Isometric Projection: Why 0.8165, and Why You Draw Full Size

Isometric projection is a pictorial drawing in which the three axes of an object are equally foreshortened (shortened because they are tilted away from you) and sit 120 degrees apart. Every edge parallel to an axis shrinks to 0.8165 of its true length, which is √(2/3). Edges not parallel to an axis shorten by different amounts, which is covered below. The object is first turned 45 degrees about a vertical axis, then tilted forward by 35°16′, as the IIT Guwahati lecture notes on isometric projections set out.

Here is the working for 0.8165 (our derivation). After the tilt, the body diagonal of a cube points straight at you, so all three edges meet your line of sight at the same angle. The direction cosines (the cosine of each edge's angle to the line of sight) must satisfy l² + m² + n² = 1 with l = m = n, so each is 1/√3. An axis-parallel edge's projected length is its true length × √(1 − 1/3) = √(2/3) = 0.8165. That is also cos 35.264°.

You can also build an isometric scale on paper. Draw a horizontal line, then draw lines at 30° and 45° to it from one point. Mark true lengths along the 45° line and drop verticals from each mark to the 30° line. The lengths read off the 30° line are the shortened ones, because cos 45° ÷ cos 30° = 0.8165 again.

Draw an isometric view at full size, unless the question tells you to use the isometric scale; that is our recommendation. An isometric view (also called an isometric drawing) has the same shape as an isometric projection but uses true lengths along the axes, so it comes out about 22.5% larger (our calculation: 1 ÷ 0.8165 = 1.2247). The same IIT Guwahati notes add that engineering drawings almost always use it because it is easier to produce.

In the November 2024 SPPU paper, you attempt one question from each of five either/or pairs: 12 + 12 + 14 + 16 + 16 = 70 marks. Q9 gives a front view and a top view, and Q10 gives a front view and a side view. Both say "Draw isometric view and show overall dimensions". Q9 is even headed "Isometric Projection", yet the drawing it asks for is an isometric view.

Scaling every length by 0.8165 gives you the identical shape, only smaller, and it adds a multiplication at every step where a slip can creep in. In our view the exam minutes are better spent on getting the shape and the dimensions right. The dimensions you write on the drawing are the true sizes (50 for a 50 mm edge, never the scaled value).

Before you start, read the two given views correctly. They are orthographic views, so if you can already draw all three views of an object (see Orthographic Projection: Draw Three Views, Find the Missing One), you will place the front, top and side faces correctly in the isometric. Check which projection method the paper uses too: Q7 of that paper states first angle, and the rule for telling it from third angle is in First Angle vs Third Angle Projection: The Rule for SPPU.

Only lines parallel to the three isometric axes keep a fixed ratio to their true length. Those are isometric lines, and you can measure them. Every other line is a non-isometric line, such as a slanted edge or a face diagonal. Each one is foreshortened by a different amount, so you plot it from its end points and never measure it.

Take a 50 mm square top face drawn as a full-size isometric view (our working). Let x run along the right-hand 30° axis and y along the left-hand 30° axis, so a point (x, y) lands at horizontal (x − y) cos 30° and vertical (x + y) sin 30°.

• The diagonal from (0, 0) to (50, 50) draws as a vertical line 50 mm long.
• The diagonal from (50, 0) to (0, 50) draws as a horizontal line 86.6 mm long.
• Both diagonals are 50√2 = 70.7 mm in real life.

One true length is drawn as 50 mm and as 86.6 mm in the same view. A student who measures a slanted edge on the front view and lays that length along an axis gets the wrong edge.

The fix is the box method:

1. Draw the box that encloses the object, using the overall length, width and height.
2. Find each end point of the slanted edge by its offsets along the three axes.
3. Join the two end points.

The box does the measuring and the slanted line follows from it.

Draw the circle's isometric square, a rhombus whose sides equal the circle's diameter. Then join each obtuse corner to the midpoints of its two far sides. The two obtuse corners and the two crossing points are the four centres of the arcs that form an approximate ellipse. This is the four-centre method. It gives a close ellipse rather than an exact one, which is accepted in engineering drawing.

For a circle of diameter 40 mm (our calculation):

The steps, shown for a circle on the top face:

1. Draw the rhombus with side 40 mm on the face that holds the circle.
2. From each of the two obtuse corners, join to the midpoints of the two opposite sides. That makes four lines, which cross at two points.
3. Centre the compass on an obtuse corner, with radius equal to the distance from that corner to a midpoint it was joined to. Draw the large arc between the two midpoints it was joined to. Repeat from the other obtuse corner.
4. Centre on each crossing point, with radius equal to the distance from that point to the nearest midpoint, and draw the small arc that joins the two large arcs.
5. A cylinder of height h is the same ellipse copied h upwards on the vertical axis, joined by two vertical lines at the sides.

The same four-centre method works on a vertical face. Build the rhombus from the vertical axis and the 30° axis that lie in that face, and the major axis of the ellipse runs perpendicular to the axis that is not in that face. On the top face that axis is vertical, so the major axis lies horizontal. If yours leans, one of your centres is misplaced.

Run these seven tests on your answer with a scale and a 30°-60° set square. Each has a pass and a fail, so you know in a few seconds whether to fix the line.

Engineering Graphics - Pune University, taught by Rahul Jewaani (RG Sir), our founder, is on our store at ₹1,399 as of October 2026 (list price ₹1,799). For how the 70 marks split across the paper, read Engineering Graphics: Where the 70 SPPU Marks Live.

Padhna Aasaan Hai, and drawing gets easier once you know which scale the question wants. See the course on the RG Lectures store.