GRE Quant · Study & Practice

Quadrilaterals & Polygons

AreaGeometry DifficultyCore GRE weightageMedium — squares and rectangles are everywhere; the polygon angle formulas are a reliable easy score

A quadrilateral is any four-sided figure, and the GRE keeps returning to a small family of them — the parallelogram and its special cases (rectangle, square) plus the trapezoid. Almost everything you need reduces to two ideas: the angle relationships (opposite angles, co-interior angles, and the (n − 2) × 180° interior-angle sum) and the area/perimeter formulas. The one formula worth memorising beyond the obvious is that the exterior angles of any polygon always add to 360°, which turns many "how many sides" questions into a one-line division. As with all GRE geometry, the figure is not necessarily drawn to scale, so a shape that looks like a square is only a square if its sides and angles are marked or stated equal.

Topics

1 Squares, Rectangles & Parallelograms 4 worked · 5 practice · 8-Q test Start → 2 Trapezoids 4 worked · 5 practice · 8-Q test Start → 3 Interior & Exterior Angles 4 worked · 5 practice · 8-Q test Start → 4 Area & Perimeter 4 worked · 5 practice · 8-Q test Start →

⚡ GRE shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Exterior angles of ANY polygon total 360°. To find the number of sides of a regular polygon, compute 360° / (exterior angle) — far faster than the interior-angle formula.
  • Interior and exterior angles at a vertex are supplementary, so flip between them with a single subtraction from 180°.
  • A trapezoid’s area is the average of the two parallel sides times the height — picture a rectangle of the average width.
  • Every parallelogram property (opposite sides equal, diagonals bisect) is inherited by rectangles and squares. Learn it once.
  • A rectangle’s diagonal is just a Pythagorean hypotenuse — a 6-8-10 or 5-12-13 triple often hides in the dimensions.
  • For a fixed perimeter, area is largest when the rectangle is closest to a square — handy on Quantitative Comparison.

⚠️ Common mistakes & traps

GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Using the slanted leg of a trapezoid or parallelogram as the height. The height is the perpendicular distance between the parallel sides.
  • Using (n − 2) × 180° when the polygon is not regular and then dividing by n anyway — that per-angle step only works for regular polygons.
  • Confusing area (square units) with perimeter (linear units), or forgetting to convert mixed units before multiplying.
  • Assuming a four-sided figure is a square or rectangle from its appearance. GRE figures are not drawn to scale.
  • Treating consecutive angles of a parallelogram as equal — they are supplementary; only opposite angles are equal.

📈 GRE exam insight & question patterns

Quantitative Comparison — a rectangle and a square each have a perimeter of 40. Quantity A: the area of the square. Quantity B: the area of the rectangle. Which is greater?

Quantity A. With a fixed perimeter, the square maximises area (10 × 10 = 100), so any non-square rectangle of perimeter 40 has less area.

A regular polygon’s interior angle is 144°. How is the number of sides found fastest?

Take the exterior angle 180 − 144 = 36°, then sides = 360 / 36 = 10. Going through the exterior angle beats the interior-sum formula.

Numeric Entry — a trapezoid has parallel sides 5 and 11 and a height of 6. Enter its area.

Area = ½ (5 + 11) × 6 = ½ × 16 × 6 = 48.

A figure looks like a square but only its four right angles are marked. Is it a square?

No — four right angles guarantee only a rectangle. Without equal sides marked or stated, you cannot call it a square, especially since the figure is not to scale.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

Sum of interior angles, n sidesTap to reveal
(n − 2) × 180°
Sum of exterior angles (any polygon)Tap to reveal
360°
Each exterior angle, regular n-gonTap to reveal
360° / n
Parallelogram consecutive anglesTap to reveal
Supplementary (sum 180°)
Rectangle area / perimeterTap to reveal
l × w / 2(l + w)
Square area / perimeter, side sTap to reveal
s² / 4s
Parallelogram areaTap to reveal
base × height (perpendicular height)
Trapezoid areaTap to reveal
½ (b₁ + b₂) × height

📌 Quick revision

Quadrilaterals and polygons come down to two toolkits. For angles: interior angles sum to (n − 2) × 180°, exterior angles always sum to 360°, and interior/exterior angles at a vertex are supplementary — that last pair is the fast route to "how many sides". For measurement: rectangle = l × w, square = s², parallelogram = base × height, and trapezoid = the average of the parallel sides times the height, always using a perpendicular height. Remember that a rectangle’s diagonal is a Pythagorean hypotenuse, that a fixed perimeter gives the most area to the most square-like shape, and that GRE figures are never guaranteed to be drawn to scale.

Chapter test

📝 Quadrilaterals & Polygons — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
📊 My GRE analytics
Track scores, accuracy by topic and progress over time.
View analytics →

🏆 Vidaara GRE success checklist

You have truly mastered Quadrilaterals & Polygons when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall8 cards