Mensuration is the branch of mathematics concerned with measuring geometric figures — how much space they cover (area), how much boundary they have (perimeter), how much space they enclose in three dimensions (volume), and how much outer surface they expose (surface area).
Think of it this way. You are tiling a kitchen floor — you need area. You are fencing a garden — you need perimeter. You are filling a water tank — you need volume. You are painting a box — you need surface area. These four ideas cover practically every mensuration question you will encounter in CTET Paper II.
The subject divides cleanly into two worlds:
2D (flat) shapes — triangles, rectangles, squares, parallelograms, trapeziums, circles, and sectors. Here you work with area and perimeter only.
3D (solid) shapes — cubes, cuboids, cylinders, cones, spheres, and hemispheres. Here you work with volume, total surface area (TSA), and curved/lateral surface area (CSA/LSA).
The conceptual leap that confuses most candidates is this: surface area is still a 2D measurement (it counts square units of outer skin), while volume counts cubic units of interior space. When you paint a box, you care about surface area. When you fill it with water, you care about volume. Keep that distinction sharp and half the confusion dissolves.
A useful analogy for the classroom: imagine wrapping a gift box. The wrapping paper you need corresponds to total surface area. The space inside the box that the gift occupies corresponds to volume. CTET questions on teaching methodology sometimes test whether a teacher can explain this distinction — so knowing it conceptually, not just formula-mechanically, matters.
Master these cold. No derivation time in the exam hall.
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × b | 2(l + b) |
| Square | a² | 4a |
| Triangle | (1/2) × b × h | sum of three sides |
| Parallelogram | b × h | 2(a + b) |
| Trapezium | (1/2)(a + b) × h | sum of all sides |
| Circle | πr² | 2πr |
| Sector | (θ/360°) × πr² | 2r + (θ/360°) × 2πr |
For CTET, trapezium appears frequently. The formula (1/2)(sum of parallel sides)(height) is the one you must know forwards and backwards — specifically, you must be comfortable rearranging it to find a missing side when area and height are given.
Here is the master reference. Every formula below has appeared directly or indirectly in CTET PYQs.
Cube (side a):
6a²a³Cuboid (length l, breadth b, height h):
2(lb + bh + lh)2h(l + b)lbhCylinder (radius r, height h):
2πr(r + h)2πrhπr²hHollow Cylinder (outer radius R, inner radius r, height h):
π(R² − r²)hCone (radius r, height h, slant height l):
l = √(r² + h²)πr(r + l)πrl(1/3)πr²hSphere (radius r):
4πr²(4/3)πr³Hemisphere (radius r):
3πr² (curved + flat base)2πr²(2/3)πr³When you scale all linear dimensions by a factor k:
k²k³Look — if every dimension of a cuboid is doubled (k = 2), volume becomes 2³ = 8 times the original. This is a recurring CTET question type. Don't compute both volumes; just apply the cube of the scale factor.
Many CTET questions give you surface area or volume and ask for a missing dimension. The approach is always the same: write the formula, substitute known values, isolate the unknown, solve. The algebraic manipulation is usually linear (first-degree), not quadratic. If you find yourself solving a quadratic, re-read the question — you have probably misidentified the unknown.
For cuboid TSA problems (a CTET favourite), the key rearrangement is:
2(lb + bh + lh) = TSA
Substitute l and b, combine the h terms, solve. This is a one-step linear equation once you expand. Do not panic — it always simplifies cleanly in CTET problems because the numbers are chosen to give integer answers.
CTET questions always specify which value of π to use. When radius or diameter is a multiple of 7, use 22/7. When it is not, use 3.14. If the question says nothing, default to 22/7 because the numbers in CTET problems are constructed to cancel cleanly with 22/7. Using 3.14 when 22/7 is intended gives a messy decimal that does not match any option.
When a cuboid TSA question gives you two dimensions and asks for volume, use the chain: expand TSA formula → collect h terms → solve for h → multiply lbh.
Example: TSA = 194, l = 8, b = 6. Step 1: 2(48 + 6h + 8h) = 194 → 96 + 28h = 194 → h = 3.5 Step 2: V = 8 × 6 × 3.5 = 168
This two-step chain takes about 25 seconds once practised. The standard approach of trial-and-error with options takes 60–90 seconds. Save 40+ seconds per such question.
When a trapezium problem says "one parallel side is longer by k cm," immediately set the shorter side as a and longer as a + k. Plug into (1/2)(2a + k)(h) = Area, solve for a, then report a + k.
Example: Area = 105, h = 7, difference = 6. (1/2)(2a + 6)(7) = 105 → (7/2)(2a + 6) = 105 → 2a + 6 = 30 → a = 12, longer = 18.
Standard guess-and-check from options: 4 substitutions, ~60 seconds. This algebraic path: ~20 seconds.
Dimensions scaled by factor k → Area scales by k², Volume by k³. No calculation needed.
If all sides double: Volume = 2³ = 8× original. If all sides triple: Volume = 3³ = 27× original. If all sides halve: Volume = (1/2)³ = 1/8 original.
Standard method (compute both volumes and divide): ~45 seconds for a cuboid. This rule: 5 seconds — read the scale factor, cube it, done.
For CSA of cone = πrl, when π = 22/7, cancel it before multiplying.
Example: CSA = 550, l = 25. Find r. 550 = (22/7) × r × 25 → 550 = (550r)/7 → r = 7.
Notice: 22 × 25 = 550, so (22/7 × 25) = 550/7. The 550 cancels immediately: r = 550 × 7/550 = 7.
Standard method (multiply 22/7 × 25 first, then divide): 3 steps, ~30 seconds. Cancellation method: 1 step, ~8 seconds.
For any prism or cylinder: Volume = Base Area × Height. So Height = Volume ÷ Base Area. This works for rectangular tanks, cylindrical tanks, any uniform cross-section solid.
Example: Volume = 140 m³, Base Area = 700 m². Height = 140/700 = 0.2 m = 20 cm.
Don't reach for the full formula. Just remember: V = A × h → h = V/A. Two-second setup, five-second division.
When a mensuration question appears, run this decision tree in your head before writing anything:
Step 1 — Identify the shape. 2D or 3D? Name it exactly (trapezium, cuboid, hollow cylinder, etc.).
Step 2 — Identify what is asked. Area / Perimeter / CSA / TSA / Volume? Pick exactly one formula.
Step 3 — Check what is given vs. what is needed. Are all variables in the formula provided, or do you need to find a missing one first? (E.g., in cuboid TSA → volume, you must find h first.)
Step 4 — Handle unit conversion before substituting. If area is in m² and volume in m³, the height comes out in metres — convert to cm only at the end if the answer choices are in cm.
Step 5 — Cancel before you multiply. With π = 22/7, look for 7s in the radius or area to cancel. With scaling questions, apply the k² or k³ rule directly — never compute both values.
If two answer options are close in value (say, 168 vs 126), do not guess — these are designed to trap arithmetic errors. Recheck your algebra for sign or coefficient mistakes in the h-term.
Why this question: The simplest application of V = A × h in reverse. Every CTET batch has at least one "find the height of water" question.
Solving path: Write V = Base Area × h. Rearrange: h = V/A = 140/700 = 0.2 m. Convert: 0.2 m = 20 cm. Match option B. Total time: under 15 seconds.
Why this question: Classic two-step cuboid: TSA → find missing dimension → compute volume. If you do not recognise the chain, this question eats 3 minutes. If you do, it takes 30 seconds.
Solving path: Expand TSA: 2(8×6 + 6h + 8h) = 194 → 2(48 + 14h) = 194 → 96 + 28h = 194 → 28h = 98 → h = 3.5. Volume = 8 × 6 × 3.5 = 168 m³. Option C.
Why this question: Trapezium reverse — given area and height, find a side. The "one side longer by k" phrasing is a CTET signature.
Solving path: Let shorter side = a, longer = a + 6. Area formula: 105 = (1/2)(a + a + 6)(7) = (7/2)(2a + 6). Multiply both sides by 2/7: 30 = 2a + 6 → a = 12. Longer side = 18 cm. Option A.
Why this question: Hollow cylinder tests whether you know V = π(R² − r²)h. Many candidates incorrectly subtract volumes as π(R−r)²h.
Solving path: V = (22/7)(7² − 5²)(10) = (22/7)(49 − 25)(10) = (22/7)(24)(10) = (22 × 240)/7 = 5280/7 ≈ 754 cm³. Option A. Note: subtract the squares of the radii, not the square of their difference.
Why this question: Tests scaling intuition. The answer is k³ = 2³ = 8 and requires zero computation if you know the rule.
Solving path: All three dimensions double → k = 2. Volume scales by k³ = 8. Confirm: original = 1200 cm³, new = 9600 cm³, ratio = 8. Option D. Using the rule takes 5 seconds; computing both volumes takes 30–40 seconds.
Hollow cylinder: (R − r)² instead of (R² − r²). The volume of material uses R² − r² (difference of squares), not (R − r)². These are different: (R − r)² = R² − 2Rr + r², which is wrong. Always write π(R² − r²)h.
Unit mismatch before the final answer. If base area is in m² and volume is in m³, height is automatically in metres. Converting too early (say, turning 700 m² into 70,000 cm² before dividing) creates messy numbers and arithmetic errors. Keep consistent units throughout; convert only the final answer.
Using TSA formula when CSA is needed (or vice versa). Cone and cylinder each have two formulas. "Curved surface area" excludes the flat circular ends. "Total surface area" includes them. Read the question word for word — CTET exploits this ambiguity.
Slant height vs. vertical height for a cone. CSA of cone = πrl where l is slant height. If the question gives vertical height h, you must first compute l = √(r² + h²). Substituting h directly into the CSA formula gives a wrong answer that will match one of the distractor options.
Forgetting that sector perimeter includes two radii. The arc length of a sector is (θ/360°) × 2πr, but the perimeter of the sector shape is arc length + 2r. Area questions do not require this, but if perimeter is asked, both radii must be added.
Applying k² instead of k³ for volume scaling. If all dimensions scale by k, area scales by k² and volume by k³. Mixing these up gives you one of the wrong options (the question designers put k² as a distractor). Always ask: area or volume? Then decide which power to use.