Geometry on the GMAT might seem like high school déjà vu, but trust me—if you know the right formulas and understand the key concepts, it's a goldmine of easy points. Geometry questions make up a smaller chunk of the GMAT Quant section, but they can be real time traps if you're not prepared. So let’s break down everything you need to know for success—formulas, concepts, and quick tips.
Why Geometry Matters on the GMAT
The GMAT Quantitative Reasoning section tests how well you can reason, not just calculate. Geometry questions are usually visual and conceptual, meaning they’re testing logic with shapes, not just memorized math. You'll encounter:
- Problem-Solving questions (standard multiple-choice)
- Data Sufficiency questions (where you decide if the info is enough to solve)
Geometry questions account for about 25% of the GMAT quantitative section. The key to tackling geometry questions is to know the properties of different shapes and several key formulas, which are reviewed in this chapter, including:
■ Angles
■ Lines
■ Triangles
■ Quadrilaterals
■ Circles
■ Coordinate geometry
Classifying Angles
In geometry, an angle is formed by two rays with a common endpoint. The two rays are the sides of the angle. The common endpoint is the vertex of the angle.
Angles that make a square corner are called right angles (see the following examples for more details about what makes an angle a right angle). In drawings, the following symbol is used to indicate a right angle: 
Opposite rays are two rays with the same endpoint that form a line. They form a straight angle. A straight angle has a 180° measure. 
An acute angle has a measure between 0° and 90°. Here are two examples of acute angles.
A right angle has a 90° measure. The corner of a piece of paper will fit exactly into a right angle. Here are two examples of right angles. 
An obtuse angle has a measure between 90° and 180°. Here are two examples of obtuse angles. 
Congruent Angles and Angle Pairs
When two angles have the same degree measure, they are said to be congruent.
Names are given to three special angle pairs, based on their relationship to each other:
■ Complementary angles: two angles whose sum is 90°.
■ Supplementary angles: two angles whose sum is 180°.
■ Vertical angles: two angles that are opposite each other when two lines cross.
When two lines cross, the adjacent angles are supplementary, and the sum of all four angles is 360°.
Perpendicular and Parallel Lines
Perpendicular lines are another type of intersecting lines. Perpendicular lines meet to form right angles. Right angles always measure 90°. In the following figure, lines x and y are perpendicular: 
Parallel lines lie in the same plane and don’t cross at any point. 
Angle–Pair Problems
Triangles
You can classify triangles by the lengths of their sides. Below are three examples of special triangles called equilateral, isosceles, and scalene triangles. 
You can also classify triangles by the measurements of their angles. Here are four examples of special triangles. They are called acute, equiangular, right, and obtuse triangles.
Area of a Triangle
To find the area of a triangle, use this formula:
Although any side of a triangle may be called its base, it’s often easiest to use the side on the bottom. To use another side, rotate the page and view the triangle from another perspective.
A triangle’s height (or altitude) is represented by a perpendicular line drawn from the angle opposite the base to the base. Depending on the triangle, the height may be inside, outside, or on the legs of the triangle.
Triangle Rules
Rule 1. The sum of the angles in a triangle is 180°:
Example: One base angle of an isosceles triangle is 30°. Find the measure of the vertex angle.
Draw a picture of an isosceles triangle. Drawing it to scale helps. Since it is an isosceles triangle, draw both base angles the same size (as close to 30° as you can) and make sure the sides opposite them are the same length. Label one base angle as 30°.
Since the base angles are congruent, label the other base angle as 30°. There are two steps needed to find the vertex angle:
■ Add the two base angles together: 30° + 30° = 60°
■ The sum of all three angles in a triangle is 180°.
To find the measure of the vertex angle, subtract the sum of the two base angles (60°) from 180°:
180° - 60° = 120°
Thus, the measure of the vertex angle is 120°. Add all 3 angles together to make sure their sum is 180°:
Rule 2. The longest side of a triangle is opposite the largest angle. This rule implies that the second longest side is opposite the second largest angle, and the shortest side is opposite the smallest angle.
Example: In the following triangle, which side is the shortest?
Determine the size of –A, the missing angle, by adding the two known angles and then subtracting their sum from 180°:
90° + 46° = 136°
180° - 136° = 44°
Thus, A is 44°. Since A is the smallest angle, side a (opposite A) is the shortest side.
Rule 3. Right triangles have a rule of their own. Using the Pythagorean theorem, you can calculate the missing side of a right triangle.
Since the perimeter is the sum of the lengths of the sides, we must first find the missing side. Use the Pythagorean Theorem:
a² + b² = c²
Substitute the given sides for two of the letters. To solve this equation, subtract 9 from both sides:
Then take the square root of both sides.
Thus, the missing side has a length of 4 units. Adding the three sides yields a perimeter of 12:
3 + 4 + 5 = 12
Quadrilaterals
A quadrilateral is four–sided polygon. Three common quadrilaterals are shown here: 
These quadrilaterals have something in common besides having four sides:
■ Opposite sides are the same length and parallel.
■ Opposite angles are the same size.
However, each quadrilateral has its own distinguishing characteristics as given on the table that follows. 
The naming conventions for quadrilaterals are similar to those for triangles: 
■ The figure is named by the letters at its four corners, usually in alphabetical order: rectangle ABCD.
■ A side is named by the letters at its ends: side AB.
■ An angle is named by its vertex letter: A.
The sum of the angles of a quadrilateral is 360°:
To find the perimeter of a quadrilateral, follow this simple rule:
Perimeter = sum of all four sides
To find the area of a rectangle, square, or parallelogram, use this formula:
Area = base * height
The base is the size of the side on the bottom. The height (or altitude) is the length of a perpendicular line drawn from the base to the side opposite it. The height of a rectangle and a square is the same as the length of its vertical side.
A parallelogram’s height is not necessarily the same as the length of its vertical side (called the slant height); it is found instead by drawing a perpendicular line from the base to the side opposite it—the length of this line equals the height of the parallelogram.
The area formula for the rectangle and square may be expressed in an equivalent form as: Area = length * width
Example: Find the area of a rectangle with a base of 4 meters and a height of 3 meters. Draw the rectangle as close to scale as possible. Label the size of the base and height.
Circles
A circle is a set of points that are all the same distance from a given point called the center.
You are likely to come across the following terms when dealing with circles:
Radius: The distance from the center of the circle to any point on the circle itself. The symbol r is used for the radius.
Diameter: The length of a line that passes across a circle through the center. The diameter is twice the size of the radius. The symbol d is used for the diameter.
The circumference of a circle is the distance around the circle (comparable to the concept of the perimeter of a polygon). To determine the circumference of a circle, use either of these two equivalent formulas:
Draw this circle and write the radius version of the circumference formula (because you’re given the radius):
Example: What is the diameter of a circle whose circumference is 62.8 centimeters? Use 3.14 for p.
The area of a circle is the space its surface occupies.
To determine the area of a circle, use this formula: Area = pi*r*r
Example: What is the diameter of a circle whose area is 9 square centimeters?
Draw a circle with its diameter (to help you remember that the question asks for the diameter), then write the area formula. (Note: Refer to Chapter 10 if you need help with square roots.) A = pi*r²
Substitute 9p for the area and solve the equation: 9pi = pi*r² =>9 = r² =>3 = r
Since the radius is 3 centimeters, the diameter is 6 centimeters.