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BT-105 · Engineering Graphics/Quick Revision Short Notes

Engineering Graphics (BT-105) - Unit 4 Short Notes

How unit 4 is examined

This unit covers cutting solids with section planes (true shape by an auxiliary view) and developing their surfaces; the sectioning drawings carry almost all the marks (14, 14, 14, 7, 7), development is a 7-mark drawing.

Prism, Cylinder, Pyramid, Cone – 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 section is the shape of the cut face made when a solid is cut by an imaginary section plane; when the plane is inclined to the reference planes, the true shape of the section is found by projecting it on an auxiliary plane parallel to the section plane.</mark>

Key points.

  1. A section plane is shown by its trace: a plane perpendicular to VP (AIP) appears as an inclined line in the front view, so the cut points are read directly in the FV.
  2. A plane parallel to HP (horizontal section) shows the true shape of the cut in the top view, so no auxiliary view is needed.
  3. Cut points are found where the trace line crosses the edges (pyramid, prism) or generators (cone, cylinder); they are projected to the top view along projectors.
  4. The part of the solid between the observer and the plane is imagined removed; the cut face in the sectional view is hatched with thin lines at $45^\circ$, equally spaced.
  5. The apparent section is the hatched cut face in the sectional TV/FV; the true shape is drawn on an auxiliary plane parallel to the trace, so new xy line is drawn parallel to the trace.
  6. For the true shape, project from each cut point perpendicular to the trace, and measure distances from the top view (widths) along these projectors from the new xy line.
  7. Standard true shapes: cylinder cut obliquely gives an ellipse, cone cut parallel to a generator gives a parabola, a pyramid or prism gives a polygon; a hollow solid gives two concentric outlines.
  8. Hatching covers only the cut material, so the hole of a hollow solid stays unhatched.

Steps.

Step 1: Draw the top view and front view of the solid in its given position.
Step 2: Draw the section trace in the FV at the given angle through the given point.
Step 3: Mark the cut points on the edges/generators and label them.
Step 4: Project the cut points down to the top view, on the same edges, and join them.
Step 5: Hatch the cut face at 45 degrees; draw the remaining solid.
Step 6: Draw a new xy line parallel to the trace, project the points perpendicular to it, take widths from the TV and join.

Example. Square pyramid, base 35 mm, height 50 mm, base edges $45^\circ$ to VP, plane $32^\circ$ to HP bisecting the axis. The corner half-diagonal is $35/\sqrt2 = 24.75$ mm. The trace passes through the axis point at 25 mm. It meets the two corner edges seen in the FV at heights about 30.9 mm and 13.8 mm above the base, and the two edges hidden behind the axis at 25 mm. Project these four points to the TV, join them for the quadrilateral, hatch it, then draw the true shape on the auxiliary plane.

Hexagonal pyramid (horizontal cut). Base side 30, axis 60, cut at 38 mm: the section is a regular hexagon of side $30\times(60-38)/60 = 11$ mm, seen true in the sectional TV.

Diagram. <figure class="ds-fig" style="margin:1.4rem 0;overflow-x:auto"><svg xmlns="http://www.w3.org/2000/svg" id="dsfig-u4-01" viewBox="0 0 467 80" width="467" height="80" role="img" aria-label="Order of the section drawing: FV with trace, sectional TV, then auxiliary view (true shape) parallel to the trace"><style>#dsfig-u4-01 .e{stroke:#454C5A;stroke-width:1.4;fill:none}#dsfig-u4-01 .e.hi{stroke:#2340B8;stroke-width:2.6}#dsfig-u4-01 .n{fill:#FFFFFF;stroke:#16181D;stroke-width:1.4}#dsfig-u4-01 .n.hi{fill:#E3E9FC;stroke:#2340B8;stroke-width:2.2}#dsfig-u4-01 .n.rb-b{fill:#16181D;stroke:#16181D}#dsfig-u4-01 .n.rb-r{fill:#BD3227;stroke:#BD3227}#dsfig-u4-01 text{font-family:"JetBrains Mono",ui-monospace,Menlo,Consolas,monospace;font-size:13px}#dsfig-u4-01 .t{fill:#16181D;font-weight:500}#dsfig-u4-01 .t.inv{fill:#FFFFFF;font-weight:700}#dsfig-u4-01 .kd{stroke:#16181D;stroke-width:1.2}#dsfig-u4-01 .dot{fill:#16181D}#dsfig-u4-01 .ann{fill:#2340B8;font-size:11px;font-weight:700}#dsfig-u4-01 .lbl{fill:#6F7787;font-family:system-ui,-apple-system,sans-serif;font-size:12px;font-weight:700}#dsfig-u4-01 .ptr{fill:#2340B8;font-size:12px;font-weight:700}#dsfig-u4-01 .ah{fill:#454C5A}#dsfig-u4-01 .ah.hi{fill:#2340B8}#dsfig-u4-01 .wl rect{fill:#FFFFFF;stroke:#DCE0E7}#dsfig-u4-01 .wl .t{font-size:12px;font-weight:700}#dsfig-u4-01 .wl.hi rect{fill:#2340B8;stroke:#2340B8}#dsfig-u4-01 .wl.hi .t{fill:#FFFFFF}html.dark #dsfig-u4-01 .e{stroke:#B1B7C3}html.dark #dsfig-u4-01 .e.hi{stroke:#8FA3FF}html.dark #dsfig-u4-01 .n{fill:#161920;stroke:#E6E8ED}html.dark #dsfig-u4-01 .n.hi{fill:#1E2748;stroke:#8FA3FF}html.dark #dsfig-u4-01 .n.rb-b{fill:#E6E8ED;stroke:#E6E8ED}html.dark #dsfig-u4-01 .n.rb-r{fill:#FF7E71;stroke:#FF7E71}html.dark #dsfig-u4-01 .t{fill:#E6E8ED}html.dark #dsfig-u4-01 .t.inv{fill:#0F1115}html.dark #dsfig-u4-01 .kd{stroke:#E6E8ED}html.dark #dsfig-u4-01 .dot{fill:#E6E8ED}html.dark #dsfig-u4-01 .ann{fill:#8FA3FF}html.dark #dsfig-u4-01 .lbl{fill:#858D9C}html.dark #dsfig-u4-01 .ptr{fill:#8FA3FF}html.dark #dsfig-u4-01 .ah{fill:#B1B7C3}html.dark #dsfig-u4-01 .ah.hi{fill:#8FA3FF}html.dark #dsfig-u4-01 .wl rect{fill:#161920;stroke:#2A2E37}html.dark #dsfig-u4-01 .wl.hi rect{fill:#8FA3FF;stroke:#8FA3FF}html.dark #dsfig-u4-01 .wl.hi .t{fill:#0F1115}</style><defs><marker id="ah3" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse"><path class="ah" d="M0,1 L9,5 L0,9 z"/></marker><marker id="ahh3" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse"><path class="ah hi" d="M0,1 L9,5 L0,9 z"/></marker></defs><path class="e" d="M59,40 L148,40" marker-end="url(#ah3)"/><path class="e" d="M188,40 L277,40" marker-end="url(#ah3)"/><path class="e" d="M317,40 L406,40" marker-end="url(#ah3)"/><g class="wl"><rect x="81" y="31" width="47.1" height="18" rx="9"/><text class="t" x="104.5" y="40" dy=".35em" text-anchor="middle">trace</text></g><g class="wl"><rect x="202.8" y="31" width="61.5" height="18" rx="9"/><text class="t" x="233.5" y="40" dy=".35em" text-anchor="middle">project</text></g><g class="wl"><rect x="321.4" y="31" width="82.2" height="18" rx="9"/><text class="t" x="362.5" y="40" dy=".35em" text-anchor="middle">true_shape</text></g><circle class="n" cx="40" cy="40" r="18"/><text class="t" x="40" y="40" dy=".35em" text-anchor="middle">FV</text><circle class="n" cx="169" cy="40" r="18"/><text class="t" x="169" y="40" dy=".35em" text-anchor="middle">Tr</text><circle class="n" cx="298" cy="40" r="18"/><text class="t" x="298" y="40" dy=".35em" text-anchor="middle">TV</text><circle class="n" cx="427" cy="40" r="18"/><text class="t" x="427" y="40" dy=".35em" text-anchor="middle">AV</text></svg><figcaption style="font-size:.82em;opacity:.72;margin-top:.45rem">Order of the section drawing: FV with trace, sectional TV, then auxiliary view (true shape) parallel to the trace</figcaption></figure>

Answer frame. Open with the definition of a section and the given position of the solid; draw the FV and TV with the trace at the given angle through the axis point; develop steps 3-6 in order, labelling cut points; hatch at $45^\circ$; close with the name of the true shape (ellipse for cylinder, parabola for cone, polygon for pyramid) and the note that it is drawn on the auxiliary plane parallel to the trace. For a cylinder lying on a generator with axis $30^\circ$ to VP, first draw the axis in the TV at $30^\circ$ to xy, project the FV, divide the circle into 12 parts, then cut. For the cone lying on a generator, first draw the upright cone, tilt it, then cut with a vertical plane parallel to the generator and draw the parabola from ordinates.

Pitfall: Hatching the hole of a hollow cylinder, or drawing the true shape with the wrong xy line (not parallel to the trace).

Asked: [14 marks] (Dec 2023, Jun 2023) Square pyramid, base 35 mm, height 50 mm, on HP with base edges equally inclined to VP; plane perpendicular to VP, inclined $32^\circ$ to HP, bisects the axis. Draw projections and true shape of the section. Asked: [14 marks] (Dec 2024) Cylinder 50 mm diameter, 65 mm long, lying on a generator on HP, axis $30^\circ$ to VP, cut bisecting the axis. Draw the apparent and true sections. Asked: [14 marks] (Jun 2024) Cylinder base 54 mm, axis 75 mm, with a central 30 mm hole, on its base on HP; plane $45^\circ$ to HP cuts the axis 20 mm from the top. Draw the sectional top view and true shape. Asked: [7 marks] (Jun 2025) Hexagonal pyramid, side 30 mm, axis 60 mm, base on HP, one base edge parallel to VP; horizontal plane 38 mm above the base. Draw the front view and sectional top view. Asked: [7 marks] (Jun 2025) Cone, base 55 mm, axis 65 mm, lying on a generator on HP with axis parallel to VP; vertical plane parallel to a generator and bisecting the axis. Draw the sectional front view and true shape.

Development of surfaces of Right Regular Solids - Prism, Pyramid, Cylinder and Cone

<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">Medium weight</span>

Definition. <mark>Development is the unrolling of the lateral surface of a solid onto a plane, so that every face is shown in its true size in one flat piece.</mark>

Key points.

  1. Development needs true lengths of edges or generators, so inclined edges are rotated until parallel to the plane of projection.
  2. A prism develops into a rectangle of length equal to the base perimeter and height equal to the axis; a square prism of edge 50 and height 65 gives $4\times50 = 200$ mm by 65 mm.
  3. A cylinder develops into a rectangle of length $\pi D$ and height $H$; for $D=30$: $\pi\times30 = 94.25$ mm.
  4. A pyramid develops into triangles on true slant edges; a cone develops into a sector of radius $L$ (slant height) and angle $\theta = 360^\circ\times r/L$.
  5. For a truncated (cut) solid, the base circle is divided into 12 equal parts, the generators are drawn, and each cut length is transferred to the development at equal spacing along the stretch-out line.
  6. Points on the development are joined by a smooth curve for cylinders and cones, and by straight lines for prisms and pyramids.
  7. An oblique cone has generators of unequal length, so a fan is built by triangulation: true length of each generator, then arcs with the base chord.
  8. A hole shows as curved cutouts, found by projecting hole points on the faces and transferring heights to the development.

Formula. $$\text{Cylinder: } L=\pi D,\qquad \text{Cone: } \theta=\frac{r}{L}\times360^\circ,\qquad \text{Prism: } L=n\times a$$

Steps.

Step 1: Draw the projections of the solid and the cutting plane, divide the base circle into 12 parts.
Step 2: Draw the stretch-out line of length pi*D (or n*a) and divide it into the same parts.
Step 3: Erect ordinates equal to the height of each generator up to the cut.
Step 4: Join the tops by a smooth curve and add the base rectangle as required.

Diagram. <figure class="ds-fig" style="margin:1.4rem 0;overflow-x:auto"><svg xmlns="http://www.w3.org/2000/svg" id="dsfig-u4-02" viewBox="0 0 338 80" width="338" height="80" role="img" aria-label="Truncated cylinder: FV with cut and 12 generators, then stretch-out rectangle of length pi D with cut heights"><style>#dsfig-u4-02 .e{stroke:#454C5A;stroke-width:1.4;fill:none}#dsfig-u4-02 .e.hi{stroke:#2340B8;stroke-width:2.6}#dsfig-u4-02 .n{fill:#FFFFFF;stroke:#16181D;stroke-width:1.4}#dsfig-u4-02 .n.hi{fill:#E3E9FC;stroke:#2340B8;stroke-width:2.2}#dsfig-u4-02 .n.rb-b{fill:#16181D;stroke:#16181D}#dsfig-u4-02 .n.rb-r{fill:#BD3227;stroke:#BD3227}#dsfig-u4-02 text{font-family:"JetBrains Mono",ui-monospace,Menlo,Consolas,monospace;font-size:13px}#dsfig-u4-02 .t{fill:#16181D;font-weight:500}#dsfig-u4-02 .t.inv{fill:#FFFFFF;font-weight:700}#dsfig-u4-02 .kd{stroke:#16181D;stroke-width:1.2}#dsfig-u4-02 .dot{fill:#16181D}#dsfig-u4-02 .ann{fill:#2340B8;font-size:11px;font-weight:700}#dsfig-u4-02 .lbl{fill:#6F7787;font-family:system-ui,-apple-system,sans-serif;font-size:12px;font-weight:700}#dsfig-u4-02 .ptr{fill:#2340B8;font-size:12px;font-weight:700}#dsfig-u4-02 .ah{fill:#454C5A}#dsfig-u4-02 .ah.hi{fill:#2340B8}#dsfig-u4-02 .wl rect{fill:#FFFFFF;stroke:#DCE0E7}#dsfig-u4-02 .wl .t{font-size:12px;font-weight:700}#dsfig-u4-02 .wl.hi rect{fill:#2340B8;stroke:#2340B8}#dsfig-u4-02 .wl.hi .t{fill:#FFFFFF}html.dark #dsfig-u4-02 .e{stroke:#B1B7C3}html.dark #dsfig-u4-02 .e.hi{stroke:#8FA3FF}html.dark #dsfig-u4-02 .n{fill:#161920;stroke:#E6E8ED}html.dark #dsfig-u4-02 .n.hi{fill:#1E2748;stroke:#8FA3FF}html.dark #dsfig-u4-02 .n.rb-b{fill:#E6E8ED;stroke:#E6E8ED}html.dark #dsfig-u4-02 .n.rb-r{fill:#FF7E71;stroke:#FF7E71}html.dark #dsfig-u4-02 .t{fill:#E6E8ED}html.dark #dsfig-u4-02 .t.inv{fill:#0F1115}html.dark #dsfig-u4-02 .kd{stroke:#E6E8ED}html.dark #dsfig-u4-02 .dot{fill:#E6E8ED}html.dark #dsfig-u4-02 .ann{fill:#8FA3FF}html.dark #dsfig-u4-02 .lbl{fill:#858D9C}html.dark #dsfig-u4-02 .ptr{fill:#8FA3FF}html.dark #dsfig-u4-02 .ah{fill:#B1B7C3}html.dark #dsfig-u4-02 .ah.hi{fill:#8FA3FF}html.dark #dsfig-u4-02 .wl rect{fill:#161920;stroke:#2A2E37}html.dark #dsfig-u4-02 .wl.hi rect{fill:#8FA3FF;stroke:#8FA3FF}html.dark #dsfig-u4-02 .wl.hi .t{fill:#0F1115}</style><defs><marker id="ah4" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse"><path class="ah" d="M0,1 L9,5 L0,9 z"/></marker><marker id="ahh4" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse"><path class="ah hi" d="M0,1 L9,5 L0,9 z"/></marker></defs><path class="e" d="M59,40 L148,40" marker-end="url(#ah4)"/><path class="e" d="M188,40 L277,40" marker-end="url(#ah4)"/><g class="wl"><rect x="70.2" y="31" width="68.7" height="18" rx="9"/><text class="t" x="104.5" y="40" dy=".35em" text-anchor="middle">12_parts</text></g><g class="wl"><rect x="188.8" y="31" width="89.4" height="18" rx="9"/><text class="t" x="233.5" y="40" dy=".35em" text-anchor="middle">stretch_out</text></g><circle class="n" cx="40" cy="40" r="18"/><text class="t" x="40" y="40" dy=".35em" text-anchor="middle">FV</text><circle class="n" cx="169" cy="40" r="18"/><text class="t" x="169" y="40" dy=".35em" text-anchor="middle">Gen</text><circle class="n" cx="298" cy="40" r="18"/><text class="t" x="298" y="40" dy=".35em" text-anchor="middle">Dev</text></svg><figcaption style="font-size:.82em;opacity:.72;margin-top:.45rem">Truncated cylinder: FV with cut and 12 generators, then stretch-out rectangle of length pi D with cut heights</figcaption></figure>

Answer frame. Open with the definition and the length of the stretch-out; draw the elevation with the cutting plane and 12 numbered generators, then the stretch-out line below it; develop steps 1-4, marking each generator height, then close with the smooth curve and note that the development shows true lengths. For the oblique cone, draw the plan and elevation, find true length of each generator by rotation, then build the fan with the chord length between the ends.

Asked: [7 marks] (Jun 2023) Develop the lateral surface of an oblique cone, base diameter 40 mm, height 40 mm, axis inclined $60^\circ$ to its base. Asked: [7 marks] (Dec 2024) Right cylinder 30 mm diameter, axis 35 mm, cut by a plane $30^\circ$ to HP passing 18 mm from the base along the axis. Draw the development of the truncated cylinder. Asked: [7 marks] (Jun 2024) Square prism edge 50 mm, height 65 mm, on a face on HP with a vertical face $45^\circ$ to VP; a 25 mm hole drilled horizontally through the centre. Draw the development of the prism surface and the hole.

Sectional orthographic views of geometrical solids, objects from industry and dwellings

<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. A sectional orthographic view is a view in which the object is imagined cut by a plane and the part in front removed, so the inside is shown with the cut face hatched.

Key points.

  1. A full section cuts the whole object along its centre line, and a half section shows one quarter removed.
  2. The cutting plane is marked by a chain line with thick ends and arrows, and the section is labelled A-A.
  3. The cut face is hatched at $45^\circ$ with thin, equally spaced lines; ribs, shafts and bolts are not hatched.
  4. In a dwelling section from foundation to slab, the cut goes through the foundation, plinth, wall, door and window openings, lintel and roof slab, showing thickness of each part.

Last-minute revision

  • Section: cut face of a solid made by a section plane; true shape needs an auxiliary plane parallel to the trace.
  • Hatch: thin lines at $45^\circ$, equally spaced, only on cut material.
  • Horizontal section: true shape seen in the top view.
  • Cylinder cut obliquely: ellipse; cone cut parallel to generator: parabola.
  • Square pyramid base 35: half-diagonal $= 35/\sqrt2 = 24.75$ mm.
  • Hexagonal pyramid 30 x 60 cut at 38: hexagon of side 11 mm.
  • Cylinder development: $\pi D$ by $H$; $D=30$ gives 94.25 mm.
  • Square prism development: $4\times50=200$ mm by 65 mm.
  • Cone development angle: $\theta = (r/L)\times360^\circ$.
  • Divide circles into 12 equal parts for cylinder and cone work.

Memory hooks

  • "Trace, Points, Project, Hatch, True": the five stages of every section.
  • New xy line runs parallel to the trace; that is the auxiliary plane.
  • Cylinder = ellipse, cone = parabola, pyramid = polygon.
  • Development = unroll: perimeter is the length, height is the height.

Coverage checklist

  • Prism, Cylinder, Pyramid, Cone – Auxiliary Views: square pyramid 32 degrees (Dec 2023, Jun 2023), cylinder on generator (Dec 2024), hollow cylinder (Jun 2024), hexagonal pyramid (Jun 2025), cone on generator (Jun 2025).
  • Development of surfaces of Right Regular Solids - Prism, Pyramid, Cylinder and Cone: oblique cone (Jun 2023), truncated cylinder (Dec 2024), prism with hole (Jun 2024).
  • Draw the sectional orthographic views of geometrical solids, objects from industry and dwellings (foundation to slab only): no past questions.
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