Isometric axes, scale and isometric length
Take a cube and tilt it so that its long diagonal points straight at your eye. All three edges from the nearest corner look equal and 120° apart. These are the isometric axes.
Because each edge is tilted away from you, it looks shorter. The shortening is the same for all three axes:
isometric length = true length × √(2/3) ≈ 0.816 × true length
A ruler made with this ratio is the isometric scale. Make it by drawing a line at 45° (true lengths) and one at 30° (isometric lengths) from the same point and dropping vertical lines from the 45° line to the 30° line.
Isometric projection uses the isometric scale. Isometric drawing (view) uses true lengths, so it is 1.225 times larger but the same shape. Lines not parallel to an axis (non-isometric lines) are never measured directly; find their end points first.
Prisms and pyramids in different positions
Method: box it in. Draw an isometric box (rectangular prism) that just holds the solid, using isometric lengths. Then locate the corners of the base and top on the faces of the box.
- Cube / square prism standing: draw the base rhombus, raise verticals, join the top.
- Hexagonal prism: draw the hexagon inside its enclosing rectangle (side a gives width 2a and depth √3·a). Mark the corners, then raise all corners by the height.
- Pyramid: draw the base, find its centre, go up the axis by the iso height, join the apex to every corner.
- Lying positions: the axis is horizontal, parallel to one of the two 30° axes. The base face is now a vertical face of the box. Everything else is the same.
Hidden edges are usually not shown in isometric.
Cylinder, cone and the four-centre ellipse
A circle of diameter d on a face is drawn inside an isometric square (rhombus) of side 0.816d.
- Draw the rhombus. Mark the midpoints of its sides.
- From each obtuse-angle corner, draw lines to the midpoints of the two opposite sides.
- These lines cross at two points. Now you have four centres: the two obtuse corners and the two crossing points.
- Draw the two big arcs from the obtuse corners and the two small arcs from the crossing points. They meet at the midpoints.
Cylinder: draw the top and bottom ellipses, join them with two vertical tangent lines. Cone: draw the base ellipse, go up the axis by the iso height, draw two tangents from the apex. In lying positions, the ellipses sit on vertical faces.
Sphere and hemisphere
A sphere looks like a circle from any direction. In an isometric projection draw a circle of the true radius R. Its centre lies on the vertical through the point of contact, at an iso height of 0.816 × R above it. (In an isometric drawing the circle radius would be 1.225 R.)
Hemisphere on its flat face: draw the base circle as an iso ellipse (four-centre method) and then the curved top as a semicircle of true radius R from the same centre. Both meet the ellipse at its ends.
Try it: With a 30°–60° set-square, draw the three axes. Mark 40 mm true = 32.7 mm iso on each. Build a 40 mm cube. Then place a 40 mm diameter ball on top: its circle has radius 20 mm, its centre 16.3 mm above the top face centre. Check with the 3D: choose Sphere.
Key formulas and definitions
- Isometric length = 0.816 × true length (√2 / √3)
- Isometric drawing size = 1.225 × isometric projection size
- Axes: 120° apart; receding axes at 30° to the horizontal
- Circle of diameter d → ellipse in rhombus of side 0.816d (four-centre method)
- Sphere in isometric projection: circle of radius R; centre 0.816R above the point of contact
Worked examples
1. Find the isometric length of a 50 mm edge.
Iso length = 0.816 × 50 = 40.8 mm.
2. An isometric length measures 24.5 mm. What is the true length?
True = 24.5 ÷ 0.816 ≈ 30 mm.
3. Draw a square prism, base 30 mm, height 60 mm, standing on its base, in isometric projection. Give the lengths you use.
Base sides: 0.816 × 30 = 24.5 mm along both 30° axes. Height: 0.816 × 60 = 49 mm vertical. Draw the base rhombus, raise four verticals of 49 mm, join the top.
4. A cone, base diameter 50 mm, height 60 mm, stands on its base. Give the steps and sizes.
Iso square for the base: side 0.816 × 50 = 40.8 mm. Draw the ellipse by four-centre method. From the centre go up 0.816 × 60 = 49 mm to the apex. Draw two tangents from apex to the ellipse.
5. A sphere of diameter 40 mm rests on the ground. What radius do you draw, and how high is the centre?
Isometric projection: circle radius = true radius = 20 mm. Centre is 0.816 × 20 = 16.3 mm vertically above the contact point (iso height).
6. A hexagonal prism (side 25 mm, length 70 mm) lies on a rectangular face with its axis parallel to a 30° axis. What box do you draw?
Hexagon across corners = 50 mm; across flats = 25√3 ≈ 43.3 mm. Lying on a face, the height is 43.3 mm and the width is 50 mm. Iso: 0.816 × 43.3 = 35.3 mm, 0.816 × 50 = 40.8 mm, and length 0.816 × 70 = 57.1 mm. Draw the hexagon on the end face of the box, then copy its corners 57.1 mm along the axis.
Common mistakes
- Using true lengths when the question says "isometric projection". Use the isometric scale (× 0.816).
- Measuring a sloping (non-isometric) line directly. Locate its end points along isometric lines, then join.
- Drawing a sphere with radius 0.816R in isometric projection. The circle uses the TRUE radius; only the centre position is scaled.
- Drawing ellipses freehand without the rhombus. Use the four-centre method so the ellipse touches the midpoints of the rhombus sides.