How unit 5 is examined
This unit covers the isometric scale and axes, isometric drawing of prisms, cylinders, cones and stacked solids, and drawing an isometric view from given orthographic views; the two solid-drawing topics carry almost all the marks.
Principles of Isometric projection – Isometric Scale, Isometric Views, Conventions
<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>Isometric projection is a single-view orthographic projection of an object tilted so that its three mutually perpendicular edges are equally inclined to the plane of projection, hence equally foreshortened and drawn $120^\circ$ apart.</mark>
Diagram.
<figure class="ds-fig" style="margin:1.4rem 0;overflow-x:auto"><svg xmlns="http://www.w3.org/2000/svg" id="dsfig-u5-01" viewBox="0 0 424 252" width="424" height="252" role="img" aria-label="Isometric axes: two axes at 30 degrees to the horizontal, one vertical, 120 degrees apart."><style>#dsfig-u5-01 .e{stroke:#454C5A;stroke-width:1.4;fill:none}#dsfig-u5-01 .e.hi{stroke:#2340B8;stroke-width:2.6}#dsfig-u5-01 .n{fill:#FFFFFF;stroke:#16181D;stroke-width:1.4}#dsfig-u5-01 .n.hi{fill:#E3E9FC;stroke:#2340B8;stroke-width:2.2}#dsfig-u5-01 .n.rb-b{fill:#16181D;stroke:#16181D}#dsfig-u5-01 .n.rb-r{fill:#BD3227;stroke:#BD3227}#dsfig-u5-01 text{font-family:"JetBrains Mono",ui-monospace,Menlo,Consolas,monospace;font-size:13px}#dsfig-u5-01 .t{fill:#16181D;font-weight:500}#dsfig-u5-01 .t.inv{fill:#FFFFFF;font-weight:700}#dsfig-u5-01 .kd{stroke:#16181D;stroke-width:1.2}#dsfig-u5-01 .dot{fill:#16181D}#dsfig-u5-01 .ann{fill:#2340B8;font-size:11px;font-weight:700}#dsfig-u5-01 .lbl{fill:#6F7787;font-family:system-ui,-apple-system,sans-serif;font-size:12px;font-weight:700}#dsfig-u5-01 .ptr{fill:#2340B8;font-size:12px;font-weight:700}#dsfig-u5-01 .ah{fill:#454C5A}#dsfig-u5-01 .ah.hi{fill:#2340B8}#dsfig-u5-01 .wl rect{fill:#FFFFFF;stroke:#DCE0E7}#dsfig-u5-01 .wl .t{font-size:12px;font-weight:700}#dsfig-u5-01 .wl.hi rect{fill:#2340B8;stroke:#2340B8}#dsfig-u5-01 .wl.hi .t{fill:#FFFFFF}html.dark #dsfig-u5-01 .e{stroke:#B1B7C3}html.dark #dsfig-u5-01 .e.hi{stroke:#8FA3FF}html.dark #dsfig-u5-01 .n{fill:#161920;stroke:#E6E8ED}html.dark #dsfig-u5-01 .n.hi{fill:#1E2748;stroke:#8FA3FF}html.dark #dsfig-u5-01 .n.rb-b{fill:#E6E8ED;stroke:#E6E8ED}html.dark #dsfig-u5-01 .n.rb-r{fill:#FF7E71;stroke:#FF7E71}html.dark #dsfig-u5-01 .t{fill:#E6E8ED}html.dark #dsfig-u5-01 .t.inv{fill:#0F1115}html.dark #dsfig-u5-01 .kd{stroke:#E6E8ED}html.dark #dsfig-u5-01 .dot{fill:#E6E8ED}html.dark #dsfig-u5-01 .ann{fill:#8FA3FF}html.dark #dsfig-u5-01 .lbl{fill:#858D9C}html.dark #dsfig-u5-01 .ptr{fill:#8FA3FF}html.dark #dsfig-u5-01 .ah{fill:#B1B7C3}html.dark #dsfig-u5-01 .ah.hi{fill:#8FA3FF}html.dark #dsfig-u5-01 .wl rect{fill:#161920;stroke:#2A2E37}html.dark #dsfig-u5-01 .wl.hi rect{fill:#8FA3FF;stroke:#8FA3FF}html.dark #dsfig-u5-01 .wl.hi .t{fill:#0F1115}</style><defs><marker id="ah5" 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="ahh5" 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="M229,203.5 L367,134.5"/><path class="e" d="M195,203.5 L57,134.5"/><path class="e" d="M212,193 L212,59"/><g class="wl"><rect x="274.5" y="160" width="47.1" height="18" rx="9"/><text class="t" x="298" y="169" dy=".35em" text-anchor="middle">30deg</text></g><g class="wl"><rect x="102.5" y="160" width="47.1" height="18" rx="9"/><text class="t" x="126" y="169" dy=".35em" text-anchor="middle">30deg</text></g><g class="wl"><rect x="188.5" y="117" width="47.1" height="18" rx="9"/><text class="t" x="212" y="126" dy=".35em" text-anchor="middle">90deg</text></g><circle class="n" cx="212" cy="212" r="18"/><text class="t" x="212" y="212" dy=".35em" text-anchor="middle">O</text><circle class="n" cx="384" cy="126" r="18"/><text class="t" x="384" y="126" dy=".35em" text-anchor="middle">X</text><circle class="n" cx="40" cy="126" r="18"/><text class="t" x="40" y="126" dy=".35em" text-anchor="middle">Y</text><circle class="n" cx="212" cy="40" r="18"/><text class="t" x="212" y="40" dy=".35em" text-anchor="middle">Z</text></svg><figcaption style="font-size:.82em;opacity:.72;margin-top:.45rem">Isometric axes: two axes at 30 degrees to the horizontal, one vertical, 120 degrees apart.</figcaption></figure>
Key points.
- The three isometric axes are one vertical axis and two axes drawn at $30^\circ$ to the horizontal, so every angle between them is $120^\circ$.
- Lines parallel to an axis are called isometric lines; lines not parallel to any axis are non-isometric lines and cannot be measured directly.
- Because the three axes are equally inclined, each edge is shortened in the same ratio, so all three dimensions use one scale.
- Derivation: if an axis makes angle $\alpha$ with the plane, its shortening is $\cos\alpha$; for three mutually perpendicular axes $\cos^2\alpha_1+\cos^2\alpha_2+\cos^2\alpha_3=2$, and equal angles give $3\cos^2\alpha=2$, so $\cos\alpha=\sqrt{2/3}=0.816$.
- Isometric projection is the true, foreshortened view (edges $\times 0.816$), whereas isometric drawing uses full true lengths and looks about $22.5\%$ larger ($1/0.816=1.225$).
- Isometric scale: a scale on which true lengths are marked and read as $0.816$ of their value, so measurements can be transferred directly without calculation.
- Construction of the isometric scale: draw a horizontal line $OA$; from $O$ draw a $45^\circ$ line for the true scale and a $30^\circ$ line for the isometric scale; mark true lengths $1, 2, 3\dots$ on the $45^\circ$ line and project them vertically down to the $30^\circ$ line to read the isometric lengths.
- The relation follows from the construction: isometric length / true length $=\cos45^\circ/\cos30^\circ=\sqrt{2/3}\approx0.816$.
- Conventions: a circle in an isometric plane becomes an ellipse (four-centre method); hidden lines are normally omitted, visible edges are drawn dark, and construction lines are kept light.
Formula. Isometric length $=0.816\times$ true length; e.g. a $25\text{ mm}$ edge is drawn $20.4\text{ mm}$ and a $50\text{ mm}$ edge $40.8\text{ mm}$.
Diagram.
<figure class="ds-fig" style="margin:1.4rem 0;overflow-x:auto"><svg xmlns="http://www.w3.org/2000/svg" id="dsfig-u5-02" viewBox="0 0 510 252" width="510" height="252" role="img" aria-label="Isometric scale: true scale on the 45 degree line, isometric scale on the 30 degree line, joined by vertical projectors."><style>#dsfig-u5-02 .e{stroke:#454C5A;stroke-width:1.4;fill:none}#dsfig-u5-02 .e.hi{stroke:#2340B8;stroke-width:2.6}#dsfig-u5-02 .n{fill:#FFFFFF;stroke:#16181D;stroke-width:1.4}#dsfig-u5-02 .n.hi{fill:#E3E9FC;stroke:#2340B8;stroke-width:2.2}#dsfig-u5-02 .n.rb-b{fill:#16181D;stroke:#16181D}#dsfig-u5-02 .n.rb-r{fill:#BD3227;stroke:#BD3227}#dsfig-u5-02 text{font-family:"JetBrains Mono",ui-monospace,Menlo,Consolas,monospace;font-size:13px}#dsfig-u5-02 .t{fill:#16181D;font-weight:500}#dsfig-u5-02 .t.inv{fill:#FFFFFF;font-weight:700}#dsfig-u5-02 .kd{stroke:#16181D;stroke-width:1.2}#dsfig-u5-02 .dot{fill:#16181D}#dsfig-u5-02 .ann{fill:#2340B8;font-size:11px;font-weight:700}#dsfig-u5-02 .lbl{fill:#6F7787;font-family:system-ui,-apple-system,sans-serif;font-size:12px;font-weight:700}#dsfig-u5-02 .ptr{fill:#2340B8;font-size:12px;font-weight:700}#dsfig-u5-02 .ah{fill:#454C5A}#dsfig-u5-02 .ah.hi{fill:#2340B8}#dsfig-u5-02 .wl rect{fill:#FFFFFF;stroke:#DCE0E7}#dsfig-u5-02 .wl .t{font-size:12px;font-weight:700}#dsfig-u5-02 .wl.hi rect{fill:#2340B8;stroke:#2340B8}#dsfig-u5-02 .wl.hi .t{fill:#FFFFFF}html.dark #dsfig-u5-02 .e{stroke:#B1B7C3}html.dark #dsfig-u5-02 .e.hi{stroke:#8FA3FF}html.dark #dsfig-u5-02 .n{fill:#161920;stroke:#E6E8ED}html.dark #dsfig-u5-02 .n.hi{fill:#1E2748;stroke:#8FA3FF}html.dark #dsfig-u5-02 .n.rb-b{fill:#E6E8ED;stroke:#E6E8ED}html.dark #dsfig-u5-02 .n.rb-r{fill:#FF7E71;stroke:#FF7E71}html.dark #dsfig-u5-02 .t{fill:#E6E8ED}html.dark #dsfig-u5-02 .t.inv{fill:#0F1115}html.dark #dsfig-u5-02 .kd{stroke:#E6E8ED}html.dark #dsfig-u5-02 .dot{fill:#E6E8ED}html.dark #dsfig-u5-02 .ann{fill:#8FA3FF}html.dark #dsfig-u5-02 .lbl{fill:#858D9C}html.dark #dsfig-u5-02 .ptr{fill:#8FA3FF}html.dark #dsfig-u5-02 .ah{fill:#B1B7C3}html.dark #dsfig-u5-02 .ah.hi{fill:#8FA3FF}html.dark #dsfig-u5-02 .wl rect{fill:#161920;stroke:#2A2E37}html.dark #dsfig-u5-02 .wl.hi rect{fill:#8FA3FF;stroke:#8FA3FF}html.dark #dsfig-u5-02 .wl.hi .t{fill:#0F1115}</style><defs><marker id="ah6" 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="ahh6" 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="M59,212 L451,212"/><path class="e" d="M57.6,204.9 L452.4,47.1"/><path class="e" d="M58.6,208.3 L451.4,129.7"/><g class="wl"><rect x="231.5" y="117" width="47.1" height="18" rx="9"/><text class="t" x="255" y="126" dy=".35em" text-anchor="middle">45deg</text></g><g class="wl"><rect x="231.5" y="160" width="47.1" height="18" rx="9"/><text class="t" x="255" y="169" dy=".35em" text-anchor="middle">30deg</text></g><circle class="n" cx="40" cy="212" r="18"/><text class="t" x="40" y="212" dy=".35em" text-anchor="middle">O</text><circle class="n" cx="470" cy="212" r="18"/><text class="t" x="470" y="212" dy=".35em" text-anchor="middle">A</text><circle class="n" cx="470" cy="40" r="18"/><text class="t" x="470" y="40" dy=".35em" text-anchor="middle">B</text><circle class="n" cx="470" cy="126" r="18"/><text class="t" x="470" y="126" dy=".35em" text-anchor="middle">C</text></svg><figcaption style="font-size:.82em;opacity:.72;margin-top:.45rem">Isometric scale: true scale on the 45 degree line, isometric scale on the 30 degree line, joined by vertical projectors.</figcaption></figure>
Epicycloid (short note). An epicycloid is the locus of a point on the circumference of a generating circle of radius $r$ that rolls without slipping on the outside of a directing circle of radius $R$; its parametric form is $x=(R+r)\cos\theta-r\cos\frac{(R+r)\theta}{r}$, $y=(R+r)\sin\theta-r\sin\frac{(R+r)\theta}{r}$. To draw it, divide the generating circle into equal parts, mark the same arc lengths on the directing circle, draw the centre arc, and join the successive positions of the point with a smooth curve.
Basic drawing commands (CAD short note). LINE draws straight segments from picked points; POLYLINE joins segments as one object; CIRCLE takes centre and radius (or diameter, 2P, 3P); ARC takes three points or start, centre and end; RECTANGLE takes two opposite corners; POLYGON takes number of sides with centre and radius; ELLIPSE takes axis endpoints and the other radius. Each is typed at the command line or chosen from the toolbar, then options are picked as prompted.
Answer frame. Open with the definition of isometric projection; draw the three axes at $30^\circ$, $90^\circ$, $30^\circ$; develop points 1, 3, 4, 5 for the projection and 6-8 for the scale; close with "Isometric length is $0.816$ of true length." For the scale question, draw the construction figure first, then the relation. For the short note, give each of the three parts about 4-5 lines.
Pitfall: Do not measure lengths along non-isometric lines; convert their end points from the isometric lines or box method first.
Asked: [14 marks] (Nov 2022) Write short notes on: i) Isometric projection ii) Epicycloid iii) Basic drawing command Asked: [7 marks] (Jun 2025) Discuss the construction and use of an Isometric Scale.
Isometric Views of lines, Planes, Simple and compound Solids
<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>An isometric view of a solid is drawn by placing its edges along the three isometric axes, using isometric lengths ($0.816\times$ true) or true lengths (isometric drawing), and drawing circles as ellipses.</mark>
Steps (box method and axes).
Step 1: Draw the isometric axes at 30, 90, 30 degrees.
Step 2: Enclose the base shape in a rectangle in the top view.
Step 3: Transfer that rectangle to the isometric plane using isometric lengths.
Step 4: Plot the base corners on the sides of the isometric rectangle.
Step 5: Erect vertical edges of the given height from every corner.
Step 6: Join the top corners; darken visible edges, drop hidden ones.
Key points.
- A prism or pyramid on its base is drawn by making the isometric of the base (square as a rhombus, hexagon by the box method), then raising vertical edges of the isometric height.
- A cylinder or cone is drawn by fitting each circular face in an isometric rhombus whose side equals the diameter, then drawing the ellipse by the four-centre method.
- Four-centre method: from the rhombus draw lines from the acute-angle corners to the midpoints of the opposite sides; these meet at two more centres, and the four centres give two large and two small arcs forming the ellipse.
- The top ellipse of a cylinder is found by shifting the bottom centres up by the height, and common vertical tangents join the two ellipses.
- A frustum uses two rhombuses of different size (bottom and top diameters) on the same vertical axis, joined by common tangents.
- For compound solids, draw the lower solid first, find the centre of its top face by the diagonals of its rhombus, and build the upper solid centrally from that point.
- A pyramid apex is fixed by measuring the isometric axis height vertically from the centre of the base and joining it to the four base corners.
- Darken visible edges, omit hidden lines, and keep construction lines light.
Example (Jun 2022, Nov 2022). Hexagonal prism, side $25\text{ mm}$, axis $65\text{ mm}$, on HP.
| Given | Isometric length ($\times0.816$) |
|---|---|
| Hexagon side $25$ | $20.4$ |
| Box (across corners) $50$ by across flats $43.3$ | $40.8$ by $35.3$ |
| Height $65$ | $53.1$ |
Draw the hexagon in a $50\times43.3$ rectangle, transfer the rectangle as an isometric rhombus, plot corners at the side midpoints and along the sides, erect vertical edges $53.1\text{ mm}$ (or $65\text{ mm}$ in isometric drawing), join the top corners. Answer: a hexagonal prism with six vertical edges and two hexagonal faces.
Data for the other drawings.
| Question | Isometric lengths |
|---|---|
| Cube $30$ on cylinder $\phi52\times20$ | $\phi52\to42.5$, height $20\to16.3$, cube $30\to24.5$ |
| Frustum $\phi50/\phi25$, height $60$ | $50\to40.8$, $25\to20.4$, height $60\to49.0$ |
| Square pyramid $25$, axis $40$ on cylinder $\phi50\times20$ | $25\to20.4$, axis $40\to32.7$, $\phi50\to40.8$, $20\to16.3$ |
| Cube $40$ on square block $60\times60\times20$, edges parallel | $60\to49.0$, $20\to16.3$, $40\to32.7$ |
Answer frame. Open with "The solid is drawn using the isometric scale ($0.816$) along axes at $30^\circ$, $90^\circ$, $30^\circ$"; draw the isometric scale or state the factor, then the lower solid, then the centre of its top face, then the upper solid; develop points 1-8 in that order; close with "Visible edges are darkened and hidden edges are omitted." For circular solids, name the four-centre method and tangents explicitly.
Pitfall: Drawing a circle as a true circle in an isometric view; it must be an ellipse from the four-centre rhombus.
Asked: [7 marks] (Jun 2022, Nov 2022) Draw the isometric view of a hexagonal prism, base side $25\text{ mm}$, axis $65\text{ mm}$, resting on its base on HP. Asked: [9 marks] (Jun 2023, Dec 2023) A cube $30\text{ mm}$ edge is placed centrally on top of a cylindrical block of $\phi52\text{ mm}$ and $20\text{ mm}$ height. Draw the isometric drawing of the solid. Asked: [7 marks] (Jun 2022) Draw the isometric projection of the frustum of a cone, base diameter $50\text{ mm}$, top diameter $25\text{ mm}$, height $60\text{ mm}$. Asked: [7 marks] (Dec 2024) A square pyramid (base $25\text{ mm}$, axis $40\text{ mm}$) rests centrally over a cylindrical block ($\phi50\text{ mm}$, $20\text{ mm}$ thick). Draw the isometric projection. Asked: [7 marks] (Jun 2024) A cube of $40\text{ mm}$ side rests centrally on a square block of $60\text{ mm}$ edges and $20\text{ mm}$ thick. Draw the isometric projection with the edges of the two blocks parallel.
Conversion of Isometric Views to Orthographic Views and Vice-versa, Conventions
<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>Conversion between isometric and orthographic views means reading the length, width and height of each part from the given views and placing them along the three isometric axes inside an isometric box.</mark>
Key points.
- Take the overall length, width and height from the front and side views, and draw an isometric box of that size along the $30^\circ$, $90^\circ$, $30^\circ$ axes.
- Cut or add each feature (base, taper, slot) by measuring its offsets from the box edges, since only isometric lines can be measured.
- Draw the base first, then the tapered upper part to its top-face dimensions, then the slot at its stated width and depth.
- Darken visible edges and erase or lighten construction lines; for the reverse conversion, project the isometric lengths back to true lengths as front, top and side views.
Example (Jun 2025). Tapered slotted block: box $72\times48\times48$; base $72\times48\times12$; top segments $12,18,12,18,12$; central slot $12\text{ mm}$ wide and $9\text{ mm}$ deep, drawn inside the box.
Asked: [7 marks] (Jun 2025) Draw the isometric view of the given figure (front and side views of a tapered block with central slot).
Last-minute revision
- Isometric axes: one vertical, two at $30^\circ$ to the horizontal, $120^\circ$ apart.
- Isometric scale factor $=\sqrt{2/3}=\cos45^\circ/\cos30^\circ\approx0.816$.
- Isometric drawing uses true lengths; isometric projection uses $0.816\times$ lengths.
- A circle in an isometric plane appears as an ellipse drawn by the four-centre method.
- Isometric lines are parallel to an axis; only they can be measured.
- Frustum: two rhombuses of different size on the same axis with common tangents.
- Compound solid: draw the lower solid, find the top-face centre, build the upper solid on it.
- Epicycloid: locus of a point on a circle rolling outside a directing circle.
- Key conversions: $25\to20.4$, $30\to24.5$, $40\to32.7$, $50\to40.8$, $60\to49.0$, $65\to53.1$.
- Jun 2025 block: box $72\times48\times48$, slot $12\times9$.
Memory hooks
- "30-90-30": the axes of every isometric drawing.
- $0.816$ is $\sqrt{2/3}$: about four-fifths of true length.
- Circles turn into ellipses; rhombus first, then four centres.
- Draw the bottom solid, find the centre, then stack the next solid.
- Isometric lines are measured; non-isometric lines are located by their end points.
Coverage checklist
- Principles of Isometric projection – Isometric Scale, Isometric Views, Conventions: isometric projection, scale construction, epicycloid, CAD commands (Nov 2022 short note, Jun 2025 scale).
- Isometric Views of lines, Planes, Simple and compound Solids: hexagonal prism, cube on cylinder, frustum, pyramid on cylinder, cube on square block.
- Conversion of Isometric Views to Orthographic Views and Vice-versa, Conventions: Jun 2025 tapered slotted block.