GRE Geometry: Core Rules, Diagrams, and Worked Problems

Geometry on the GRE combines familiar formulas with careful interpretation. A diagram can make a relationship visible, but it may not be drawn to scale. Read the labels and constraints first, then mark the figure with what is known. The strongest solution often uses one angle relationship or one area formula rather than a long chain of calculations.

Start with the stated facts

A diagram is a visual aid, not a measuring instrument. A segment that looks twice as long may not be twice as long. An angle that looks acute could be obtuse unless its type or measure is given. Equal tick marks, right-angle boxes, parallel arrows, and written lengths carry mathematical meaning. Apparent symmetry or alignment carries no meaning unless stated or proved.

Translate each sentence into a mark. If AB is parallel to CD, mark both lines. If a triangle is isosceles, mark the equal sides and infer that the opposite angles are equal. If two angles are complementary, write that their sum is 90 degrees. Keeping a small list of facts prevents assumptions from slipping into a proof.

Angles and parallel lines

Angles on a straight line sum to 180 degrees. Angles around a point sum to 360 degrees. Vertical angles formed by intersecting lines are equal. These facts often reduce a diagram to a simple equation. If one linear-pair angle is 67 degrees, the adjacent angle is 113 degrees; the opposite angle to the 67-degree angle is also 67 degrees.

When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles sum to 180 degrees. These rules require parallel lines. Two lines that merely look parallel in a sketch do not justify using them.

A useful solving habit is to mark one angle at a time. If the diagram gives 58 degrees and a pair of parallel lines, copy the corresponding 58-degree angle, then use a straight-line or triangle sum to find the next one. Avoid naming several relationships from memory at once; write the equation that directly connects the angles.

Triangles

The interior angles of every triangle sum to 180 degrees. An exterior angle equals the sum of the two remote interior angles. In an isosceles triangle, two equal sides face two equal angles. In an equilateral triangle, all sides are equal and every angle is 60 degrees.

The triangle inequality says the sum of any two side lengths must exceed the third. A triangle with sides 3, 4, and 8 is impossible because 3 + 4 is not greater than 8. If two sides have lengths 5 and 9 and the third is an integer x, then 4 < x < 14, so x may be any integer from 5 through 13. Check both inequalities, not just that the longest side is less than the sum of the other two.

A right triangle has one 90-degree angle. The Pythagorean theorem is a² + b² = c², where c is the hypotenuse opposite the right angle. The common triples 3-4-5, 5-12-13, and their multiples can save time. In a 45-45-90 triangle, the legs are equal and the hypotenuse is a leg times √2. In a 30-60-90 triangle, the sides opposite 30, 60, and 90 degrees are in the ratio 1:√3:2.

Area is one-half base times height. The height must be perpendicular to the selected base; it is not necessarily a slanted side. If a triangle has base 10 and perpendicular height 6, its area is 30. Choosing a different side as the base changes the corresponding height but not the area.

Quadrilaterals and polygons

A rectangle has four right angles, opposite sides equal, and area length times width. A square is a rectangle with all four sides equal. A parallelogram has opposite sides parallel and equal; its area is base times perpendicular height, not base times slanted side. A rhombus has four equal sides, while its angles need not be right angles.

The interior-angle sum of an n-sided polygon is (n − 2) × 180 degrees. A quadrilateral's interior angles sum to 360 degrees, and a pentagon's sum is 540 degrees. For a regular polygon, all sides and angles are equal, so each interior angle is ((n − 2) × 180)/n. A regular hexagon's interior angles are each 120 degrees.

Perimeter is the total distance around a figure. Area measures the enclosed two-dimensional region. Doubling every side length doubles perimeter but multiplies area by four. This scaling distinction appears in comparisons: similar figures with scale factor k have perimeter ratio k and area ratio k².

Circles

A circle's circumference is 2πr or πd, and its area is πr². The diameter is twice the radius. If radius doubles, circumference doubles but area quadruples. Keep radius and diameter distinct; a frequent trap is substituting a diameter into a formula that requires r.

A central angle has its vertex at the center. An arc's length is the same fraction of the full circumference as its central angle is of 360 degrees. A 90-degree arc is one quarter of the circle, with arc length one quarter of 2πr and sector area one quarter of πr². The arc length is a distance; the sector area is a surface.

An inscribed angle has its vertex on the circle and measures half the central angle that subtends the same arc. A diameter subtends a right angle at a point on the circle. These properties can turn a circle diagram into a triangle problem. Confirm which arc an angle intercepts before applying the half-angle relationship.

A tangent is perpendicular to the radius at the point of tangency. Tangent segments drawn from the same exterior point are equal in length. If an exterior point connects to a circle at two tangent points, the two outside segments match even if the sketch looks asymmetric.

Coordinate geometry

The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). This is the Pythagorean theorem applied to horizontal and vertical changes. For (1, 2) and (4, 6), the changes are 3 and 4, so the distance is 5.

The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). It averages the x-coordinates and the y-coordinates separately. A horizontal line has constant y; a vertical line has constant x. The slope between two points is (y₂ − y₁)/(x₂ − x₁) when x₂ differs from x₁. Parallel nonvertical lines have equal slopes; perpendicular nonvertical lines have slopes whose product is −1.

A circle centered at (h, k) with radius r has equation (x − h)² + (y − k)² = r². A point lies on the circle if its distance from the center is r. To test a point, substitute its coordinates and compare with r², avoiding a square root when possible.

Three-dimensional figures

A rectangular prism has volume length × width × height. A cylinder has volume πr²h. A prism's volume is base area times perpendicular height. Surface area adds the areas of all exterior faces. Keep these quantities separate: volume uses cubic units, surface area uses square units, and a linear measurement uses ordinary units.

If every dimension of a solid is multiplied by 3, its surface area is multiplied by 9 and its volume by 27. Similarity scaling can answer many questions without computing every face. When a problem asks about capacity, identify volume; when it asks about material covering the outside, identify surface area.

Original practice problems

Question 1: triangle angle reasoning

An isosceles triangle has a vertex angle of 38 degrees between its two equal sides. What is each base angle?

1. 38°
2. 71°
3. 142°
4. 151°

Answer: B, 71°. The two base angles are equal and together measure 180° − 38° = 142°. Each is 142°/2 = 71°. The equal sides meet at the vertex angle; the equal angles lie opposite those sides.

Question 2: perimeter and area scaling

A rectangle has side lengths 4 and 7. Each side of a similar rectangle is 3 times as long. By what factor does its area increase?

1. 3
2. 6
3. 9
4. 21

Answer: C, 9. The original area is 4 × 7 = 28. The new dimensions are 12 and 21, giving area 252. The ratio is 252/28 = 9. A scale factor of 3 applies to each of two dimensions, so area scales by 3².

Question 3: circle arc

A circle has radius 6. What is the length of a 60-degree arc?

1. π
2. 2π
3. 4π
4. 6π

Answer: B, 2π. A 60-degree arc is 60/360 = 1/6 of the circle. The circumference is 2π(6) = 12π, and one sixth of that is 2π. Do not calculate a sector area because the question asks for arc length.

Question 4: coordinate distance

What is the distance from (−2, 3) to (4, 11)?

1. 8
2. 10
3. 12
4. 14

Answer: B, 10. The horizontal difference is 4 − (−2) = 6 and the vertical difference is 11 − 3 = 8. Distance is √(6² + 8²) = √100 = 10. The signs matter when subtracting coordinates, but the squared differences are positive.

Question 5: triangle inequality

Two sides of a triangle have lengths 7 and 12. Which of the following could be the third side?

1. 4
2. 6
3. 19
4. 20

Answer: B, 6. If the third side is x, the triangle inequality requires |12 − 7| < x < 12 + 7, or 5 < x < 19. Only 6 satisfies the strict bounds. Length 4 is too short, and 19 is degenerate because the sum of 7 and 12 equals 19 rather than exceeding it.

Corrected practice choice set: 4, 6, 19, 20. The answer is 6 because it is strictly between 5 and 19. Length 4 is too short to connect the two segments, and 19 is degenerate because the sum of 7 and 12 equals 19 rather than exceeding it. In a triangle, inequalities are strict.

How to solve geometry under time pressure

1. Read the entire prompt and identify the requested quantity before choosing a formula.
2. Mark only stated equalities, right angles, parallel lines, and lengths on the diagram.
3. Look for a direct relationship such as an angle sum, similar-triangle ratio, or area formula.
4. Use a familiar special triangle or a Pythagorean triple when the numbers support it.
5. Check units, scale factors, and whether the answer should be length, area, angle, or volume.

If a diagram feels crowded, redraw only the relevant portion. A clean sketch can make a transversal or triangle relation easier to see. Avoid drawing to scale and avoid adding marks based on appearance. Your redraw is a reasoning aid, not additional evidence.

Common geometry traps

• Assuming a figure is to scale and estimating an unmarked angle or length.
• Using a slanted side as a triangle or parallelogram height when the formula needs a perpendicular height.
• Confusing radius with diameter in a circle formula.
• Treating an area scale factor as the same as the linear scale factor.
• Forgetting that a triangle side inequality is strict.
• Using parallel-line angle rules when parallelism was never given.
• Adding lengths when a question asks for area or volume.

What to study and what not to memorize blindly

Know the core formulas well enough to recognize when they apply: triangle and rectangle area, circle area and circumference, Pythagorean theorem, prism and cylinder volume, and coordinate distance. Memorizing a formula without identifying its variables often leads to substituting diameter for radius or a slanted edge for height. Practice naming what each symbol represents before inserting values.

Geometry is one component of the GRE Quantitative Reasoning measure. The current test has 27 quantitative questions across two sections, but ETS does not publish a fixed geometry-question count. Build fluency across geometry, algebra, arithmetic, and data interpretation rather than assigning preparation time based on an assumed quota.

Practice routine

First solve mixed geometry questions without time pressure and write one sentence for each inference: ‘These angles sum to 180 because they form a line’ or ‘The area uses perpendicular height.’ Then time a short set and review every item that required guessing from the picture. Redraw it and list the facts explicitly. This separates knowledge gaps from diagram-reading mistakes.

ETS's free Math Review can refresh geometry fundamentals. After reviewing a topic, use new problems to test transfer: a rule learned in a triangle should still be recognizable when the triangle is part of a larger diagram. Keep an error log of the relationship you missed, rather than only the formula you forgot.