Visualising Solid Shapes - Drawing Solids on a Flat Surface (Oblique & Isometric Sketches)
Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
2D Representation of 3D Solids: Since paper is a two-dimensional surface, we use two main techniques to represent three-dimensional objects: Oblique Sketches and Isometric Sketches. These methods create a visual illusion of depth and volume on a flat plane.
Oblique Sketches: These are drawn on squared or grid paper. In an oblique sketch, the front face and its opposite back face are drawn as their actual shapes (e.g., a square for a cube). However, the edges receding into the background are drawn at an angle (usually ) and are not drawn to scale, appearing shorter than their true length to provide perspective.
Isometric Sketches: These are drawn on isometric dot paper, where the dots form a pattern of equilateral triangles. In this representation, the measurements of the edges are proportional to the actual solid. For example, if a cube has a side of units, all edges in the sketch will span exactly unit distances on the dot grid.
The Isometric Grid: Visualizing the isometric grid involves seeing dots arranged such that connecting any three adjacent dots forms an equilateral triangle. Horizontal lines of the solid are drawn at angles to the horizontal base of the paper, while vertical lines remain vertical.
Comparison of Proportions: In an oblique sketch, the dimensions of the front and back faces are preserved, but other edges are distorted. In an isometric sketch, all parallel edges of the solid remain parallel in the drawing, and all three dimensions (length, width, height) can be measured according to the scale of the dots.
Hidden and Visible Edges: To successfully visualize a solid, we must distinguish between visible edges (solid lines) and hidden edges (often represented by dashed or dotted lines). For example, a solid cube has edges, but from a single perspective, only are typically visible.
Mapping 3D to 2D: When drawing, we imagine the object as a skeleton of edges. For a cuboid of dimensions , we translate these into units along the length axis, units along the width axis, and units along the vertical height axis on the isometric paper.
📐Formulae
Euler's Formula for Polyhedrons: , where
Dimensions of a Cuboid:
Total edges in a Cube or Cuboid:
Total vertices in a Cube or Cuboid:
Total faces in a Cube or Cuboid:
💡Examples
Problem 1:
Draw an oblique sketch of a cube with an edge length of units.
Solution:
- Draw the front face as a square with side length units on squared paper. 2. Draw the back face of the same size ( units) but offset slightly upwards and to the right. 3. Connect the corresponding four corners of the front face to the back face using slanted lines. 4. Use solid lines for visible edges and dashed lines for the three edges that are hidden inside the solid.
Explanation:
In oblique sketches, the front face appears in its true shape (a square), while the depth edges are drawn at an angle to create a 3D effect without being strictly proportional.
Problem 2:
Draw an isometric sketch of a cuboid with dimensions (Length , Breadth , Height ).
Solution:
- Start at a dot and draw a vertical line downwards of units to represent the height. 2. From the top point, draw a line units long along the right-slanted dot path to represent length. 3. From the same top point, draw a line units long along the left-slanted dot path to represent breadth. 4. Complete the top face by drawing parallel lines of and units. 5. Draw vertical lines of units down from each corner and connect the bottom points to form the base of the cuboid.
Explanation:
On isometric paper, we use the dots to maintain the exact ratio of . Every line follows the grid of dots, ensuring that parallel edges in the real object are parallel in the drawing.