Study guide
This chapter covers the geometry and data analysis content that appears alongside arithmetic and algebra across the two Quantitative Reasoning sections, again in the four formats of Quantitative Comparison, Select-One, Select-One-or-More, and Numeric Entry. Every figure is described in words rather than shown, matching how you must be ready to reason from a verbal description alone if a figure is described but not drawn to scale, and every worked solution below is recomputed by two independent methods.
Triangles and the Pythagorean Theorem
A triangle's three interior angles always sum to 180 degrees, and the side opposite the largest angle is always the longest side. The Pythagorean theorem, a^2 + b^2 = c^2, applies only to right triangles, where c is the hypotenuse, the side opposite the right angle. Picture a right triangle with legs of length 9 and 12; find the hypotenuse. Method one, direct formula: 9^2 + 12^2 = 81 + 144 = 225, and sqrt(225) = 15. Method two, recognize the 3-4-5 family: 9 = 3x3 and 12 = 3x4, so this triangle is a 3-4-5 triangle scaled by 3, meaning the hypotenuse is 5x3 = 15, matching the direct computation. The triangle inequality states that the sum of any two sides must exceed the third side, which is useful for Quantitative Comparison items about whether a triangle with given side lengths can exist at all. For a 30-60-90 triangle, sides are in ratio 1 : sqrt(3) : 2, with the side opposite 30 degrees shortest and the side opposite 90 degrees the hypotenuse. For a 45-45-90 triangle, sides are in ratio 1 : 1 : sqrt(2). Picture an isosceles right triangle with legs of length 7; the hypotenuse is 7 x sqrt(2). Recheck with the Pythagorean theorem directly: 7^2 + 7^2 = 49 + 49 = 98, and sqrt(98) = sqrt(49 x 2) = 7 x sqrt(2), confirming the special ratio.
Circles, Polygons, and Area, Perimeter, Volume
For a circle of radius r, circumference is 2 x pi x r and area is pi x r^2. Picture a circle with radius 6; find its area. Method one, direct formula: pi x 6^2 = 36 x pi, approximately 36 x 3.14159 = 113.1. Method two, use the diameter form, area = pi x (d/2)^2 with d = 12: pi x (12/2)^2 = pi x 6^2 = 36 x pi, the same result confirmed from the diameter. For polygons, the sum of interior angles is (n-2) x 180 degrees, where n is the number of sides; a hexagon has (6-2) x 180 = 4 x 180 = 720 degrees total, and recheck by dividing evenly for a regular hexagon: 720 / 6 = 120 degrees per interior angle, matching the known value for a regular hexagon. Rectangular volume is length x width x height. Picture a rectangular box 4 units long, 5 units wide, and 3 units tall; find its volume. Method one, direct multiplication: 4 x 5 x 3 = 60. Method two, group differently: (4 x 3) x 5 = 12 x 5 = 60, the same result by a different grouping order, confirming 60 cubic units. Cylinder volume is pi x r^2 x h; picture a cylinder with radius 3 and height 10. Method one: pi x 3^2 x 10 = pi x 9 x 10 = 90 x pi, approximately 282.7. Method two, compute the base area first, then multiply by height: base area = pi x 9 = 28.27, and 28.27 x 10 = 282.7, matching.
Descriptive Statistics and Probability
The mean is the sum of values divided by the count of values; the median is the middle value when data is ordered, or the average of the two middle values for an even count; the mode is the most frequent value; the range is the maximum minus the minimum. Picture a data set of exam scores: 72, 85, 90, 90, 68. Method one for the mean: sum = 72+85+90+90+68 = 405, and 405 / 5 = 81. Method two, adjust from an estimated average: guess an average of 80, compute each value's deviation from 80 (-8, +5, +10, +10, -12), sum the deviations (-8+5+10+10-12 = 5), and add the average deviation (5/5 = 1) to the guess, 80 + 1 = 81, matching. For this ordered data (68, 72, 85, 90, 90), the median is the middle value, 85, and the mode is 90, since it appears twice. Basic probability of an event is favorable outcomes divided by total possible outcomes. Picture a jar with 4 red marbles, 3 blue marbles, and 5 green marbles, 12 total; find the probability of drawing a blue marble. Method one, direct ratio: 3/12 = 1/4. Method two, complement check: probability of not blue is (4+5)/12 = 9/12 = 3/4, and 1 - 3/4 = 1/4, matching the direct calculation. For independent events, multiply individual probabilities; for mutually exclusive events, add them, but never both in the same step without checking which rule the situation requires.
Counting Methods and a Linked Data Interpretation Set
The fundamental counting principle states that if one choice can be made in m ways and a second, independent choice can be made in n ways, the two together can be made in m x n ways. Permutations count ordered arrangements, using n! / (n-r)!, while combinations count unordered selections, using n! / (r! x (n-r)!). Picture choosing 2 books from a shelf of 5 distinct books, where order does not matter. Method one, combination formula: 5! / (2! x 3!) = (5x4x3x2x1) / ((2x1) x (3x2x1)) = 120 / (2 x 6) = 120/12 = 10. Method two, direct listing logic: the first book can be any of 5, the second any of the remaining 4, giving 5x4 = 20 ordered pairs, but each unordered pair is counted twice, so 20/2 = 10, matching. Data Interpretation sets on the GRE link two or three questions to a single shared figure, described here in words: imagine a bar graph titled Annual Widget Sales by Region showing four regions, North, South, East, and West, with bar heights of 120, 90, 150, and 140 thousand units respectively for one year. A linked question might ask for the percentage of total sales from the East region. Method one: total = 120+90+150+140 = 500, and East's share = 150/500 = 0.30, or 30 percent. Method two, pairwise reduction: North+South = 210, East+West = 290, total = 210+290 = 500, confirming the same total, and 150/500 reduces to 3/10, or 30 percent, matching. Always compute the shared total once per linked set and reuse it, checking the total by adding in a different order or grouping, since an error in the shared total propagates through every question tied to that figure.
Quantitative Comparison with Geometry and Data
Quantitative Comparison items in this content area frequently withhold a figure's scale intentionally, since ETS figures are not guaranteed to be drawn to scale, and the comparison must be provable from the stated facts alone, not from how the description looks. Picture two shapes described in words only: Quantity A is the area of a square with perimeter 20, and Quantity B is the area of a rectangle with perimeter 20 and one side of length 8. Method one, direct computation: the square has side 20/4 = 5, so its area is 5^2 = 25; the rectangle has two sides of 8 and 8, so the remaining two sides sum to 20 - 16 = 4, meaning each remaining side is 2, and its area is 8 x 2 = 16. Quantity A, 25, is greater than Quantity B, 16. Method two, general principle check: among all rectangles with a fixed perimeter, the square always encloses the maximum possible area, a geometric fact that independently confirms Quantity A must be greater without recomputing the specific numbers. For data-based Quantitative Comparison, a common setup withholds one figure, such as asking about a mean when the underlying data set is only partially described; if a value could reasonably vary depending on unlisted data, the correct choice is that the relationship cannot be determined, and you should test at least two different plausible completions of the missing data before concluding a comparison is fixed, exactly as with algebraic variables in the previous chapter.
Key terms
- Pythagorean theorem
- — The relationship a^2 + b^2 = c^2 for a right triangle, where c is the hypotenuse.
- Triangle inequality
- — The rule that the sum of any two sides of a triangle must exceed the length of the third side.
- Interior angle sum
- — The total measure of a polygon's interior angles, computed as (n-2) x 180 degrees for n sides.
- Mean, median, mode
- — Three measures of central tendency: the arithmetic average, the middle value, and the most frequent value in a data set.
- Range (statistics)
- — The difference between the maximum and minimum values in a data set.
- Fundamental counting principle
- — The rule that independent choices with m and n options combine to give m x n total outcomes.
- Permutation
- — An ordered arrangement of items, counted using n! / (n-r)!.
- Combination
- — An unordered selection of items, counted using n! / (r! x (n-r)!).
- Data Interpretation set
- — A group of two or more linked questions sharing one figure, table, or graph.
- Not drawn to scale
- — A note indicating a geometric figure's visual proportions cannot be trusted; only stated facts may be used.
- Complement (probability)
- — The probability that an event does not occur, equal to 1 minus the probability that it does.
Exam tips
- Never trust a figure's apparent proportions; use only the numbers and relationships explicitly stated, since GRE figures may not be drawn to scale.
- Memorize the 3-4-5, 30-60-90, and 45-45-90 side ratios; recognizing them is usually faster than applying the Pythagorean theorem from scratch.
- In linked Data Interpretation sets, compute the shared total or baseline once, verify it with a second grouping, and reuse it across every question in the set.
- On statistics or data Quantitative Comparison items with incomplete information, test at least two plausible data completions before concluding the relationship is fixed.
- Recompute every geometry and counting answer with a second method, such as a formula check against a special ratio or a listing-based sanity check, before moving on.