Drawing a solid on the isometric grid

The cuboid below is drawn on the isometric grid. Every step from one dot to the next dot is 1 unit, and the hidden edges are drawn as dotted lines. How many unit cubes are needed to build this cuboid? Try it first. Pause if you like. Start at the lowest corner. This edge passes 5 dots. But it has only 4 steps between them: 1, 2, 3, 4. So the length is 4 units. Count the steps for the breadth: 1, 2. So the breadth is 2 units. Count the steps up for the height: 1, 2, 3. So the height is 3 units. The number of unit cubes is length times breadth times height. 4 times 2 times 3 is 24. So the cuboid needs 24 unit cubes.

A cuboid is built from 12 unit cubes. Its base is 3 units by 2 units. How many units tall is the cuboid? Draw it on the isometric grid below, with the hidden edges as dotted lines. Try it first. Pause if you like. One layer of cubes covers the base. 3 times 2 is 6, so one layer has 6 unit cubes. 12 divided by 6 is 2, so there are 2 layers. Start at the lowest corner. Count 3 steps for the length and 2 steps for the breadth. There are 2 layers, so count 2 steps up for the height. Complete the solid with 6 more edges. Each one is parallel to an edge already drawn. Draw the three edges that meet at the back-bottom corner as dotted lines, because they are behind the solid. So the cuboid is 2 units tall. Start at the lowest corner, count the steps between the dots, and draw the hidden edges as dotted lines.