What is an axonometric projection?
A solid has three directions: width, depth and height. A drawing on paper has only two. In an axonometric projection we turn and tilt the solid, then send parallel lines from every corner straight onto the paper. The result is one picture that shows three faces.
The three edges that meet at one corner become the three axonometric axes. How far we turn and tilt the solid decides the angles between these axes and how much each axis is shortened.
- Isometric: all three axes shortened equally; angles 120°, 120°, 120°.
- Dimetric: two axes shortened equally, the third differently.
- Trimetric: all three shortened by different amounts.
Isometric projection and the isometric scale
Turn a cube 45° about its vertical axis, then tilt it about 35.26° forward, so you look along its long diagonal. Each edge now makes the same angle with the paper. On paper the height axis is vertical and the width and depth axes go off at 30° to the horizontal. So the three axes are 120° apart.
Because each edge slopes away from you, it looks shorter. Its seen length is √(2/3) ≈ 0.82 of the true length. This ratio is the isometric scale. A 100 mm edge is drawn about 82 mm long in a true isometric projection.
Isometric drawing vs isometric projection
In practice, people often skip the 0.82 and draw every edge at full length. This is an isometric drawing (or isometric view). It has the same shape, just about 22% bigger (1 ÷ 0.82 ≈ 1.22). Only lines parallel to the axes (isometric lines) can be measured; slanted lines must be drawn by joining points.
Oblique views: cavalier and cabinet
In an oblique view the front face is drawn flat, at true size and true shape. Depth lines go back at an angle, usually 45°.
- Cavalier: depth drawn at full length. Easy, but the object looks too deep.
- Cabinet: depth drawn at half length. It looks more natural.
Put the face with circles or the most detail at the front: in oblique view those circles stay true circles. Oblique views are quick for packaging sketches: draw the box front, add depth, and you can also lay out its net (the flat unfolded box) to check the flaps.
Circles, solids and technical sketches in isometric
A circle on any face of an isometric drawing becomes an ellipse. Draw it in four steps (four-centre method): draw the square around the circle as an isometric rhombus; mark the midpoints of its sides; from the two obtuse corners draw lines to the opposite midpoints; use the four crossing points as centres for four arcs.
To draw a solid from its front, top and side views: draw a "crate" (box) with the overall width, depth and height along the isometric axes, then cut away parts and add details. Hidden edges are usually left out.
A technical sketch is a freehand isometric drawing, often on isometric dot paper, with light construction lines first and dark outlines last.
Try it: an isometric sketch of a matchbox
Take a matchbox (or any small box). Measure its width, depth and height in mm. On paper draw a vertical line for the nearest corner, then two lines at 30° for width and depth. Mark the lengths and complete the box with lines parallel to the axes. Then press Isometric and Cabinet in the 3D and compare with your sketch.
Key formulas and definitions
- Isometric axes: 120° apart; width and depth at 30° to horizontal
- Isometric scale = √(2/3) ≈ 0.816 (true projection length ÷ true length)
- Isometric drawing size ≈ 1.22 × isometric projection size
- Cavalier: depth × 1 at 45°; Cabinet: depth × ½ at 45°
- Tilt for isometric = arctan(1/√2) ≈ 35.26°, turn = 45°
Worked examples
1. An edge of a cube is 50 mm. How long is it drawn in (a) an isometric drawing (b) an isometric projection?
(a) Full length: 50 mm. (b) Use isometric scale: 50 × 0.816 ≈ 40.8 mm.
2. A box is 60 mm wide, 40 mm deep and 30 mm high. Give the depth line length in cavalier and cabinet oblique.
Cavalier keeps depth full: 40 mm at 45°. Cabinet halves it: 20 mm at 45°. Width 60 mm and height 30 mm stay true in both.
3. In an isometric projection an edge measures 65.3 mm on paper. Find its true length.
True length = seen length ÷ 0.816 = 65.3 ÷ 0.816 ≈ 80 mm.
4. A circle of diameter 40 mm is on the top face. How do you draw it in isometric drawing?
Draw a 40 mm isometric rhombus (sides along the two horizontal isometric axes). Mark midpoints, join the obtuse corners to opposite midpoints, and draw four arcs from the crossing points. The result is an ellipse, not a circle.
5. Why does an isometric drawing look bigger than an isometric projection of the same object?
The drawing uses true lengths (×1), while the projection uses 0.816. 1 ÷ 0.816 ≈ 1.22, so the drawing is about 22% larger in every direction. The shape is identical.
6. Which view would you use for a round clock face on a box, and why?
An oblique (cabinet) view with the clock face at the front. In oblique views the front face is true shape, so the circle stays a circle and is easy to draw with a compass.
Common mistakes
- Drawing width and depth at 45° in isometric. In isometric they are at 30° to the horizontal; 45° is for oblique.
- Measuring along a slanted (non-isometric) line. Only lines parallel to the three axes keep the scale; find slanted edges by joining their end points.
- Drawing a circle on an isometric face with a compass. It must be an ellipse.
- Mixing up cavalier and cabinet: cavalier = full depth, cabinet = half depth.