Architectural and Engineering Graphics: Drawing, Scales, Curves, Projection, Perspective and Shadow
2. Scales: representative fraction, plain and diagonal scales
The representative fraction (RF) is the length of an object on the drawing divided by its actual length, both in the same unit. A scale of 1 : 100 has RF = 1/100: 1 mm on paper stands for 100 mm (0.1 m) on site. 1 : 1 is full size, 1 : n with n > 1 is a reduction and n : 1 an enlargement; 1 : 50 is a larger scale than 1 : 100 because its fraction is larger. The length of scale to be drawn is RF × the longest length to be measured. Lengths scale by RF, areas by RF² and volumes by RF³.
- At 1 : 50 a 6.4 m wall is 6400/50 = 128 mm long on paper.
- A plan area of 24 cm² measured on a 1 : 500 drawing is 24 × 500² = 6,000,000 cm² = 600 m² on site. Square the RF; do not multiply by 500 once.
| Scale | What it reads | Least count and note |
|---|---|---|
| Plain scale | two units, e.g. metres and decimetres: main divisions plus one subdivided division | one subdivision, e.g. 1 dm |
| Diagonal scale | three units, e.g. m, dm and cm: a diagonal across ten horizontal steps splits the subdivision by similar triangles | subdivision ÷ 10, e.g. 1 cm |
| Graphical (bar) scale | a drawn bar marked in ground units | stays correct when the sheet is enlarged or reduced; the written RF does not |
Design of a diagonal scale: RF 1 : 50, to read metres, decimetres and centimetres up to 6 m. Length = 6000/50 = 120 mm; 1 m = 20 mm main division; 1 dm = 2 mm; 1 cm = 0.2 mm. The first 20 mm division is cut into ten 2 mm parts, ten horizontal lines are drawn above the scale, and a diagonal across the strip makes each line add a further tenth, 0.2 mm, which stands for 1 cm.
3. Engineering curves I: the conics
A conic is the locus of a point that moves so that the ratio of its distance from a fixed point (the focus) to its distance from a fixed straight line (the directrix) stays constant. That ratio is the eccentricity e. If e < 1 the curve is an ellipse, if e = 1 a parabola and if e > 1 a hyperbola. A plane cutting a right circular cone produces the same family, which is why they are called conic sections.
| Curve | e | Cutting plane (α = angle of a generator with the base) | Facts |
|---|---|---|---|
| Circle | 0 | parallel to the base | all points equidistant from the centre |
| Ellipse | < 1 | inclined to the base at less than α, so it cuts every generator | sum of the distances from the two foci = major axis; foci at ± ae from the centre |
| Parabola | = 1 | parallel to one generator (inclined at exactly α) | y² = 4ax, focus at a from the vertex; arch and reflector profiles |
| Hyperbola | > 1 | inclined at more than α, up to parallel to the axis | rectangular hyperbola xy = c has e = √2 and perpendicular asymptotes; PV = constant |
Construction methods to recognise: the concentric-circles and foci (arcs) methods for the ellipse, the rectangle (tangent) method for the parabola, and the asymptote method (xy = c) for the hyperbola. The isometric circle is drawn by the four-centre approximation to an ellipse. Worked: an ellipse with major axis 100 mm and minor axis 60 mm has a = 50 and b = 30, so its foci lie c = √(a² − b²) = 40 mm either side of the centre (80 mm apart), e = c/a = 0.8, and the focal distances of every point add to 100 mm. The foci are found by swinging an arc of radius a = 50 mm from an end of the minor axis.
4. Engineering curves II: cycloid, involute, helix and spirals
The remaining engineering curves are traced by rolling, unwinding or advancing along a path. What is tested is how each is generated and one or two defining numbers.
| Curve | Generation | Key facts |
|---|---|---|
| Cycloid | a point on the circumference of a circle rolling without slipping along a straight line | one arch has base 2πr and height 2r (the rolling diameter), with cusps on the line; rolling outside a circle gives an epicycloid, inside a hypocycloid, and a point off the circumference gives a trochoid |
| Involute | the free end of a taut string unwound from a circle or a polygon | the string is always tangent to the circle and its unwound length equals the arc rolled off; the normal at any point is tangent to the base circle, so the radius of curvature is the free string length; gear-tooth profiles |
| Helix | a point moving round and along a cylinder or cone with uniform angular and axial speeds | pitch = advance along the axis in one turn; plan is a circle, elevation a sine curve; unrolled, one turn is a straight line, the hypotenuse of a triangle of base 2πr and height = pitch, so tan λ = pitch/2πr; helical stairs, ramps and screws |
| Archimedean spiral | a point moving away from the pole at uniform speed while the radius vector turns uniformly | r = aθ: equal increase of radius for equal angles, so the gain per revolution is constant (2πa); cams and flat coils |
| Logarithmic spiral | a point whose radius vector grows in geometric progression over equal angles | r = a e^(bθ): the angle between radius vector and tangent is constant (equiangular) and the shape is self-similar; the golden spiral grows by φ every quarter turn |
- Cycloid from a rolling circle of diameter 60 mm: base of one arch = π × 60 = 188.5 mm, height 60 mm.
- Involute of a circle of radius 20 mm unwound through one full turn: free string = 2π × 20 = 125.7 mm.
- Helical stair of mean radius 1.0 m rising 3.0 m in a turn: one turn of rail = √((2π × 1.0)² + 3.0²) = √48.48 = 6.96 m, and helix angle = tan⁻¹(3.0/6.283) = 25.5°.
- Archimedean spiral gaining 24 mm of radius per revolution: after 120° the radius is 24 × 120/360 = 8 mm; after 2.5 revolutions it is 60 mm.
5. Orthographic projection: views, true length and true shape
In orthographic projection the projectors are perpendicular to the plane of projection. The principal planes are the horizontal plane (HP: plan, the top view), the vertical plane (VP: elevation, the front view) and the profile plane (side view); the HP and VP divide space into four quadrants. Indian (BIS) practice uses first-angle projection, in which the object lies between the observer and the plane. The United States and Canada use third-angle, in which a transparent plane lies between the observer and the object.
| Feature | First angle | Third angle |
|---|---|---|
| Object lies | in the first quadrant, between observer and plane | in the third quadrant, behind the transparent plane |
| Plan is drawn | below the elevation | above the elevation |
| Left-side view appears | to the right of the elevation | to the left of the elevation |
| Used in | India and most of Europe | USA and Canada |
For a line of true length L inclined θ to the HP and φ to the VP: the top view has length L cos θ and the front view L cos φ; the ends differ in height by L sin θ (seen in the front view) and in depth by L sin φ (seen in the plan). A line shows its true length only on a plane to which it is parallel, its apparent inclinations are never smaller than the true ones, and θ + φ cannot exceed 90°. A plane figure of area A tilted β from a plane of projection has projected area A cos β and its true shape appears only on a parallel plane, which is why an auxiliary view is drawn for an inclined face.
- L = 100 mm, θ = 30°, φ = 45°: top view = 100 cos 30° = 86.6 mm; front view = 100 cos 45° = 70.7 mm; ends differ in height by 100 sin 30° = 50 mm. Check: θ + φ = 75°, which is below 90°.
- A rectangular plate of 2400 mm² tilted 60° to the HP has a top-view area 2400 cos 60° = 1200 mm²; its true area is seen only in a view normal to the plate.
6. Isometric projection and isometric drawing
In axonometric projection the object is tilted so that its three principal directions are all inclined to the plane of projection. In isometric projection they are inclined equally, each edge making 35°16′ with the plane, so all three axes are foreshortened equally and appear 120° apart: one vertical and two rising at 30° to the horizontal. The foreshortening factor is cos 35°16′ = √(2/3) = 0.816. An isometric drawing (view) ignores it and uses true lengths along the axes. It has the same shape, 1/0.816 = 1.225 times larger, and is what is normally drawn.
- Measure only along isometric lines, those parallel to the three axes. An inclined or oblique line is drawn by locating its end points with the help of an enclosing box.
- Angles are not true: a right angle in a face appears as 120° or 60°.
- Hidden lines are normally omitted, and a circle on any of the three faces becomes an ellipse.
| Item | Isometric projection (scale 0.816) | Isometric drawing (true lengths) |
|---|---|---|
| Cube edge of true length a | 0.816 a | a |
| Circle of diameter d on an isometric face | ellipse: major axis d, minor axis 0.577 d | ellipse: major axis 1.225 d, minor axis 0.707 d |
| Sphere of diameter D | circle of diameter D | circle of diameter 1.225 D |
Worked: a box with 150 mm edges drawn as a true isometric projection has edges 150 × 0.8165 = 122.5 mm long; drawn as an isometric drawing they are 150 mm. The ellipse of a circle always has minor : major = 1 : √3 = 0.577, because a face is inclined at 54°44′ to the plane of projection and cos 54°44′ = 0.577. Its major axis is perpendicular to the isometric axis that is normal to the face, so it is horizontal for the top face.
7. Development of surfaces
Development is the unfolding of a surface onto one plane, in true size, with no stretching or overlap. It gives the templates for sheet-metal ducts, roofing sheets, cladding panels, hoppers and paper models. Only plane-faced and single-curved surfaces can be developed exactly (prism, pyramid, cylinder, cone and their frusta). Double-curved surfaces such as the sphere, a dome or a torus cannot: they are approximated by gores or zones, the same difficulty as flattening the globe into a map. The three methods are parallel-line (prisms, cylinders), radial-line (pyramids, cones) and triangulation (transition pieces and oblique surfaces).
| Solid | Development | Lateral area |
|---|---|---|
| Right prism | a strip of rectangles: length = base perimeter, height = h | perimeter × h |
| Right cylinder | one rectangle 2πr × h | 2πrh |
| Right pyramid | triangles meeting at the apex | ½ × perimeter × slant height |
| Right cone | a sector of radius l = √(r² + h²) and arc 2πr, with angle 360° × r/l | πrl |
| Frustum of a cone | part of a sector, an annular sector | π(r₁ + r₂) l |
- A conical roof on a 6 m diameter drum rising 4 m: l = √(3² + 4²) = 5 m; sector angle = 360° × 3/5 = 216°; roof sheet = π × 3 × 5 = 47.1 m² before laps.
- Frustum with end radii 40 and 16 mm and height 7 mm: slant l = √(24² + 7²) = 25 mm; lateral area = π × (40 + 16) × 25 = 1400π = 4398 mm².
- Hexagonal prism, base side 20 mm, height 50 mm: development strip 6 × 20 = 120 mm long and 50 mm high, lateral area 6000 mm².
8. One-point and two-point perspective
Perspective projects the object onto a picture plane (PP) with straight visual rays converging on one station point (SP), the eye. Its parts are the ground plane (GP), the ground line (GL) where the PP meets the GP, the horizon line (HL) at eye level, the centre of vision (CV), the point of the HL directly in front of the eye, and the vanishing points (VP). Rules: a set of parallel horizontal lines converges to one VP on the HL; lines parallel to the PP stay parallel in the image; with a vertical PP, vertical lines stay vertical; image size falls with distance; and an edge lying in the PP is drawn true size.
| Type | Set-up | Vanishing points | Typical use |
|---|---|---|---|
| One-point (parallel) | one face of the object parallel to the PP; lines perpendicular to the PP converge | one, at the CV | interiors, corridors, streets, courtyards; the face parallel to the PP keeps its true shape |
| Two-point (angular) | object turned so that no face is parallel to the PP; vertical edges parallel to the PP | two, on the HL | building exteriors and corner views; no face keeps its shape |
| Three-point | PP tilted, so verticals converge as well | three | very tall buildings seen from above or below (bird's-eye, worm's-eye) |
Let D be the distance from the SP to the PP. A horizontal line making angle α with the PP vanishes on the HL at distance D cot α from the CV (α = 90° gives the CV itself, α → 0 gives infinity). Two perpendicular sets, at α and 90° − α, have their VPs D cot α and D tan α from the CV on opposite sides, and the lines from the SP to them are at right angles. The usual 30°/60° layout therefore puts the VPs at 1.73 D and 0.58 D, 2.31 D apart. A vertical edge of height h lying d behind the PP appears on the PP with height h × D/(D + d). The cone of vision is kept near 30° (never beyond about 60°) to avoid distortion, a high HL gives a bird's-eye view and a low one a worm's-eye view.
- SP 5 m from the PP, edges at 30° to the PP: VP at 5 cot 30° = 8.66 m from the CV; the perpendicular set at 60° has its VP at 5 cot 60° = 2.89 m on the other side.
- A 3 m vertical edge stands 2 m behind the PP with the SP 6 m in front of it: image height = 3 × 6/(6 + 2) = 2.25 m (at drawing scale).
9. Light, shade and shadow
Shade is the unlit part of an object itself, while a shadow is the dark area it casts on another surface. The line that separates the lit and unlit faces is the shade line, and its shadow outlines the cast shadow. Architectural drawing uses conventional light: parallel rays along the body diagonal of a cube, so the ray appears at 45° in both plan and elevation, arriving from the top left front. Its true inclination to the ground is 35°16′ (tan⁻¹ 1/√2), because it makes equal angles with all three principal planes. Every shadow can then be drawn by offsets of equal horizontal and vertical measure.
- The shadow of a point on a plane is where the ray through the point meets the plane.
- The shadow of a line on a plane parallel to it is parallel and equal to the line.
- A vertical line casts a vertical shadow on a vertical wall, and on the ground a shadow along the plan of the ray, at 45° in plan.
- In elevation, the shadow of an element on the surface behind it is displaced across and down by the projection of the element from that surface: a 600 mm chajja casts a band 600 mm deep on the wall below its edge, and a vertical fin projecting 300 mm casts a 300 mm shadow on the wall beside it.
- A circle parallel to the receiving plane casts an equal circle, otherwise an ellipse. A recess is shaded on the sides nearest the light (top and left), a projection casts its shadow on the sides away from it (bottom and right).
With real sunlight the shadow of a vertical object of height h on level ground has length h / tan(solar altitude): a 9 m pole at 30° altitude casts 9/tan 30° = 15.6 m, and at 45° the shadow equals the height. Under conventional light a vertical pole of height h casts a ground shadow of true length h√2, which appears as h in elevation and as h√2 at 45° in plan. In rendering, shadows add depth and make projections and recesses readable, and cast shadows are made darker than the shade on the object.
Key takeaways
- RF = drawing length ÷ actual length; lengths scale by RF, areas by RF², volumes by RF³; a plain scale reads two units, a diagonal scale three, and a bar scale survives enlargement or reduction.
- Conics: e < 1 ellipse, e = 1 parabola, e > 1 hyperbola; a plane parallel to a generator gives a parabola and one parallel to the axis a hyperbola; an ellipse with semi-axes a and b has foci at ± √(a² − b²).
- Cycloid: base 2πr, height 2r. Involute: radius of curvature = free string. Helix: unrolls to a straight line, tan λ = pitch/2πr. Archimedean spiral: equal radius gain per angle; logarithmic spiral: equal ratio.
- First angle (India): plan below elevation. A line of length L shows L cos θ in the top view and L cos φ in the front view, apparent angles are never smaller than true ones, and projected area is A cos β.
- Isometric: axes 120° apart, factor 0.816 in projection and 1.0 in drawing (1.225 times larger); circle becomes an ellipse with minor : major 0.577. Cone development: sector angle 360° × r/l, area πrl, with l the slant height.
- Perspective: the horizon line is eye level; one-point has one VP at the CV and a true-shape face parallel to the PP; two-point VPs lie D cot α and D tan α from the CV; conventional light is 45° apparent, 35°16′ true; sun shadow = h / tan(altitude).
Practice questions (18)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
In a floor plan, the heaviest line weight is normally used for
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Answer: B — walls and columns that are cut by the horizontal cutting plane
The cut elements are the nearest and most important surfaces in a plan, so they carry the heaviest line and often a solid fill. Elements above the cutting plane are dashed, dimension and projection lines are thin, and centre lines are thin chain lines. The tempting choice, beams above the cut, is drawn dashed, not heavy.On a site plan drawn at 1 : 500, a plot measures 24 cm² on the paper. Its actual area, in m², is ____.
Numerical answer — type the value.
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Answer: 600
Areas scale by the square of the RF: actual area = 24 × 500² = 24 × 250,000 = 6,000,000 cm² = 600 m² (1 m² = 10,000 cm²). Check with sides: a 4 cm × 6 cm plot is 2000 cm × 3000 cm = 20 m × 30 m = 600 m². Multiplying by 500 once gives only 1.2 m² of ground, which is far too small.Which statement about plain and diagonal scales is correct?
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Answer: D — A diagonal scale reads three units, because similar triangles split its smallest division into ten parts
The diagonal across ten horizontal steps divides the smallest subdivision into tenths by similar triangles, so the scale reads, for example, metres, decimetres and centimetres. The least count is therefore one tenth of the smallest subdivision, not the subdivision itself, which is the plain-scale case. Neither scale measures angles.A right circular cone is cut by a plane parallel to one of its generators. The section is
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Answer: A — a parabola, with eccentricity equal to one
A plane inclined at exactly the generator's angle, that is parallel to a generator, cuts the cone in a parabola (e = 1). Steeper planes, up to parallel to the axis, give hyperbolas, which is the tempting wrong choice; flatter planes give ellipses, and a plane parallel to the base gives a circle.An ellipse has a major axis of 100 mm and a minor axis of 60 mm. Its eccentricity is
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Answer: B — 0.80
Semi-axes a = 50 and b = 30, so the focal distance from the centre is c = √(2500 − 900) = 40 mm and e = c/a = 40/50 = 0.80. The value 0.60 is b/a, 0.75 is b/c and 0.64 is e², each a slip in one step.A helical stair has a mean radius of 1.0 m and rises 3.0 m in one full turn. The length of handrail for one turn, measured along the mean radius, in m to two decimal places, is ____.
Numerical answer — type the value.
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Answer: 6.96
Unrolled, one turn is the hypotenuse of a right triangle with base 2πr = 6.283 m and height equal to the pitch, 3.0 m. Length = √(6.283² + 3.0²) = √(39.478 + 9) = √48.478 = 6.96 m. Adding the two sides, 9.28 m, or using the plan circumference alone, 6.28 m, ignores the rise.A taut string is unwound from a circle to trace an involute. At a point P of the curve, the radius of curvature equals
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Answer: A — the length of the free string, from its point of tangency on the circle to P
The string is tangent to the circle, and at each instant the curve turns about the point of tangency, so the radius of curvature is the free string length. The distance from P to the centre is larger, √(r² + l²), and the base radius and the full circumference 2πr are constant, whereas the radius of curvature grows from zero as the string is paid out.In first-angle projection, as followed in Indian drawing practice, the plan is placed
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Answer: D — below the elevation, on the same vertical projector
The object lies between the observer and the planes, so the view projected on the HP is rotated down into the plane of the VP and lands below the elevation. Placing the plan above the elevation is third-angle practice, the tempting choice. The plan always lies on the vertical projector through the elevation, never beside it.A straight line 100 mm long is inclined at 30° to the HP and at 45° to the VP. The length of its front view is
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Answer: C — 70.7 mm
The front view is the projection on the VP, so its length is L cos φ with φ the inclination to the VP: 100 cos 45° = 70.7 mm. The value 86.6 mm is the top-view length, 100 cos 30°, which uses the inclination to the HP, and 50 mm is the height difference of the ends, 100 sin 30°.A cube of edge 150 mm is drawn as a true isometric projection, using the isometric scale. The length of each edge on the drawing, in mm to one decimal place, is ____.
Numerical answer — type the value.
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Answer: 122.5
In true isometric projection every edge along an axis is foreshortened by cos 35°16′ = √(2/3) = 0.8165. Edge = 150 × 0.8165 = 122.47, which is 122.5 mm to one decimal place. An isometric drawing, which uses true lengths, would show 150 mm.A sphere of diameter D is shown in an isometric drawing, with true lengths used along the axes. It appears as
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Answer: C — a circle of diameter 1.225 D
Any orthographic projection of a sphere is a circle, never an ellipse. In true isometric projection its diameter is D. An isometric drawing enlarges everything by 1/0.816 = 1.225, so the circle has diameter 1.225 D. The circle of diameter D is the projection answer, which is the tempting one.A conical roof has a base diameter of 6 m and a vertical rise of 4 m. Its sheet is developed as a sector. The angle of the sector, in degrees, is ____.
Numerical answer — type the value.
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Answer: 216
Base radius r = 3 m and vertical height h = 4 m, so the slant height is l = √(9 + 16) = 5 m. The arc of the sector is the base circumference 2π × 3, on a circle of radius 5, so the angle = 360° × 3/5 = 216°. Using h in place of l gives 270°, the usual error.An interior view of a long rectangular room, looking straight towards the end wall, is most simply drawn in
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Answer: B — one-point perspective, with the lines running along the room converging at the centre of vision
The end wall is parallel to the picture plane, so it keeps its true shape and every line perpendicular to the picture plane, the floor, ceiling and side-wall edges, converges to the single vanishing point at the centre of vision. This is the one-point set-up. Two-point perspective turns the room so that no wall is parallel to the picture plane, which is unnecessary here.A vertical edge 3 m high stands 2 m behind the picture plane, and the station point is 6 m in front of the picture plane. In a perspective drawn at full scale, the height of the edge on the picture plane, in m to two decimal places, is ____.
Numerical answer — type the value.
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Answer: 2.25
Visual rays from the eye pass through the picture plane at the fraction D/(D + d) of their length to the edge, where D = 6 m is the eye-to-plane distance and d = 2 m the distance behind it. Image height = 3 × 6/(6 + 2) = 2.25 m. Ignoring the 2 m set-back would give the full 3.00 m, which is right only for an edge lying in the picture plane.Under conventional 45° light, a horizontal chajja projects 600 mm from a plain wall. In elevation, the depth of its shadow on the wall below the lower edge of the chajja is
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Answer: D — 600 mm
The ray appears at 45° in the vertical plane perpendicular to the wall, so the shadow drops by exactly the projection: 600 mm. The value 849 mm is 600√2, the length of the ray as it appears in that side view between the chajja edge and the wall, not the depth of the shadow on the wall.Which of the following statements about conventional light and shadows are correct?
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Answer: A — The conventional ray appears at 45° in both plan and elevation, but its true inclination to the ground is about 35°16′; B — The shadow of a line on a plane parallel to that line is parallel to it and equal in length
The ray is the cube diagonal: 45° in plan and in elevation, but 35°16′ true, so a vertical pole's true ground shadow is h√2, not h. A line parallel to a plane casts a parallel, equal shadow on it. Shade is the unlit side of the object itself, whereas a shadow falls on another surface.Which of the following statements about two-point (angular) perspective are correct?
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Answer: A — Both vanishing points lie on the horizon line; B — Vertical edges remain vertical when the picture plane is vertical
Both sets of horizontal lines vanish on the horizon, and a vertical picture plane keeps verticals vertical. No face is parallel to the picture plane, so none keeps its true shape. The vanishing points are equidistant from the centre of vision only when both sets are turned 45°; at 30° and 60° they lie D cot 30° and D cot 60° away.Which of the following surfaces can be developed exactly, without stretching or tearing?
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Answer: A — A right circular cone; B — A hexagonal prism; C — A square pyramid
The cone is single-curved and unrolls into a sector, and the prism and pyramid are made of plane faces that unfold into rectangles or triangles. A sphere is double-curved: any flat sheet must be stretched or cut into gores to cover it, so it can only be approximated.