Mensuration and geometry for CLAT sounds heavier than it is. Nobody is asking you to prove a theorem. You are handed a short, factual passage — a plot of land, a water tank, a tiled floor, a garden to be fenced — and asked to read off a number using a formula you already met in Class 10. Quantitative Techniques is only about 10% of the paper, but it is the most learnable section: the formulas are fixed, the question styles repeat, and a few weeks of drilling turns this into your most dependable scoring area.
📌 The one idea to carry through this chapter
Every mensuration question is just 'pick the right formula, then put the numbers in'. The hard part is never the arithmetic — it is reading the passage carefully enough to know which measurement is wanted (area or perimeter? surface or volume?) and in which units. Get those two things right and the answer is one substitution away.
How mensuration appears in CLAT
CLAT Quantitative Techniques is data-interpretation style: a short passage or a simple figure sits at the top, followed by a set of questions that all draw on it. Mensuration shows up when that passage describes something physical — a field, a room, a tank, a wheel, a roll of fencing. You are then asked to find its size, its cost, how many tiles cover it, or how it changes when a dimension changes.
- ✓Area & cost problems — find the floor area, then multiply by a rate per square metre (flooring, painting, carpeting, turfing).
- ✓Fencing & perimeter — find the boundary length of a plot, then cost it per metre, or count posts/wire needed.
- ✓Tiling & 'how many' — divide a large area by the area of one tile/slab to count how many fit.
- ✓Capacity & volume — find how much a tank, cylinder or container holds, often converting between cm³, litres and m³.
- ✓Scaling effects — what happens to area or volume when a side, radius or height is doubled, halved or increased by a percentage.
ℹ️ No prior maths beyond Class 10
Everything in this chapter is on the NCERT Class 9–10 syllabus. There is no calculus, no trigonometry beyond a stray Pythagoras step, and no exotic solids. If you can find the area of a rectangle and the volume of a cylinder, you already have most of the marks here.
Perimeter and area of 2D shapes
Start with the flat shapes, because most CLAT mensuration questions live here. Perimeter is the distance round the boundary (a length, in m or cm). Area is the surface covered (in square units, m² or cm²). Confusing the two is the single commonest error, so fix the difference now.
- Rectangle — area = length × breadth; perimeter = 2(length + breadth).
- Square — area = side²; perimeter = 4 × side. (A square is just a rectangle with equal sides.)
- Triangle — area = ½ × base × height; perimeter = sum of the three sides.
- Parallelogram — area = base × height (the perpendicular height, not the slanting side).
- Trapezium — area = ½ × (sum of the two parallel sides) × height between them.
- Circle — area = πr²; circumference (perimeter) = 2πr, where r is the radius.
💡 Heron's formula — area of any triangle from its three sides
When you are given the three sides of a triangle (a, b, c) but no height, use Heron's formula. First find the semi-perimeter s = (a + b + c) ÷ 2, then area = √[s(s−a)(s−b)(s−c)]. CLAT keeps the numbers friendly — the classic 3–4–5, 5–12–13 or 13–14–15 triangles — so the square root comes out clean. For 13–14–15, s = 21 and area = √(21×8×7×6) = √7056 = 84.
⚠️ Area is NOT perimeter — and π ≈ 22/7
Read the stem twice. Fencing, edging, a border, wire, posts → perimeter. Flooring, painting, turf, carpet, glass → area. The two have different units (m vs m²) and getting them swapped is CLAT's favourite trap. Second trap: when a circle is involved, use π ≈ 22/7 if the radius is a multiple of 7, and π ≈ 3.14 otherwise — the question is usually set so one of these gives a clean answer.
🧩 Worked example
A rectangular school playground is 80 metres long and 50 metres wide. The school wants to put a fence all the way around its boundary. Fencing costs 120 rupees per metre.
What is the total cost of fencing the playground?
A16,000 rupees
B31,200 rupees
C48,000 rupees
D62,400 rupees
▸ Show solution
Answer: B. Fencing follows the boundary, so use perimeter, not area. Perimeter = 2(length + breadth) = 2(80 + 50) = 2 × 130 = 260 m. Cost = 260 × 120 = ₹31,200. Option A is the trap for students who use only one length + one breadth; C and D over-count. B is correct.
Surface area and volume of 3D solids
Three-dimensional questions describe a box, a pipe, a tank, a cone or a ball, and ask either how much surface it has (surface area, for painting or wrapping) or how much it holds (volume, for capacity or filling). Surface area is measured in square units; volume in cubic units (and 1000 cm³ = 1 litre, 1 m³ = 1000 litres).
- Cube (side a) — volume = a³; total surface area = 6a².
- Cuboid (l × b × h) — volume = l × b × h; total surface area = 2(lb + bh + hl).
- Cylinder (radius r, height h) — volume = πr²h; curved surface = 2πrh; total surface = 2πr(r + h).
- Cone (radius r, height h, slant l) — volume = ⅓πr²h; curved surface = πrl; total surface = πr(r + l).
- Sphere (radius r) — volume = ⁴⁄₃πr³; surface area = 4πr². (A hemisphere is half the volume plus its flat circular face.)
ℹ️ Cone slant height links to Pythagoras
A cone's slant height l, its vertical height h and its base radius r form a right-angled triangle, so l² = r² + h². If a question gives you r and h but the surface-area formula needs l, find l first with Pythagoras — the numbers are usually a 3–4–5 pattern (r = 3, h = 4 → l = 5).
🧩 Worked example
A closed cylindrical water tank has a radius of 7 metres and a height of 10 metres. Take pi as 22 by 7.
What is the volume of the tank?
A440 cubic metres
B1,540 cubic metres
C2,200 cubic metres
D4,400 cubic metres
▸ Show solution
Answer: B. Volume of a cylinder = πr²h. Here r = 7, so r² = 49, and with π = 22/7 the 7 cancels neatly. Volume = (22/7) × 49 × 10 = 22 × 7 × 10 = 1,540 m³. Option A uses curved surface logic, C and D mis-multiply. B is correct — note how choosing π = 22/7 with a radius of 7 makes the arithmetic effortless.
The complete formulas reference table
Memorise this table cold. Almost every CLAT mensuration question is one row of it applied to a real scenario. Keep π as 22/7 or 3.14 depending on the numbers; keep units consistent before you substitute.
| Shape / solid | Area / Curved or Total surface | Perimeter / Volume |
|---|---|---|
| Rectangle (l, b) | Area = l × b | Perimeter = 2(l + b) |
| Square (side a) | Area = a² | Perimeter = 4a |
| Triangle (base b, height h) | Area = ½ × b × h | Perimeter = sum of 3 sides |
| Triangle (sides a, b, c) | Area = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2 | Perimeter = a + b + c |
| Parallelogram (base b, height h) | Area = b × h | Perimeter = 2(sum of adjacent sides) |
| Trapezium (parallel sides p, q; height h) | Area = ½ × (p + q) × h | Perimeter = sum of 4 sides |
| Circle (radius r) | Area = πr² | Circumference = 2πr |
| Cube (side a) | Total surface = 6a² | Volume = a³ |
| Cuboid (l, b, h) | Total surface = 2(lb + bh + hl) | Volume = l × b × h |
| Cylinder (r, h) | Curved = 2πrh; Total = 2πr(r + h) | Volume = πr²h |
| Cone (r, h, slant l) | Curved = πrl; Total = πr(r + l) | Volume = ⅓πr²h |
| Sphere (radius r) | Surface = 4πr² | Volume = ⁴⁄₃πr³ |
📌 Units checklist before every substitution
Convert everything to the same unit first. If a tile is given in cm and a floor in metres, switch one of them (1 m = 100 cm, so 1 m² = 10,000 cm²). For capacity remember 1000 cm³ = 1 litre and 1 m³ = 1000 litres. A right formula with mismatched units is the second-commonest reason a correct method scores zero.
Basic geometry facts that show up in DI
Pure geometry is rare in CLAT, but a handful of facts keep appearing inside data sets and figure-based questions. Know these on sight; you will rarely need more.
- ✓Angles on a straight line add up to 180°; angles round a point add up to 360°.
- ✓Angles in a triangle add up to 180°; in any quadrilateral they add up to 360°.
- ✓Pythagoras theorem — in a right-angled triangle, hypotenuse² = base² + height² (the 3–4–5 and 5–12–13 triples come up again and again).
- ✓Similar triangles — same shape, different size: their angles are equal and their corresponding sides are in the same ratio. This is the engine of every scaling and shadow-height question.
- ✓Isosceles / equilateral — equal sides face equal angles; an equilateral triangle has every angle 60°.
ℹ️ Similar triangles power 'height of the tower' questions
When a passage compares a pole and its shadow with a tower and its shadow, the two right-angled triangles are similar, so their sides share one ratio: pole ÷ pole-shadow = tower ÷ tower-shadow. One proportion gives the unknown height — no trigonometry needed.
🧩 Worked example
A vertical pole 2 metres tall casts a shadow 3 metres long on level ground. At the same moment, a nearby tower casts a shadow 45 metres long.
How tall is the tower?
A20 metres
B30 metres
C45 metres
D67.5 metres
▸ Show solution
Answer: B. The pole-and-shadow and tower-and-shadow form similar triangles, so heights are in the same ratio as shadows: height ÷ shadow is constant. For the pole that ratio is 2 ÷ 3. So tower height = (2/3) × 45 = 30 m. Option D wrongly flips the ratio (3/2); A and C ignore the proportion. B is correct.
Cost, tiling and 'how many' problems
This family of questions is where mensuration meets the arithmetic you have already drilled. The recipe is always two steps: first find an area or perimeter, then attach a rate or a count. Keep the two steps separate and labelled and you will not slip.
- 1
Find the sizeWork out the area (for flooring, painting, tiling) or the perimeter (for fencing, edging). Write down the number with its unit.
- 2
Match the unitsMake the floor and the tile, or the wall and the rate, share one unit. Convert before you divide or multiply — never after.
- 3
Apply the rate or countFor cost, multiply size × rate. For 'how many tiles', divide total area ÷ area of one tile (round up if asked for whole tiles).
- 4
Sense-check the answerIs the number the right order of magnitude? A room cannot need three tiles or three million; a quick estimate kills silly options.
🧩 Worked example
A rectangular hall floor measures 12 metres by 8 metres. It is to be covered completely with square tiles, each measuring 40 centimetres by 40 centimetres. There is no wastage and tiles can be cut to fit.
How many tiles are needed to cover the floor?
A240
B300
C600
D1,500
▸ Show solution
Answer: C. Floor area = 12 × 8 = 96 m². Each tile is 40 cm = 0.4 m, so one tile's area = 0.4 × 0.4 = 0.16 m². Number of tiles = 96 ÷ 0.16 = 600. Option D forgets to square the conversion (treats the tile as 0.4 m²-ish); A and B mis-divide. C is correct — the whole trap is in the cm-to-m conversion of an area.
Drill mensuration & geometry now
10 drills, 150 questions — real CLAT-style data sets on area, cost, fencing, tiling and capacity, each with full working in the answer.
Scaling: what happens when a dimension changes
CLAT loves to ask how a measurement changes when you stretch a shape. There is one rule that answers nearly all of these, and it surprises most students: area scales with the square, volume with the cube.
- Double a side of a square → the area becomes 4 times (2²), not 2 times.
- Triple a radius of a circle → the area becomes 9 times (3²).
- Double an edge of a cube → the volume becomes 8 times (2³), the surface area 4 times.
- Increase a side by 10% → area rises by about 21% (1.1² = 1.21), not 10%.
⚠️ Doubling a side quadruples the area
This is the most-set scaling trap in CLAT. Students see 'the side is doubled' and answer 'the area doubles'. It does not — area depends on side², so doubling the side makes the area four times larger. For solids it is worse: doubling an edge makes the volume eight times larger. Whenever a dimension is multiplied by k, area × k² and volume × k³.
🧩 Worked example
A square garden has a side of 10 metres. The owner enlarges it so that each side becomes 30 metres.
By what factor does the area of the garden increase?
A3 times
B6 times
C9 times
D12 times
▸ Show solution
Answer: C. Each side is multiplied by 30 ÷ 10 = 3. Area depends on side², so the area is multiplied by 3² = 9. Check directly: old area 10² = 100 m², new area 30² = 900 m², and 900 ÷ 100 = 9. Option A is the classic trap (matching the side factor, not its square). C is correct.
Multiply a length by k and the area grows by k², the volume by k³. Never just k.
A repeatable method for the exam screen
Under the clock, do not improvise. Run the same short loop on every mensuration question and the answer falls out cleanly.
- 1
Name what is askedLength, area, surface or volume? Underline the verb — fence/edge → perimeter, cover/paint → area, fill/hold → volume.
- 2
Pick the formulaMatch the shape to its row in the reference table. Write the formula before touching numbers.
- 3
Fix the units and πConvert everything to one unit; choose π = 22/7 if a 7 is in play, else 3.14.
- 4
Substitute and attach the ratePlug the numbers in, then — for cost or count — multiply by the rate or divide by the unit size. Sense-check the size of the answer.
💡 Read all the questions on a passage first
Quant is data-interpretation: several questions share one passage. Glance over them before you start — one early calculation (say, the area of the plot) often feeds two or three later questions, so you compute it once and reuse it. This is where the time saving in the section really comes from.
🎯 Mensuration & geometry in a nutshell
- Quant is data-interpretation: read the passage, then pick a formula and substitute. The arithmetic is the easy part.
- Perimeter = boundary (fencing, edging); area = surface (flooring, painting); volume = capacity (filling). Match the verb to the measure.
- Memorise the formulas table — rectangle, square, triangle (incl. Heron's), parallelogram, trapezium, circle, plus cube, cuboid, cylinder, cone, sphere.
- Use π ≈ 22/7 when a radius is a multiple of 7, else 3.14; the question is set for clean numbers.
- Convert all units before substituting — remember 1 m² = 10,000 cm² and 1 m³ = 1000 litres.
- Scaling: multiply a length by k and the area grows by k², the volume by k³ — doubling a side quadruples the area.
Common mistakes to stop making
- ✓Using area when perimeter is wanted (or vice versa) — read the verb in the stem first.
- ✓Forgetting to square or cube the unit when converting (cm² to m², cm³ to litres).
- ✓Using the slant side of a parallelogram or cone as its height instead of the perpendicular height.
- ✓Assuming doubling a side doubles the area — it quadruples it; for volume it multiplies by eight.
- ✓Choosing the wrong value of π, turning a clean answer into an ugly one and second-guessing a correct method.
📐
Quantitative Techniques hub
All CLAT Quantitative Techniques chapters, the section strategy and every drill in one place.
Section guide
🧮
Practise the drills
10 mensuration & geometry drills with 150 CLAT-style data-interpretation questions and worked solutions.
Guide + 10 drills
Back to all Quantitative Techniques chapters
Mensuration is one slice of CLAT Quant. Browse every chapter — ratios, percentages, averages, data interpretation and more — on the section hub.
Frequently asked questions
How much of CLAT is Quantitative Techniques, and how much is mensuration?
Quantitative Techniques is about 10% of the CLAT UG paper, the smallest section. Mensuration and geometry are a regular slice of it, appearing inside data-interpretation sets about plots, tanks, floors and fences. Because the formulas are fixed and the styles repeat, it is one of the most learnable and reliable scoring areas in the whole exam.
Do I need maths beyond Class 10 for CLAT mensuration?
No. CLAT Quant uses only NCERT Class 9 to 10 maths. For mensuration that means areas, perimeters, surface areas and volumes of standard shapes, plus a few geometry facts like the angle sums, Pythagoras theorem and similar triangles. There is no calculus or trigonometry. If you know the Class 10 formulas, you have nearly all the marks.
When should I use pi as 22 by 7 and when as 3.14?
Use 22/7 when the radius or diameter is a multiple of 7, because the 7 cancels and the arithmetic stays clean. Use 3.14 otherwise. CLAT usually sets the numbers so one of these gives a tidy answer, so if your result looks ugly, you have probably chosen the wrong value of pi. The question sometimes tells you which to use.
What is the difference between area and perimeter in these questions?
Perimeter is the distance around the boundary, measured in metres, and is what you need for fencing, edging, borders and wire. Area is the surface covered, measured in square metres, and is what you need for flooring, painting, tiling and turfing. Swapping the two is CLAT's most common mensuration trap, so always read the verb in the stem first.
Why does doubling a side not double the area?
Because area depends on the side squared, not the side itself. Double the side and the area becomes 2 squared, that is 4 times larger; triple it and it becomes 9 times larger. For solids, volume depends on the cube, so doubling an edge makes the volume 8 times larger. The rule: multiply a length by k and area grows by k squared, volume by k cubed.
How do similar triangles help in CLAT data interpretation?
Similar triangles have the same shape, so their corresponding sides share one ratio. This solves height and shadow problems instantly: a pole and its shadow form a triangle similar to a tower and its shadow, so height divided by shadow is the same for both. One proportion gives the unknown height, with no trigonometry needed at all.
How should I revise mensuration formulas for the exam?
Memorise the single reference table covering area and perimeter of the 2D shapes and surface area and volume of the 3D solids. Then drill applied questions until matching a scenario to its formula is automatic. Practise the unit conversions too, since cm to m squared and cubed conversions trip up more correct methods than the formulas themselves. The drills cover all of this.