What does "placed centrally" mean?
Two solids are placed centrally when their axes are in one straight line. The centre of the upper solid's base sits exactly on the centre of the lower solid's top face.
Common pairs: a cone or cylinder on a square or hexagonal slab, a cube on a cylinder, a sphere on a frustum, a pyramid on a prism. Usually the lower one is wider so the upper one fits on it.
Steps to draw the combination
- Read the sizes and change every length to isometric length (× 0.816), or use the isometric scale.
- Draw the lower solid completely, with its top face clearly drawn.
- Find the centre of the top face by joining its diagonals (for a circle, the ellipse centre).
- On this top face, draw the base of the upper solid around the centre (rhombus, hexagon or ellipse) using iso lengths.
- Go up the common vertical axis by the upper solid's iso height and complete it (apex, top face or tangents).
- For a sphere on top: circle of TRUE radius, centre 0.816 × R above the top face centre.
- Darken visible edges; leave out hidden lines. Write the dimensions if asked.
Total iso height = 0.816 × (h₁ + h₂).
Direction of viewing
The question often shows the front view (or an arrow) and says "from the direction of the arrow". This tells you which side of the object is the front. In the isometric picture, that face goes on the left or right lower front face.
- If an edge or corner of a square slab points at you, you see two side faces equally.
- If a face points straight at you, one face is in front and its sides go back along the axes.
- For round solids (cylinder, cone, sphere) the direction changes nothing in the outline, but it still matters for the base or slab under them.
Always check: features that the arrow says are in front must appear on the front faces of your drawing.
Exam tips and Try it
- Start the lower solid low on the sheet so the tall total height fits.
- Keep the construction lines thin; final lines thick.
- Mark the origin corner or point and the arrow direction on your drawing.
Try it: Stack a small cup upside down centrally on a matchbox. Look from a corner of the box. Sketch the box in isometric, mark the centre of its top by crossing its diagonals, and draw the cup's ellipse around that point. Then turn the box 45° and look again: which face now comes to the front?
Key formulas and definitions
- Common axis: centre of upper base = centre of lower top face
- Total iso height = 0.816 × (h₁ + h₂)
- Sphere on top: radius R (true), centre 0.816R above contact point
- Top-face centre: crossing point of the diagonals of the iso rhombus
- Direction of viewing decides the front faces
Worked examples
1. A cylinder (Ø40, height 50) stands centrally on a square slab 80 × 80 × 20. Find the iso sizes.
Slab: 0.816 × 80 = 65.3 mm sides; thickness 0.816 × 20 = 16.3 mm. Cylinder: ellipse in rhombus of side 0.816 × 40 = 32.6 mm; height 0.816 × 50 = 40.8 mm. Total iso height = 0.816 × 70 = 57.1 mm.
2. A cone (base Ø50, height 60) rests centrally on a cylinder (Ø70, height 30). Give the steps.
Draw the cylinder: ellipses in rhombus of side 57.1 mm, iso height 24.5 mm. At the top ellipse centre, draw the cone base ellipse in rhombus of side 40.8 mm. Go up the axis 0.816 × 60 = 49 mm to the apex. Draw two tangents from the apex. Total iso height 73.5 mm.
3. A sphere of Ø50 rests centrally on a cube of side 50. Where is the sphere centre?
Draw the cube with iso sides 40.8 mm. Find the top face centre (diagonals). Sphere centre is 0.816 × 25 = 20.4 mm above it. Draw a circle of TRUE radius 25 mm.
4. A square pyramid (base 40, height 60) stands centrally on a hexagonal slab (side 40, thickness 20). Find the total iso height.
Total true height = 20 + 60 = 80 mm. Iso height = 0.816 × 80 = 65.3 mm.
5. A square slab is viewed so that one corner points at the viewer. Then it is viewed with a face straight in front. What changes in the drawing?
Corner in front: two side faces show equally, both edges along the 30° axes. Face in front: the slab is turned 45°, so its base edges are no longer along the axes; you box it and find corners from the enclosing square.
6. A cube (side 30) sits centrally on a cylinder (Ø60, height 25). The cube's faces are parallel to the axes. Find the total iso height and the cube's iso side.
Cube side iso = 0.816 × 30 = 24.5 mm. Total height = 25 + 30 = 55 mm → iso 0.816 × 55 = 44.9 mm.
Common mistakes
- Putting the upper solid on a corner of the top face instead of its centre. Always join diagonals to find the centre.
- Adding true heights but forgetting to multiply by 0.816 for projection.
- Shrinking a sphere's radius on top of a solid. The circle keeps the true radius; only the centre height is scaled.
- Ignoring the arrow. The face shown by the direction of viewing must be in front.