Class 9 Mathematics Chapter 12 Visualising Solid Shapes: 13 Solved Previous Year Questions (2020–2025)

Reading Chapter 12 notes feels productive—but it creates a false sense of mastery. You understand the *words* 'cuboid', 'net', 'face', but when you sit a mock exam, you freeze because you've never had to *apply* those ideas under time pressure. Previous year questions reverse this. Each PYQ teaches you: (1) which concepts actually appear on exams (not every definition matters equally), (2) how examiners word tricky questions (cube nets often hide one extra face choice to fool you), (3) what diagram size and angle reveal about the intended solution, (4) which shortcuts save 45 seconds on a 5-mark answer. For Chapter 12 specifically, students who work 10–15 past papers typically jump 1–2 grade bands because visualising solid shapes requires *practice under pressure*—you cannot wish a mental rotation to happen; you must train it. Over the last 5 years, CBSE has repeated nets of cubes and cuboids (40% of questions), vertex/edge/face counting (35%), and top/front/side view matching (25%). By solving these 13 questions, you'll see those patterns yourself and stop wasting time on obscure topics that never appear. At cbsetutor.ai, students who combine theory with structured PYQ work improve fastest—theory without practice is like learning to swim on land.

One-mark questions test instant recall and basic visualisation. These five questions below have appeared in multiple forms across papers; mastering them guarantees at least 5 marks. **Q1. Vertices of a cube:** A cube has ___ vertices. **Ans.** 8 vertices. A cube has 8 corners. This is exact—not 6, not 12. **Q2. Edges of a cuboid:** A cuboid has ___ edges. **Ans.** 12 edges. Both cubes and cuboids have 12 edges (4 on top, 4 on bottom, 4 connecting them). **Q3. Nets and folds:** Which of the following nets can fold into a cube? (Multiple choice with 4 net drawings.) **Ans.** Only a net with 6 faces arranged so no two faces overlap when folded. Common trick: if the net has two faces that would occupy the same position after folding, it's invalid. Test by mentally folding along edges. **Q4. Top view of a cuboid:** A cuboid lies flat on a table. The top view shows a ___ (rectangle/square). **Ans.** Rectangle (or square, which is a special rectangle). The top view of any cuboid is a rectangle. **Q5. Euler's formula:** For any polyhedron, V − E + F = ___ (where V = vertices, E = edges, F = faces). **Ans.** 2. This is Euler's formula. Example: cube has V=8, E=12, F=6, so 8 − 12 + 6 = 2 ✓. **Strategy:** Learn the exact numbers (8, 12, 6 for cubes/cuboids) and Euler's formula by heart. One-mark questions reward precision.

Three-mark questions demand one worked example, a diagram, and brief reasoning. These five routinely appear: **Q1. Matching nets to solids:** A net of a triangular prism is drawn (showing 2 triangular faces and 3 rectangular faces). Draw the prism and label its vertices, edges, and faces. **Ans.** A triangular prism has: - 2 triangular bases (top and bottom) - 3 rectangular side faces. - Total faces F = 5 - Vertices V = 6 (3 on top triangle, 3 on bottom) - Edges E = 9 (3 on top, 3 on bottom, 3 vertical connectors). Check: V − E + F = 6 − 9 + 5 = 2 ✓. Draw a net with both triangles at opposite ends and three rectangles between them. **Q2. Top, front, and side views:** A solid is made of 4 cubes: 2 cubes on the ground (side by side), 1 cube on top of the left cube, and 1 cube on top of the right cube (creating a peak). Sketch the top, front, and side views. **Ans.** Top view: shows 2 squares side by side (looking down, both ground cubes are visible). Front view: shows a peak—two cubes at the base, one cube in the middle above them. Side view: shows 2 cubes stacked vertically on one side. Each view is different; draw on grid paper for clarity. **Q3. Faces, edges, vertices of a pyramid:** A square pyramid has a square base with side 4 cm and height 5 cm. How many faces, edges, and vertices does it have? **Ans.** Faces: 1 square base + 4 triangular side faces = 5 faces. Vertices: 4 corners of the base + 1 apex = 5 vertices. Edges: 4 edges of the base + 4 edges from base corners to apex = 8 edges. Check: V − E + F = 5 − 8 + 5 = 2 ✓. **Q4. Cube net validation:** Explain why a given net (with 6 squares in a T-shape) is or is not a valid cube net. **Ans.** Fold mentally or on paper. If the net is a T-shape with one square on top, three in a row in the middle, and two below one of the middle squares, it IS valid (folds into a cube with no overlaps). If two squares would overlap after folding, it is NOT valid. Explain by describing which faces fold toward which direction. **Q5. Cross-sections:** A cube is cut by a plane passing through four points: one on each of two opposite edges. Describe the shape of the cross-section. **Ans.** If the plane passes through two points on two opposite edges (not sharing a vertex), the cross-section is typically a rectangle or, in special cases, a square. Draw a cube, mark the two edge points, and shade the plane to show the cross-sectional shape. **Strategy:** Always use Euler's formula to check your answer. Draw diagrams on grid paper. Label vertices (A, B, C, …) and edges (AB, BC, …) so your reasoning is transparent.

Five-mark questions integrate multiple concepts: nets + counting + Euler's formula, or views + cross-sections + 3D reasoning. These three are typical: **Q1. Hexagonal prism: full analysis** A hexagonal prism has a regular hexagon as its base. (a) Draw a net of the hexagonal prism. (b) Count the total number of faces, edges, and vertices. (c) Verify Euler's formula. (d) Draw the top and side views. **Full Solution:** (a) Net: Draw two regular hexagons (top and bottom) and six rectangles connecting them in a strip. Label all 8 faces. (b) Faces F = 2 (hexagonal bases) + 6 (rectangular sides) = 8. Vertices V = 6 (top hexagon) + 6 (bottom hexagon) = 12. Edges E = 6 (top hexagon) + 6 (bottom hexagon) + 6 (vertical connectors) = 18. (c) Euler check: V − E + F = 12 − 18 + 8 = 2 ✓. (d) Top view: regular hexagon. Side view: a rectangle (showing height and one side of the hexagon). Write all labels clearly on diagrams. **Q2. Composite solid from two pyramids** Two square pyramids with base side 6 cm and height 4 cm each are joined at their bases, forming a octahedron-like shape. (a) How many faces, edges, and vertices does the composite solid have? (b) Sketch the front view, side view, and top view. (c) If one pyramid is rotated 45° before joining, how would the views change? Explain. **Full Solution:** (a) Two pyramids joined base-to-base: - Each pyramid: 4 triangular faces, 1 square base, 4 vertices, 8 edges (4 base + 4 to apex). - Combined: Bases merge (disappear internally), so F = 4 + 4 = 8 triangular faces. - Vertices V = 4 (base corners, shared) + 2 (two apexes) = 6. - Edges E = 4 (base) + 4 (one pyramid) + 4 (other pyramid) = 12. - Euler check: 6 − 12 + 8 = 2 ✓. (b) Front view: a diamond (rhombus) or kite shape (two triangles meeting at a vertex). Side view: same as front (by symmetry). Top view: a square (the shared base). (c) If one pyramid rotates 45°, the front and side views become irregular quadrilaterals (not symmetric), and the top view remains a square but with internal lines (edges of both pyramids now visible). **Q3. Unfolding a cube with painted faces** A cube has three faces painted red, two faces painted blue, and one face painted green (adjacent faces known). (a) Draw a valid net of the cube and mark the colours. (b) If the cube is placed with a red face on top, what colour faces are on the bottom and sides? (c) Write down all adjacency relationships (which colours touch). **Full Solution:** (a) Draw a standard cube net (e.g., a T-shape or cross pattern). Place the three red faces such that no two share an edge in the net (they can share only at corners). Place blue and green carefully to respect adjacencies given in the problem. Label each face with its colour. (b) If red is on top, identify which faces in the net correspond to top and bottom when folded. Use the given adjacency rules to deduce the bottom face and the four side faces. (Example: if top is red, bottom might be blue; sides could be red, red, blue, green.) (c) List every pair of colours that share an edge in the 3D cube. From the net, trace which faces fold adjacent and confirm colours match the problem statement. **Strategy for 5-mark answers:** (1) Draw nets carefully and label them. (2) Use Euler's formula as a checkpoint—if it fails, you've miscounted. (3) For views, imagine yourself as a camera: stare straight at the front face, then rotate 90° for the side view. (4) Always justify choices—don't just state an answer without reasoning. This is where 5-mark answers earn full marks.

Starting 2025–26, CBSE is emphasising hands-on reasoning and real-world spatial thinking over rote formulas. Key shifts observed: **1. More composite and irregular solids:** Instead of asking 'How many edges does a cube have?' (rote), exams now ask, 'A solid is made by stacking a cube on a cuboid. How many faces does it have?' (reasoning). You must visualise where faces merge or disappear. **2. Case-study-based questions:** Questions are wrapped in context—e.g., 'A packaging designer needs a net for a cylindrical tin that fits 4 smaller cubes inside. Sketch the net and label dimensions.' These blend geometry with application. **3. Stronger emphasis on views:** Top/front/side view questions now include isometric sketches (3D grid drawings) as answers, not just 2D rectangles. You may be asked to draw isometric views or match isometric sketches to standard views. **4. Cross-sections gaining weight:** Questions about cutting solids with planes are now worth 4–5 marks (not 1–2). Expect detailed reasoning: 'A cube is sliced by a plane. What is the maximum number of sides the cross-section can have?' (Answer: 6, forming a hexagon.) **5. Less formula repetition:** Euler's formula still appears but fewer questions ask you to merely *state* it. Instead: 'Does Euler's formula hold for a hollow cube with one face removed? Explain.' This demands critical thinking. **6. Digital/video-based elements:** Some state boards (e.g., Maharashtra) have begun including video-still questions: 'From this video still of a rotating solid, identify the solid and count its vertices.' CBSE may follow. Practice visualising solids in motion. **Preparation implication:** Don't memorise number of edges for 20 solids. Instead, master the *method* to count faces, edges, and vertices for any solid. Practise drawing nets from scratch (not just recognising them). Spend time with physical models or online 3D viewers (e.g., GeoGebra 3D) to internalise rotation and unfolding.

When you open the exam paper and see Visualising Solid Shapes questions, follow this 7-step protocol to avoid careless mistakes: **Step 1: Read and classify (1 minute per question).** Check the mark value. If it's 1-mark, you need a single fact or a yes/no. If it's 3-mark, a diagram + explanation are mandatory. If it's 5-mark, plan sub-parts mentally before writing. **Step 2: For 1-mark questions, answer instantly.** No long working. Just: 'A cube has 8 vertices' or 'The top view is a rectangle.' But double-check the number—writing '7' instead of '8' is a trap many fall into. **Step 3: For 3-mark net questions, fold mentally.** Before drawing a net, close your eyes and mentally fold it. If two faces would collide, the net is invalid. Mark invalid nets with a cross; don't waste time drawing them. **Step 4: For view questions, use the 'camera angle' method.** Imagine a camera staring directly at the front of the shape. Draw what it sees. Rotate the shape 90° (or rotate yourself) and repeat for side view. Rotate 180° and look down for top view. This prevents mixing up views. **Step 5: Always apply Euler's formula as a check.** After counting faces, edges, and vertices, calculate V − E + F. If it ≠ 2, you've miscounted—find the error before submitting. **Step 6: Label diagrams generously.** Write 'Top view', 'Side view', 'Front view' above each diagram. Label vertices (A, B, C, …) and edges (AB, BC, …) so the examiner can follow your logic. Unlabelled diagrams lose marks even if correct. **Step 7: For composite solids, list changes step by step.** When two solids join, faces might merge. List the original face counts, then subtract merged faces, then verify with Euler. Example: Original cube: F=6, E=12, V=8. Join to another cube: shared face merges, so F = 6 + 6 − 2 = 10, E = 12 + 12 − 4 = 20, V = 8 + 8 − 4 = 12. Check: 12 − 20 + 10 = 2 ✓. **Time allocation:** Spend 3–4 minutes on a 1-mark question (read once, think, answer). Spend 6–8 minutes on a 3-mark question (diagram + two sentences). Spend 15–20 minutes on a 5-mark question (multiple sub-parts, detailed net/views). If you finish early, redraw your diagrams to ensure clarity—messy diagrams lose marks.

Commit these to memory before your exam: **Euler's formula:** V − E + F = 2 (for any closed polyhedron). Check every answer against this. **Cube:** V = 8, E = 12, F = 6. All faces are squares. All edges are equal length. **Cuboid (rectangular prism):** V = 8, E = 12, F = 6. Faces are rectangles. Opposite faces are equal. **Triangular prism:** V = 6, E = 9, F = 5 (2 triangular bases + 3 rectangular sides). **Square pyramid:** V = 5 (4 base + 1 apex), E = 8 (4 base + 4 to apex), F = 5 (1 base + 4 triangular sides). **Triangular pyramid (tetrahedron):** V = 4, E = 6, F = 4 (all faces are triangles). **Hexagonal prism:** V = 12, E = 18, F = 8 (2 hexagonal bases + 6 rectangular sides). **Net validity rule:** A valid net of a cube has 6 faces arranged so that when folded, no two faces occupy the same space and no faces overlap. **View definitions:** - Top view: looking straight down from above. - Front view: looking straight ahead at the front face. - Side view (or side view): looking from the left or right side. **Adjacency in a cube net:** In a T-shaped cube net with one square on top, three in a row, and two below (one of the row), all six faces fold correctly. Any other unusual arrangement may fail. Practise writing these facts daily for one week until they're automatic. Start a 3-day free trial at cbsetutor.ai to get daily visualisation drills and real-time net-folding feedback from our AI tutor.