Sketching & visualising prisms
A clear drawing helps you see all of a solid — including the parts hidden at the back. One easy way to draw a rectangular prism is oblique projection: draw the front face, copy it shifted up and across for the back, then join matching corners.
Sketch
Q1. Following the three steps above, sketch a rectangular prism that is 3 wide, 2 high and 2 deep. Show the hidden edges as dashed lines.
Q2. Sketch a cube. How is sketching a cube the same as sketching any other rectangular prism?
Visualise the hidden cubes
Q3. A prism is drawn as a solid block 3 long, 2 wide and 2 high. In the drawing you can only see the cubes on the outside. How many unit cubes are hidden inside or behind, and how many are there altogether?
Q4. Look at a solid built from a 4 × 3 base, 2 layers high. Without building it, how many cubes touch the table (the bottom layer)? How many cubes are in the whole solid?
Q5. Reasoning. Why is it easy to miss cubes when you only look at a drawing of a solid? Describe one thing you can do to make sure you count them all.
Answer key (teacher use)
Suggested answers for the questions in this worksheet. Sketching answers are examples — accept any correct, clearly drawn equivalent.
Q1. A correct oblique sketch of a 3 × 2 × 2 prism: a 3-wide by 2-high front rectangle, an identical rectangle shifted up and to the right, and four lines joining matching corners. The three hidden edges (the back-left vertical and the two edges meeting the hidden back-bottom corner) should be dashed.
Q2. A cube is sketched exactly the same way; the only difference is that all three dimensions (length, width, height) are equal, so the front face is a square and the depth shift equals the side length.
Q3. The solid is 3 × 2 × 2 = 12 cubes altogether. The number hidden depends on the viewpoint, but in a standard oblique view roughly 3–4 cubes are fully hidden (the back-bottom corner cubes); accept any answer that reasons "total 12 minus the ones I can see." The key understanding is total = 12.
Q4. Bottom layer (touching the table) = 4 × 3 = 12 cubes. Whole solid = 12 × 2 = 24 cubes.
Q5. A drawing only shows the faces pointing toward you, so cubes at the back, underneath, or in the middle are hidden. Sample strategy: count one layer and multiply by the number of layers, or build the solid with real cubes to reveal the hidden ones.