A solid in one picture
A plan and two elevations show a solid in three flat views. Isometric dot paper shows it in a single picture. Its dots sit in rows, and every other row is shifted along by half a gap, so each dot has six nearest neighbors, all the same distance away. A step from one dot to a nearest neighbor is called a hop.
Every edge of a cube is drawn as one hop, and every hop runs in one of three directions. Edges that are parallel on the cube are drawn parallel on the paper, so each of the three directions stands for one of the cube’s three directions: its length, its width and its height.
A cube seen from one corner. The corner nearest you is the dot in the middle, where three edges meet, one in each of the three directions.
What the picture shows
The cube is seen from a corner, so three of its faces can be seen, and each is drawn as a rhombus with sides of one hop. The picture has 7 corners and 9 edges. A cube has 8 corners and 12 edges: the eighth corner is at the back, hidden behind the one nearest you, and the 3 edges that meet there are hidden too, so they are not drawn.
Along the three directions one hop is always one unit, so every edge is drawn at its true length, and lengths can be counted in hops. The angles are not true. Every corner of a real cube is a right angle, but on the paper the faces are rhombuses whose angles are 60° and 120°.
Stretch one direction
To draw a cuboid, make the edges in one direction longer. Two hops in one direction and one hop in each of the other two draws a cuboid 2 cubes long, 1 cube wide and 1 cube high. All four edges in the long direction are two hops, because they are parallel and equal on the solid.
The number of hops along each direction counts the cubes along that edge, so a drawing on isometric paper can be read as a count of cubes.
The same drawing with one direction stretched to two hops: a cuboid two cubes long. The three edges in the long direction that can be seen are each two hops.
Counting hidden cubes
A stack of cubes drawn from one corner hides some of its cubes: the cubes under a taller column, and the cubes behind it. Counting only the cubes you can see gives too few.
Count column by column instead. A column’s height in the drawing tells you how many cubes it holds, and every one of them is there, seen or not. A plan with the height of each column written in its square is a quick record of the count.
Worked example: Counting the Cubes in a Stack
Question Identical cubes of edge 2 cm are stacked on a table. Seen from above, the stack covers 3 by 2 positions; the back row has columns 2, 3 and 2 cubes high, and the front row has columns 1, 1 and 1 high. How many cubes are there, and what is the volume of the stack?
1.Back row: 2 + 3 + 2 = 7 cubes.
Back row columns: 2 + 3 + 2 = 7 cubes. 2.Front row: 1 + 1 + 1 = 3 cubes.
Front row: 1 + 1 + 1 = 3 cubes. 3.Total = 7 + 3 = 10 cubes; the cubes under the tall columns are counted even though the view hides them.
7 + 3 = 10 cubes, hidden ones included. 4.Each cube is 2 × 2 × 2 = 8 cm3: volume = 10 × 8 = 80 cm3.
Each cube 8 cm³: 80 cm³.
Answer: 10 cubes; 80 cm3
Common mistakes
- Counting only the cubes that can be seen in the drawing.
- Using 2 cm3 for a cube of edge 2 cm.