25=r2

r=5 in

Then calculate circumference:

C=2πr (approximating as 3.14)

To find the perimeter of the square, we can use P=4s , where is the perimeter and is the side length:

P=4×7 in=28 in

Example 5

Circle A has an area of 121π. What is the perimeter of an enclosed semi-circle with half the radius of circle A?

Possible Answers:

  • 11π
  • 5.5π+11
  • 22π
  • 11π+5.5
  • 5.5π+5.5

Correct answer:

5.5π+11

Explanation:

Based on our information, we know that 121π = πr2; 121 = r2; r = 11.

Our other circle with half the radius of A has a diameter equal to the radius of A. Therefore, the circumference of this circle is 11π. Half of this is 5.5π.

We have to include the diameter in the perimeter. Therefore, the total perimeter of the semi-circle is 5.5π + 11.

Example 6

x≥1

Quantity A: The circumference of a circle with radius 24x

Quantity B: The area of a circle with a diameter one fourth the radius of the circle in Quantity A

Which of the following is true?

Possible Answers:

  • The two quantities are equal.
  • Quantity B is larger.
  • Quantity A is larger.
  • The relationship between the two values cannot be determined.

Correct answer:

The relationship between the two values cannot be determined.

Explanation:

Let's compute each value separately. We know that the radii are positive numbers that are greater than or equal to 1. This means that we do not need to worry about the fact that the area could represent a square of a decimal value like 0.0000001.

Quantity A

Since c=2πr, we know:

a=2⋅π⋅24x=48xπ

Quantity B

If the diameter is one-fourth the radius of A, we know:

d=14⋅24x=6x

Thus, the radius must be half of that, or 3.

Now, we need to compute the area of this circle. We know:

A=πr2

Therefore, A=π⋅(3x)2=9x2

Now, notice that if x=1, Quantity A is larger.

However, if we choose a value like x=1000, we have:

Quantity A: 48 x 1000π=48000π

Quantity B: 9 x 10002π=9000000π

Therefore, the relation cannot be determined!

Example 7

Quantitative Comparison

Quantity A: Area of a circle with radius r

Quantity B: Perimeter of a circle with radius r

Possible Answers:

  • The two quantities are equal.
  • The relationship cannot be determined from the information given.
  • Quantity A is greater.
  • Quantity B is greater.

Correct answer:

The relationship cannot be determined from the information given.

Explanation:

Try different values for the radius to see if a pattern emerges. The formulas needed are Area = πr2 and Perimeter = 2πr.

If r = 1, then the Area = π and the Perimeter = 2π, so the perimeter is larger.

If r = 4, then the area = 16π and the perimeter = 8π, so the area is larger.

Therefore the relationship cannot be determined from the information given. (5)

References:

  1. https://www.prepscholar.com/gre/blog/gre-geometry-review-practice/
  2. https://gre.economist.com/gre-advice/gre-quantitative-section/geometry-gre/geometry-concepts-tested-gre-overview
  3. https://www.kaptest.com/study/gre/whats-tested-on-the-gre-geometry/
  4. https://magoosh.com/gre/gre-geometry-formulas/
  5. https://www.varsitytutors.com/gre_math-help/geometry