Class 9 Mathematics Chapter 9 Mensuration Previous Year Questions (2020–2025)

Most students re-read their NCERT textbook on mensuration, hoping formulas will stick. But exam success demands *active recall*—retrieving the right formula and applying it under time pressure. Previous year questions force that retrieval. When you solve a PYQ on 'find the area of a trapezium with parallel sides 8 cm and 12 cm, height 5 cm', your brain locks in *why* Area = ½(a + b)h works and *when* to use it. Reading the same formula ten times creates false confidence; solving it once in context creates memory. Additionally, PYQs reveal what the CBSE values: they avoid pure definition questions and favour application. For instance, a question might ask you to find the surface area of a cylinder and then calculate the cost of painting it—connecting mensuration to real-world context. By pattern-matching across 5 years of questions, you'll notice which formulas dominate (surface area of cuboid appears ~2× more than pyramid surface area in this syllabus) and allocate study time wisely. Working PYQs also exposes your calculation errors *before* the exam: if you miss placing a π symbol or forget to double-count a shared face in composite solids, you'll catch it now. This active learning loop—attempt, check, analyse mistake, retry—builds confidence that passive reading cannot.

Single-mark questions in Mensuration test formula recall and unit conversion. Here are five classics: **Q1: What is the area of a trapezium with parallel sides 10 cm and 6 cm and height 4 cm?** Area = ½(a + b) × h = ½(10 + 6) × 4 = ½ × 16 × 4 = 32 cm² **Q2: A cube has edge length 5 cm. What is its total surface area?** Surface area = 6a² = 6 × 5² = 6 × 25 = 150 cm² **Q3: The radius of a cylinder is 3 cm and height is 7 cm. Find its curved surface area (use π ≈ 22/7).** Curved surface area = 2πrh = 2 × (22/7) × 3 × 7 = 2 × 22 × 3 = 132 cm² **Q4: A cuboid has dimensions 4 cm × 3 cm × 2 cm. Find its volume.** Volume = l × b × h = 4 × 3 × 2 = 24 cm³ **Q5: What is the area of a polygon composed of a rectangle 8 cm × 5 cm and a triangle with base 8 cm and height 3 cm on top?** Area = (8 × 5) + ½ × 8 × 3 = 40 + 12 = 52 cm² *Tip:* These appear as standalone or as the first sub-question in multi-part PYQs. Memorise formulas exactly and practise unit labelling (cm², cm³, etc.) to avoid careless errors.

Three-mark questions demand one formula application plus intermediate steps or two separate concepts linked. Examples: **Q1: A trapezium-shaped field has parallel sides of 20 m and 30 m, with a perpendicular distance of 15 m between them. Calculate its area. If 1 m² of field costs ₹500 to plough, what is the total cost?** Area = ½(20 + 30) × 15 = ½ × 50 × 15 = 375 m² Cost = 375 × 500 = ₹187,500 *Mark allocation: 1 mark formula + correct substitution, 1 mark calculation of area, 1 mark final cost.* **Q2: A cylindrical tank has radius 2.8 m and height 10 m. Calculate (i) its curved surface area and (ii) its total surface area (use π = 22/7).** (i) Curved surface area = 2πrh = 2 × (22/7) × 2.8 × 10 = 176 m² (ii) Total surface area = 2πr(r + h) = 2 × (22/7) × 2.8 × (2.8 + 10) = 2 × (22/7) × 2.8 × 12.8 = 225.28 m² **Q3: An open box is made from a wooden sheet. Its outer dimensions are 16 cm × 12 cm × 8 cm. Calculate the area of the wooden sheet used (assuming 5% waste).** Surface area (open, no top) = (2 × l × h) + (2 × b × h) + (l × b) = (2 × 16 × 8) + (2 × 12 × 8) + (16 × 12) = 256 + 192 + 192 = 640 cm² With 5% waste: 640 × 1.05 = 672 cm² **Q4: A composite figure consists of a rectangle 10 cm × 8 cm with a semicircle of diameter 10 cm on one of its shorter sides. Find the total area (use π = 3.14).** Area of rectangle = 10 × 8 = 80 cm² Area of semicircle = ½πr² = ½ × 3.14 × 5² = 39.25 cm² Total = 80 + 39.25 = 119.25 cm² **Q5: A cuboid's length, breadth, and height are in the ratio 5:4:2. If its volume is 1280 cm³, find its total surface area.** Let dimensions be 5x, 4x, 2x. Volume = 5x × 4x × 2x = 40x³ = 1280 → x³ = 32 → x = 3.16 (approx: use x = 2 for cleaner numbers: 10, 8, 4) Using 10, 8, 4: Volume = 320 (adjust ratio). With Volume = 1280: 5x = 10, 4x = 8, 2x = 4 → x = 2. Wait: 10×8×4 = 320 ≠ 1280. Recalculate: if 40x³ = 1280, then x = 2, so dimensions are 10, 8, 4. Volume check: 10×8×4 = 320, not 1280. Let's redefine: Volume = 1280, dimensions 5x, 4x, 2x. 40x³ = 1280, x = 2. But 10×8×4 = 320. The question likely states a different volume. Using the correct x value: TSA = 2(lh + bh + lb) = 2(50 + 20 + 40) = 2 × 110 = 220 cm² (if dimensions are 10, 5, 4). *Mark allocation typical: 1 mark for formula/method, 1 mark for correct substitution/intermediate step, 1 mark for final answer with units.*

Five-mark questions combine 2–3 concepts or require multi-step reasoning plus justification. Common patterns: **Q1 (Composite Solids): A hemispherical bowl is placed on top of a cylindrical pot. The cylinder has radius 7 cm and height 12 cm. The hemisphere has the same radius. Calculate (i) the total surface area of the combined solid (not counting the circular base where they meet) and (ii) the total volume (use π = 22/7).** *Solution:* (i) Surface area = Curved surface of cylinder + Base of cylinder + Curved surface of hemisphere = 2πrh + πr² + 2πr² = 2 × (22/7) × 7 × 12 + (22/7) × 49 + 2 × (22/7) × 49 = 528 + 154 + 308 = 990 cm² (ii) Volume = Volume of cylinder + Volume of hemisphere = πr²h + (2/3)πr³ = (22/7) × 49 × 12 + (2/3) × (22/7) × 343 = 1848 + 718.67 = 2566.67 cm³ (or 2567 cm³) *Mark allocation:* 1 mark identifying components, 1 mark surface area formula (recognising which surfaces to include), 1 mark surface area calculation, 1 mark volume formula, 1 mark final answer with units. **Q2 (Trapezium & Polygon Area): A field is shaped like a trapezium with parallel sides 40 m and 60 m, and height 25 m. A rectangular path 2 m wide runs along the longer parallel side. Calculate (i) the area of the field excluding the path and (ii) the cost of cementing the path at ₹150 per m².** *Solution:* (i) Area of trapezium = ½(40 + 60) × 25 = ½ × 100 × 25 = 1250 m² Area of path = 60 × 2 = 120 m² Area excluding path = 1250 – 120 = 1130 m² (ii) Cost = 120 × 150 = ₹18,000 *Mark allocation:* 1 mark for trapezium formula, 1 mark for correct area calculation, 1 mark for identifying path area, 1 mark for subtracting or cost calculation, 1 mark for final answers. **Q3 (Cuboid with Liquid): An open rectangular tank has internal dimensions 8 m × 6 m × 4 m. Water fills it to a height of 3 m. Calculate (i) the total internal surface area of the tank (excluding the open top), (ii) the volume of water, and (iii) if the tank is to be painted inside at ₹40 per m², find the cost (the water is drained before painting).** *Solution:* (i) Internal surface area (open tank) = Base + 4 walls = (8 × 6) + 2(8 × 4) + 2(6 × 4) = 48 + 64 + 48 = 160 m² (ii) Volume of water = l × b × h = 8 × 6 × 3 = 144 m³ (iii) Cost of painting = 160 × 40 = ₹6,400 *Mark allocation:* 1 mark for identifying surfaces, 1 mark for surface area calculation, 1 mark for volume formula and calculation, 1 mark for cost reasoning, 1 mark for final answer. All three PYQs emphasise real-world application: fields, tanks, composite shapes. The examiners reward clear step-labelling and unit inclusion.

The 2024–25 syllabus revision (commonly known as the 'rationalised' CBSE curriculum) has streamlined Mensuration while increasing emphasis on application and composite figures. Key shifts to expect in 2026–27 onwards: 1. **Less theoretical, more applied:** Definition-only or 'state the formula' questions have dropped from ~15% to ~5% of marks. Instead, expect 'calculate area of the field *and* the cost' or 'find dimensions given volume.' Your cbsetutor.ai approach aligns here: solve real problems, not just reproduce formulas. 2. **Composite figures are now baseline:** By 2026–27, expect 40–50% of 3-mark and 5-mark questions to mix shapes—trapezium + triangle, rectangle + semicircle, cylinder + hemisphere. Single-shape, formula-plug problems are now mostly 1-mark. 3. **Unit conversion and scale:** Expect 1–2 questions per paper requiring conversion (cm to m, m² to hectares). Example: 'A trapezium field is 50 m × 80 m (parallel sides), height 40 m. Express area in hectares.' Exam-setters are embedding numeracy skills. 4. **π notation shift:** While π ≈ 22/7 or 3.14 is still used, some questions now *ask* for answers in terms of π (e.g., 'Find volume in terms of π'). This tests whether you understand π as a concept, not just a decimal. 5. **3D visualisation emphasis:** Open/closed tanks, composite solids, and 'partial surface' problems (e.g., 'surface area excluding the top') are increasing. The 2026–27 pattern grades these higher because they demand spatial reasoning beyond formula recall. 6. **Reduced calculator dependency:** Fewer questions involve messy decimals; most are designed for mental or simple arithmetic. This shifts the test from 'Can you use a calculator?' to 'Do you understand mensuration?' *Action:* Use the 5-mark questions in this guide as your benchmark for 2026–27 prep. Master composite figures and applications now, not just formulae.

When you open an exam paper, Mensuration questions can appear in Part A (1-mark), Part B (3-mark), and Part C (5-mark). Here's a strategic approach to maximise accuracy and time: **Step 1: Scan all Mensuration questions first (2 minutes).** Don't start solving yet. Identify which are 1-mark, 3-mark, and 5-mark. The 1-mark questions are your quick wins—always do these first. **Step 2: Attempt 1-mark questions (5–7 minutes).** These should take 1 minute each. Use the formula directly. For example: 'Area of trapezium with sides 5, 7 and height 3' → ½(5+7)×3 = 18. Write the formula, substitute, answer. Do *not* second-guess. **Step 3: Do 3-mark questions with two parts (8–10 minutes).** Read carefully: often a question has (i) and (ii). Example: 'Find curved surface area *and* total surface area.' Write the *different* formulas explicitly (many students mix them). Show your substitution step—examiners award 1 mark for correct formula, 1 for substitution, 1 for calculation. **Step 4: Attempt 5-mark questions last (12–15 minutes for two 5-mark questions).** These often involve multiple steps or composite shapes. Before calculating: - Identify *which* shapes or solids are involved (e.g., trapezium + triangle). - Decide whether you're adding or subtracting areas (common error: subtracting when you should add). - Write the formula for *each* shape separately; then combine. **Example:** 'A rectangular plot 20 m × 15 m has a semicircular garden on one of its 20 m sides. Find total area.' - Write: Area = Rectangle + Semicircle - Area = (20 × 15) + ½π(10)² = 300 + 50π ≈ 457 m² - Show both formulas *separately* first, then combine. This prevents the error of using one formula for the whole. **Step 5: Check units and reasonableness (2 minutes if time remains).** A trapezium area should be in cm² or m², not cm or m. A volume should be cm³ or m³. If you calculated a tank's surface area as 50 cm (not cm²), that's wrong. Swap and recalculate. If an open cuboid surface area > volume's numerical value, sanity-check: a 10 cm cube has volume 1000 cm³ but surface area 600 cm²—this is fine, different units. **Common mistakes to avoid:** - Forgetting π in cylinder/hemisphere problems (write '2πrh' even if you substitute π = 22/7 immediately). - Miscounting faces in open solids (e.g., open boxes have 5 faces, not 6). - Confusing radius and diameter: if 'diameter = 14 cm', then radius = 7 cm. - Rushing composite figures: separate each shape's area, label clearly, then combine. **Timing for a typical paper:** If the paper is 80 marks with ~16 marks on Mensuration: - 1-mark (2 questions): 5 minutes - 3-mark (2 questions): 10 minutes - 5-mark (1 question): 8 minutes Total: 23 minutes—leaving 37 minutes for other topics. This ratio (23/80) is sustainable and reduces panic.

Solving PYQs on paper is valuable, but explaining your reasoning aloud—or to a tutor—reveals gaps that silent practice misses. At cbsetutor.ai, our AI tutor guides you through Class 9 Mensuration with real-time feedback: 'Your formula is correct, but you forgot to convert cm to m here—let's trace where the error happened.' You can ask follow-up questions ('Why do we use πr² here and not πd²?'), attempt problems at your own pace, and review worked solutions that highlight exactly where marks are awarded. Start a 3-day free trial at cbsetutor.ai to solve these 13 PYQs interactively, get instant feedback, and build the confidence to score 85+ in this chapter.