GRE Quant · Study & Practice

3-D Figures (Solids)

AreaGeometry DifficultyCore GRE weightageMedium — boxes, cubes and cylinders recur; the space-diagonal formula is a favourite twist

Three-dimensional figures on the GRE stay refreshingly narrow: rectangular boxes, cubes, and cylinders — no cones, spheres or pyramids, and no trigonometry. Nearly everything reduces to two questions, "how much fits inside" (volume) and "how much surface covers it" (surface area), each with a short formula per shape. The one genuinely 3-D idea worth drilling is the space diagonal of a box, which is just the Pythagorean theorem applied twice and lands as √(l² + w² + h²). Keep cylinder answers in terms of π, watch that every dimension shares the same unit, and — as with all GRE geometry — read the given measurements rather than trusting a figure that is not drawn to scale.

Topics

1 Rectangular Solids & Cubes 4 worked · 5 practice · 8-Q test Start → 2 Cylinders 4 worked · 5 practice · 8-Q test Start → 3 Volume 4 worked · 5 practice · 8-Q test Start → 4 Surface Area 4 worked · 5 practice · 8-Q test Start → 5 Diagonals of a Box 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.

  • Only three solids appear on the GRE: box, cube and cylinder. Learn their volume and surface-area formulas and you have covered the topic.
  • Volume is a product of dimensions (cubic units); surface area is a sum of face areas (square units). Check which the question wants.
  • Scale every linear dimension by k and volume scales by k³ — doubling a cube’s edge makes it 8 times bigger.
  • The space diagonal of a box is Pythagoras twice: d = √(l² + w² + h²). Look for hidden triples (3-4-12 → 13, 1-2-2 → 3).
  • A cube’s space diagonal is just s√3; its face diagonal is s√2.
  • Keep cylinder answers in terms of π, and halve a stated diameter before using r in πr²h.

⚠️ Common mistakes & traps

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

  • Confusing volume (cubic units, fills the inside) with surface area (square units, covers the outside).
  • Scaling volume linearly — doubling a dimension multiplies the volume by 2³ = 8, not by 2.
  • Using the diameter as the radius in a cylinder formula. Halve it first.
  • Mixing the space diagonal √(l² + w² + h²) with a face diagonal √(l² + w²) — the space diagonal uses all three dimensions.
  • Multiplying dimensions given in different units without converting them to a common unit first.

📈 GRE exam insight & question patterns

Quantitative Comparison — cube A has edge 2 and cube B has edge 4. Quantity A: the volume of B. Quantity B: 4 times the volume of A. Which is greater?

Quantity A. B = 4³ = 64, and 4 × A = 4 × 8 = 32. Doubling the edge multiplies volume by 8, not 4, so B is far larger.

A rectangular box is 3 by 4 by 12. What is the longest straight rod that fits inside it?

The space diagonal: √(9 + 16 + 144) = √169 = 13. It is longer than any edge or face diagonal.

Numeric Entry — a cylinder has radius 5 and height 2. Enter its volume in terms of π.

V = πr²h = π × 25 × 2 = 50π.

A cube has a surface area of 54. What is its volume?

6s² = 54 → s² = 9 → s = 3, so volume = 3³ = 27.

🎴 Flashcards — instant recall

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

Volume of a boxTap to reveal
V = l × w × h
Volume of a cube, edge sTap to reveal
V = s³
Volume of a cylinderTap to reveal
V = πr²h
Surface area of a boxTap to reveal
SA = 2(lw + lh + wh)
Surface area of a cubeTap to reveal
SA = 6s²
Total surface area of a cylinderTap to reveal
2πr(r + h)
Space diagonal of a boxTap to reveal
√(l² + w² + h²)
Space diagonal of a cubeTap to reveal
s√3
Scale dimensions by k ⇒ volumeTap to reveal
Scales by k³

📌 Quick revision

GRE solids are limited to boxes, cubes and cylinders, so a handful of formulas covers everything: volumes l×w×h, s³, and πr²h; surface areas 2(lw+lh+wh), 6s², and 2πr(r+h). Keep the two families straight — volume fills the inside in cubic units, surface area covers the outside in square units — and remember that scaling a linear dimension by k scales volume by k³. The signature 3-D move is the space diagonal, √(l² + w² + h²), which is nothing more than the Pythagorean theorem applied twice and often hides a whole-number answer. Keep cylinder results in terms of π, unify your units, and trust the stated dimensions over the drawing.

Chapter test

📝 3-D Figures (Solids) — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara GRE success checklist

You have truly mastered 3-D Figures (Solids) 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 (5 topics)5/5
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall9 cards