Question 1
A rectangular box measures 2 by 4 by 5.
What is its volume?
- 44
- 76
- 11
- 8
- 40
Show the worked solution
- Volume = length × breadth × height = 2 × 4 × 5 = 40 cubic units.
- Answer: 40 cubic units.
Mensuration
Mensuration questions — the formulas, and when each one applies
Mensuration questions ask for the area, perimeter, surface area or volume of a standard shape. The marks are lost to two things and not to the formulas: mixing units within one question, and reading total surface area as curved surface area. Fix the units first, then confirm which surface the question actually wants.
Mensuration is a GRE Quant topic and is entirely recall plus care. Nothing in it requires insight: a candidate who knows the formulas and reads carefully finishes quickly, and one who half-remembers misses questions they understood.
Try it against the clock. Mensuration runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.
How to solve mensuration questions without a unit slip
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Convert every measurement to one unit before starting
A question mixing feet and inches is not testing your formula, it is testing whether you noticed. Convert everything as you read, and write the unit beside each number rather than trusting yourself to remember it.
-
Decide which surface the question wants
Curved surface area excludes the ends; total surface area includes them. For a cylinder that is 2πrh against 2πr(r + h). Both appear in the option list of a well-set question, so read the phrase before reaching for a formula.
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Keep π symbolic — the answer usually is
GRE and GMAT options are written in terms of π: 49π, not 154. Carry π through the working as a symbol and the answer is the coefficient you already have. Substituting 3.14 creates decimals that never needed to exist and matches none of the options.
-
Use ratio reasoning when a dimension changes
If a radius doubles, area scales by four and volume by eight — no recomputation is needed. A large share of mensuration questions ask exactly this, and answering by scaling is a few seconds against a minute.
The formulas worth having instantly
| Shape | Formula |
|---|---|
| Circle | Area πr², circumference 2πr |
| Cylinder | Volume πr²h · CSA 2πrh · TSA 2πr(r + h) |
| Cone | Volume ⅓πr²h · CSA πrl · slant l = √(r² + h²) |
| Sphere | Volume ⁴⁄₃πr³ · surface 4πr² |
| Cuboid | Volume lbh · TSA 2(lb + bh + hl) |
| Scaling a linear dimension by k | Area × k², volume × k³ |
3 sample mensuration questions, with worked solutions
Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.
Question 2
A square has side length 19.
What is its perimeter?
- 38
- 76
- 46
- 361
- 90
Show the worked solution
- Perimeter = 4 × side = 4 × 19 = 76 units — perimeter is a length, area is a square.
- Answer: 76 units.
Question 3
A circle has radius 3.
Its area is kπ. What is k?
- 36
- 6
- 9
- 18
- 3
Show the worked solution
- Area = πr² = π × 3² = 9π, so k = 9.
- Answer: 9.
Common mistakes this drill is built from
The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.
Mixing units inside one question
Feet with inches, or square feet with square yards. Convert at the point of reading, not at the point of computing, and the error becomes impossible rather than merely unlikely.
Using the height where the slant height is required
A cone's curved surface uses the slant l, not the vertical height h. Where only h and r are given, l = √(r² + h²) comes first. The answer using h is always among the options.
Recomputing instead of scaling
When a question doubles or halves a dimension, the answer follows from k² or k³ directly. Rebuilding the whole calculation is slower and introduces arithmetic that was never needed.
Why your answer came out wrong
The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here.
The radius is in inches and the height in feet — do I really have to convert?
Always, and before you substitute, not after. A stray foot among inches inside a formula that cubes the length puts the answer out by a factor of 1,728, and inside an area formula by 144. Convert both to one unit on the first line of your working.
When does a cone question need the slant height rather than the height?
Curved and total surface area use the slant height l; volume uses the vertical height h; and l² = h² + r² connects them. Substituting h where l belongs gives a smaller, entirely plausible answer that the paper prints. If the question asked for a surface area and you never used Pythagoras, check which one you used.
The radius doubles — should I recompute the volume from scratch?
No, scale it. Volume grows with the cube of the radius, so it multiplies by 8; surface area grows with the square, so by 4. Recomputing with π and fractions takes the best part of a minute, and scaling takes five seconds.
FAQ
What is the difference between curved and total surface area?
Curved surface area covers only the curved part — for a cylinder, 2πrh. Total surface area adds the flat ends, giving 2πr(r + h). Questions ask for both and the option list carries both values, so the deciding step is reading which one is wanted, not knowing the formula.
Should I substitute a value for π?
Almost never. GRE and GMAT answer choices are written in terms of π — a circle of radius 7 has area 49π, and 49π is the option. Keep π as a symbol from the first line, and substitute 3.14 only when a question asks you to estimate or compare against a whole number.
If the radius of a sphere doubles, what happens to its volume?
It multiplies by eight. Volume scales with the cube of any linear dimension, and surface area with the square, so doubling the radius gives 2³ = 8 times the volume and 2² = 4 times the surface area. Recognising this answers a whole family of questions without recomputation.
Practise the next topic
- All 51 drill types — the shared daily cycles 36 of 51, with every published time limit.
- Simplification questions — BODMAS — simplification questions, timed.
- Percentage questions — percentage questions, timed.
- Speed tricks — the seven mental-math shortcuts that make a 15-second answer possible.
- How to improve calculation speed — the six-week arc these drills fit into.
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