Exercise 13.1 — 3-D to 2-D
Problems related to representation of 3-D figures on 2-D.
What is Exercise 13.1 About?
Exercise 13.1 of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) belongs to Chapter 13 — Visualizing 3-D Shapes in 2-D. This exercise trains you to represent solid three-dimensional objects on flat paper and to "read" a 2-D drawing and mentally reconstruct the 3-D solid it represents.
The four core skills practised here are: drawing 3-D figures on an isometric dot sheet, drawing a cuboid with given dimensions, counting unit cubes hidden inside a solid structure, and sketching the front view, top view, and side view of 3-D figures. These skills are tested in Class 8 board exams in Telangana and Andhra Pradesh, and they build the spatial reasoning foundation used in Class 9 surface area & volume and Class 10 geometry.
Question 1 — Draw the Following 3-D Figures on an Isometric Dot Sheet
This question gives you four different 3-D solid structures and asks you to redraw each one accurately on an isometric dot sheet. The isometric dot sheet preserves the shape and proportions of cubes and cuboids because the dots align along three axes at 60° angles.
The key technique is to start with the top face of the solid, then draw the front-left edge and front-right edge going diagonally downward from the corners, and finally close the bottom edges. For compound shapes (figures made of multiple cubes), draw each cube systematically from front to back.
Draw one cube — top face as a rhombus, then two side faces going down.
A rectangular block = 2 unit cubes placed left-to-right. The top face becomes a wider rhombus.
3 cubes in the bottom row + 1 cube on top of the leftmost cube. Draw bottom row first, then the stacked cube.
3 cubes across the top row + 2 cubes forming the vertical stem below the middle. Resembles the letter T.
- Identify the top face — it always appears as a rhombus (parallelogram) on the isometric sheet.
- Draw the top face first — connect 4 dots to form the rhombus shape.
- Drop vertical edges — from the front corners, draw edges straight downward (or diagonally as per the dot alignment).
- Close the bottom — connect the bottoms of the vertical edges.
- For compound shapes — draw one unit cube at a time, sharing faces where cubes touch.
Question 2 — Draw a Cuboid with Measurements 5 × 3 × 2 Units
A cuboid is a 3-D shape with 6 rectangular faces. Unlike a cube (where all sides are equal), a cuboid has three different dimensions: length (l), breadth (b), and height (h). This question asks you to draw a cuboid with length = 5 units, breadth = 3 units, height = 2 units on the isometric dot sheet.
Cuboid dimensions: Length (l) = 5 units | Breadth (b) = 3 units | Height (h) = 2 units
- Count 5 dot-gaps along the right-diagonal for the length.
- Count 3 dot-gaps along the left-diagonal for the breadth.
- Count 2 dot-gaps straight down for the height.
- Draw the top rhombus first, then drop four vertical edges (2 units each), then close the base.
Question 3 — Find the Number of Unit Cubes in the 3-D Figures
A unit cube is a cube with each edge measuring exactly 1 unit. When 3-D figures are built by stacking unit cubes, you count them layer by layer — starting from the bottom layer and working upward. Some cubes may be hidden inside the structure, so a systematic approach is essential.
The figure shows a staircase shape. Looking carefully:
- The bottom layer has 3 unit cubes arranged in a row.
- On top of the middle cube in the bottom row, there are 2 more cubes stacked vertically.
The figure forms a cross or plus (+) shape when viewed from above. Counting systematically:
- There is a central row of 5 cubes running from left to right.
- On each side of the central (middle) cube, there are 2 additional cubes extending outward — one on each of the two perpendicular directions.
- That gives 2 + 2 = 4 extra cubes for the cross arms.
This figure looks like a stepped platform or pyramid with a flat top. Count by layer:
- Bottom layer: 4 horizontal rows × 4 vertical columns = 4 × 4 = 16 cubes.
- Top layer: 2 horizontal rows × 2 vertical columns = 2 × 2 = 4 cubes sitting in the centre of the base.
This figure is a step pyramid with three distinct layers, each smaller than the one below it:
- Bottom layer: 3 × 3 = 9 cubes (widest layer).
- Middle layer: 2 × 2 = 4 cubes centred on the bottom layer.
- Top layer: 1 cube at the very centre top.
Quick Answer Table — Unit Cube Count
| Figure | Shape Description | Layer Breakdown | Total Cubes |
|---|---|---|---|
| (i) | Staircase | 3 (bottom) + 2 (stacked on middle) | 5 |
| (ii) | Plus / Cross | 5 (centre row) + 4 (side arms) | 9 |
| (iii) | Two-layer platform | 4×4 (bottom) + 2×2 (top) | 20 |
| (iv) | Three-step pyramid | 3×3 + 2×2 + 1×1 | 14 |
Question 4 (Part A) — Find the Area of Shaded Regions
This part uses the same four 3-D figures from Question 3. The shaded region (shown in teal/cyan in the textbook) represents only the top faces of the visible unit cubes. Since every cube face is a square of side 1 unit, each shaded square has area = 1 square unit.
The method is simple: count the number of shaded square faces, then multiply by 1 sq. unit each.
3 top faces are shaded (one per cube in the staircase view)
9 top faces are shaded (all cubes in the cross)
16 top faces of the bottom layer are shaded
11 top faces visible and shaded across all 3 layers
Area of shaded region = (Number of shaded unit squares) × 1 sq. unit
Question 4 (Part B) — Draw Front View, Top View and Side View
When you look at a 3-D solid from different directions, each view gives a flat 2-D picture. There are three standard views used in engineering and mathematics:
- Front View — what you see when you look at the solid straight from the front.
- Top View (or Plan View) — what you see when you look directly down from above.
- Side View (or Elevation) — what you see when you look from the right or left side.
The dot spacing = 1 cm in this question, so each unit square in the view corresponds to exactly one unit cube dimension.
| 3-D Shape | Front View | Top View | Side View |
|---|---|---|---|
| Staircase tower: 3-wide base + 2-high column in middle + 1 extra on top | Cross/T shape — 2 columns wide at base row, narrowing upward | Single column of 4 squares (1 wide × 4 tall) | Single column of 6 squares (1 wide × 6 tall) |
| 3-D Shape | Front View | Top View | Side View |
|---|---|---|---|
| Staircase: 1-high left block + lower right row of 3 | L-shaped: 2 squares tall on left, 1 square tall extending right | Plus/cross pattern — wider than tall | Row of 4 squares × 1 tall on bottom, with 1 extra square on top left |
| 3-D Shape | Front View | Top View | Side View |
|---|---|---|---|
| L-shaped block: tall 2-wide column on left + wide flat 4-wide row on bottom | L-shape — 2 columns wide and 3 tall on left, extending 2 more columns wide at the base | L-shaped footprint: 2 columns wide at top, extending to 6 columns wide at bottom | Tall rectangle: 2 wide × 3 tall on left portion + 6 wide at bottom |
Front View = shadow from front | Top View = shadow from above | Side View = shadow from side
Common Mistakes to Avoid in Visualizing 3-D Shapes
- Using plain graph paper instead of isometric paper: On normal graph paper, cubes look distorted. Always use isometric dot sheets for 3-D figures to get the correct perspective.
- Forgetting hidden cubes: When counting unit cubes in a compound structure, students often miss cubes that are sandwiched inside or hidden behind the front layer. Count layer by layer systematically.
- Confusing top view with front view: The top view shows the floor plan (what you see from directly above). The front view shows the elevation (what you see from directly in front). Many students draw them the wrong way around.
- Not counting shaded faces carefully: In the shaded area questions, only the teal/coloured top faces count. Do not count side faces or hidden faces.
- Drawing cuboid edges without checking dimensions: Count the dot-gaps on the isometric sheet carefully — 5 dot-gaps = 5 units. Miscounting by 1 is a very common error.
Quick Reference — All Answers at a Glance
| Question | Task | Answer / Key Result |
|---|---|---|
| Q1(i) | Draw single cube on isometric sheet | Top face = rhombus; 2 side faces going down |
| Q1(ii) | Draw 2 cubes side by side | Wider top face + 3 vertical faces |
| Q1(iii) | Draw staircase shape (3 + 1 stacked) | L-staircase: 3 bottom + 1 on top of left cube |
| Q1(iv) | Draw T-shape (3 top + 2 stem) | Inverted T: 3 wide across + 2 tall below centre |
| Q2 | Cuboid 5 × 3 × 2 units | Long wide cuboid; length = 5, breadth = 3, height = 2 |
| Q3(i) | Unit cubes — staircase | 5 cubes (3 + 2) |
| Q3(ii) | Unit cubes — plus shape | 9 cubes (5 + 4) |
| Q3(iii) | Unit cubes — 4×4 + 2×2 | 20 cubes (16 + 4) |
| Q3(iv) | Unit cubes — 3-step pyramid | 14 cubes (9 + 4 + 1) |
| Q4a(i) | Shaded area — staircase | 3 sq. units |
| Q4a(ii) | Shaded area — plus shape | 9 sq. units |
| Q4a(iii) | Shaded area — platform | 16 sq. units |
| Q4a(iv) | Shaded area — pyramid | 11 sq. units |
| Q4b | Front, Top & Side views of 3-D figures | 2-D silhouette from each direction (drawn on dot sheet) |
What This Exercise Prepares You For
Mastering Chapter 13 / Exercise 13.1 directly supports your understanding of surface area and volume in higher classes. When you can visualize how a 3-D solid looks from different angles, calculating how much paint covers its surface or how much water fits inside becomes intuitive rather than mechanical.
The skill of counting unit cubes is exactly the reasoning used when you calculate the volume of a cuboid using Volume = l × b × h — each unit cube occupies exactly 1 cubic unit of space. In Telangana and Andhra Pradesh board exams, isometric drawing and view questions regularly appear as 2-mark or 4-mark questions in Chapter 13 papers.
For further practice, explore the introduction to this topic at Introduction to Visualizing 3-D Shapes, and see how surface area is calculated for the same shapes in Class 9 — Surface Areas and Volumes.