Isometric scale

The essentials

When a cube is tilted so that its body diagonal is perpendicular to the plane of projection, each of its edges makes the same angle (about 35°16′) with that plane. Every edge therefore appears shorter than its true length by the same factor, cos 35°16′ ≈ 0.816. The isometric scale is a ready-made ruler that converts true lengths into these shortened isometric lengths so that you do not have to multiply each dimension.

Key points

• Draw a horizontal line. From its left end draw one line at 45° and another at 30° to the horizontal.
• Mark true lengths on the 45° line: main divisions of 10 mm, with the first division subdivided into 1 mm parts.
• From each mark drop a line perpendicular to the horizontal (a vertical line) to cut the 30° line. The cut points form the isometric scale.
• The angle between the 45° (true) and 30° (isometric) lines is 15°.
• Ratio: isometric length ÷ true length = cos 45° ÷ cos 30° = √2 : √3 ≈ 0.816 (about 82%).
• Examples: 50 mm true → about 40.8 mm isometric; 60 mm true → about 49 mm isometric.
• Label the scale clearly, with 0 placed after the subdivided part, as on an ordinary diagonal or plain scale.

Exam tips

• The 45° and 30° angles are the two leading angles of the construction; remember which line carries which scale.
• State in your drawing whether you used the isometric scale (projection) or full size (view/drawing).

Common mistakes

• Swapping the true and isometric lines (the true scale is on the 45° line).
• Drawing the projectors parallel to the 45° line instead of perpendicular to the horizontal.
• Using the isometric scale for non-isometric lines such as the slant edges of a pyramid; these are located by their end points instead.