How unit 3 is examined
Projections of prisms, pyramids and cones tilted to HP and VP by the stage method (high weight, every paper), then dimensioning and scale (unasked) and a 1-BHK floor plan (low).
Projections of Regular Solids inclined to both the Planes - Auxiliary Views
<span style="display:inline-block;padding:.16em .6em;border:1.5px solid currentColor;border-radius:999px;font-size:.68em;font-weight:700;letter-spacing:.06em;text-transform:uppercase;opacity:.75">High weight</span>
Definition. <mark>A solid inclined to both planes is drawn in stages: first in a simple position (axis perpendicular or parallel to a plane), then tilted for the first inclination, then rotated for the second inclination.</mark>
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Key points.
- In the simple position a solid standing on its base has the axis perpendicular to HP, so the top view shows the true shape of the base and the front view is a rectangle or triangle.
- A solid lying on a face or generator has its axis parallel to HP, so the front view shows the true base shape when the axis is perpendicular to VP.
- Stage 2 tilts the view that shows the inclination to HP; the resting edge, face or generator is placed on or at the given angle to the xy line, with the shape unchanged and only redrawn at the new angle.
- Stage 3 rotates the top view of stage 2 so the resting edge makes the VP angle with xy, with the shape unchanged and only redrawn at the new angle; heights are projected across from stage 2.
- Each new view is drawn by projecting vertically from the view just turned and horizontally from the view that keeps the same height.
- Rule of thumb: an inclination to HP is set in the front view, an inclination to VP is set in the top view.
- Visible edges are continuous and hidden edges dashed; an edge is hidden if it is covered by a nearer face in that view.
- Label all corners a, b, c ... in the top view and a', b', c' ... in the front view, and always mark xy.
Formula. Triangle altitude $= s\frac{\sqrt3}{2}$; tetrahedron height $H=\sqrt{\tfrac23}\,s$; cone slant $l=\sqrt{h^2+r^2}$; semi-apex angle $\alpha=\tan^{-1}(r/h)$.
Example. Tetrahedron, edge 30 mm, resting on an edge, edge $45^\circ$ to VP, face $30^\circ$ to HP (Jun 2024).
| Stage | Work |
|---|---|
| Given | $s=30$; $H=\sqrt{2/3}\times30=24.5$ mm |
| 1 | Triangle abc in TV with edge ab perpendicular to xy; apex o at centroid; FV a'b'c' base with o' at 24.5 mm |
| 2 | Tilt FV so the face through the resting edge makes $30^\circ$ with xy and the edge lies on xy; project new TV from stage 1 TV |
| 3 | Rotate stage 2 TV so resting edge makes $45^\circ$ with xy; project final FV |
Answer: final FV and TV of the tetrahedron with the resting edge on HP.
Other given data.
| Question | Shape values | Method |
|---|---|---|
| Triangular prism, face on HP, axis parallel to VP | Altitude $30\times\frac{\sqrt3}{2}=25.98$ mm | Single stage: FV rectangle 55 x 25.98 on xy; TV rectangle 55 x 30 with centre line; end view equilateral triangle side 30 |
| Pentagonal prism 25 x 50, face on HP, axis $45^\circ$ to HP | Pentagon side 25 | Stage 1 axis parallel to HP and perpendicular to VP; stage 2 tilt axis $45^\circ$ in FV; project TV |
| Cone dia 40, h 80, on a generator | $l=\sqrt{80^2+20^2}=82.46$ mm | Stage 1 upright; stage 2 tilt FV so a generator lies on xy; project TV (ellipse for base, apex) |
| Cone dia 50, h 62, generator perpendicular to HP | $\alpha=\tan^{-1}(25/62)\approx22^\circ$ | Stage 2 tilt so one generator is vertical, rim on xy; project TV |
| Pentagonal prism 30 x 90, base edge on HP, edge $30^\circ$ VP, face $45^\circ$ HP | Pentagon side 30 | Stage 1 upright; stage 2 tilt face to $45^\circ$; stage 3 rotate resting edge to $30^\circ$ |
| Hexagonal prism 30 x 75, axis $30^\circ$ to VP, parallel to HP | Hexagon side 30 | Stage 1 hexagon in FV, edge vertical; stage 2 turn TV axis to $30^\circ$ with xy; project FV |
Answer frame. Open with "The solid is drawn in stages: simple position, then the HP inclination, then the VP inclination"; draw the stage 1 views, then the tilted views, then the final views with the given angles marked; develop points 1-4 in order; finish with visible and hidden lines and the sentence "Hence the final projections are complete".
Pitfall: Setting the VP angle in the front view or the HP angle in the top view; the tilt to HP is done in the front view, the rotation to VP in the top view.
Asked: [14 marks] (Dec 2023, Jun 2023) A triangular prism of side of base 30 mm and axis 55 mm long lies on one of its rectangular faces in HP with its axis parallel to VP. Draw its projection. Asked: [14 marks] (Dec 2024) A right pentagonal prism 90 mm high with each side of the base 30 mm is resting on one of the base edges on the HP and inclined at $30^\circ$ to VP and the face containing that edge is inclined at $45^\circ$ to the HP. Draw the projections. Asked: [14 marks] (Jun 2024) A regular tetrahedron edge of base 30 mm is resting on one of its edges on the HP. The resting edge makes $45^\circ$ to VP and the face containing the edge makes $30^\circ$ to HP. Draw its projections. Asked: [7 marks] (Nov 2022) Draw the projections of a pentagonal prism of base 25 mm side and 50 mm long resting on one of its rectangular faces in HP with its axis inclined at $45^\circ$ to HP. Asked: [7 marks] (Nov 2022) A right circular cone of axis height 80 mm is resting on one of its generators in HP. Draw its projections. Base is 40 mm dia. Asked: [7 marks] (Jun 2023) A right circular cone, base diameter 50 mm and axis 62 mm, rests on its base rim on HP with axis parallel to VP and one of the elements perpendicular to HP. Draw the projections. Asked: [7 marks] (Jun 2025) Draw the projection of a hexagonal prism base 30 mm and axis 75 mm long when its axis is inclined at $30^\circ$ to the VP and parallel to HP and edge of the base is perpendicular to HP.
Annotation, dimensioning and scale
<span style="display:inline-block;padding:.16em .6em;border:1.5px solid currentColor;border-radius:999px;font-size:.68em;font-weight:700;letter-spacing:.06em;text-transform:uppercase;opacity:.75">Not asked since 2022</span>
Definition. <mark>Dimensioning is the writing of sizes on a drawing with dimension lines, extension lines, arrowheads and figures so that the object can be made without asking.</mark>
Key points.
- Extension lines are thin lines that project out from the outline and do not touch it, and dimension lines with arrowheads lie between them.
- Figures are written above the dimension line, read from the bottom or the right, in millimetres with no unit shown.
- Each dimension is given once only and never repeated; a centre line is never used as a dimension line.
- Scale is the ratio drawing size to object size: 1:2 reduces, 2:1 enlarges, 1:1 is full size, and it is written in the title block.
Floor plans: windows, doors and fixtures
<span style="display:inline-block;padding:.16em .6em;border:1.5px solid currentColor;border-radius:999px;font-size:.68em;font-weight:700;letter-spacing:.06em;text-transform:uppercase;opacity:.75">Low weight</span>
Definition. <mark>A floor plan is a horizontal section of a building taken about 1 m above the floor, showing walls, doors, windows, room sizes and fixtures.</mark>
Key points.
- A 1-BHK plan has one hall, bedroom, kitchen, bath/WC and a verandah, for example hall 3.5 x 4.0 m, bedroom 3.0 x 3.5 m, kitchen 2.5 x 3.0 m, bath/WC 1.5 x 2.0 m.
- Walls are shown hatched or thick, external 200 or 300 mm and internal 100 or 150 mm thick.
- Doors D1, D2 are drawn with a swing arc and windows W1, W2 with a thin double line across the wall; a ventilator is V.
- Fixtures are the kitchen slab and sink, WC, wash basin, and shower; draw dimension lines and label each room.
Asked: [7 marks] (Jun 2025) Draw the floor plan of a single BHK house showing all details like window, doors and other fixtures. Assume suitable dimension for plan.
Last-minute revision
- Solids inclined to both planes: draw in 2-3 stages, simple position first.
- HP angle is set in the front view; VP angle is set in the top view.
- Equilateral triangle altitude $=0.866s$, so 30 mm gives 25.98 mm.
- Tetrahedron height $=\sqrt{2/3}\,s$, so 30 mm gives 24.5 mm.
- Cone slant $=\sqrt{h^2+r^2}$; dia 40, h 80 gives 82.46 mm.
- Semi-apex angle for dia 50, h 62 is $22^\circ$.
- Hidden edges dashed, visible continuous.
- Scale 1:2 reduces, 2:1 enlarges.
- Floor plan: walls, D/W symbols, room sizes, fixtures, dimensions.
Memory hooks
- "Tilt in front, turn in top": HP angle in FV, VP angle in TV.
- Stage 1 simple, stage 2 tilt, stage 3 turn.
- Cone on a generator: tilt the front view until that generator lies on xy.
- D for door with an arc, W for window with a double line.
Coverage checklist
- Projections of Regular Solids inclined to both the Planes- Auxiliary Views: triangular prism (Dec 2023, Jun 2023), pentagonal prism (Dec 2024, Nov 2022), tetrahedron (Jun 2024), cones (Nov 2022, Jun 2023), hexagonal prism (Jun 2025).
- Draw simple annotation, dimensioning and scale: none asked.
- Floor plans that include: windows, doors, and fixtures such as WC, bath, sink, shower, etc.: 1-BHK plan (Jun 2025).