Add one direction — height — and a flat figure stands up into a solid.
Hold a sheet of paper flat on the table. It has length and width — two directions — and you can draw any flat figure on it: a point, a line, a triangle, a circle. Now lift the page and stack a thousand of them: you have a block with a height as well, and you have stepped out of the plane into space. That one new direction is the whole story of this stage. In this lesson we learn to see the basic solids — to name them, to sort them, and to know how each is born from a flat figure that stands up or spins. Then we learn to draw them honestly on a flat page, using the textbook oblique (cabinet, 斜二测) trick, where the trickiest rule is that depth is halved and slanted and the edges you cannot see are dashed, never erased.
A flat page is two-dimensional: every point on it is fixed by two numbers, how far right and how far up. Geometry in the plane — Stages 13 through 18 — lived entirely here. To build space we add one more independent direction, straight out of the page: height (or depth). Now a point needs three numbers, and we write it as a triple (x, y, z) — x to the right, y into the page, z up.
The objects climb a ladder of dimension. A point has no size at all (0-D). Drag a point and it sweeps out a line (1-D). Sweep a line sideways and it sweeps out a plane (2-D, like our page). Sweep a plane figure in the new third direction and it sweeps out a solid (3-D) — something with a definite volume, a chunk of space you could hold. The cube below is exactly that: the flat square base ABCD lifted straight up by its side length to make the top A₁B₁C₁D₁.
Space is the plane plus one more direction. A point is 0-D, a line 1-D, a plane 2-D, a solid 3-D. Drawing a solid means projecting those three directions back down onto a flat page — and that projection must keep parallel edges parallel and tell the truth about which edges are hidden.
Empty a child's box of wooden blocks onto the table and you can name almost everything you will ever meet in this stage. There is the long prism (a box is a prism; so is a hexagonal pencil); the pyramid that tapers to a point; the frustum, a pyramid with its tip sliced off; the round cylinder (a can); the cone (an ice-cream cone); the truncated cone (a flowerpot, a lampshade); and the sphere (a ball). Spin the hero figure at the top of the page and you can pick each one out.
There is one clean way to sort them. Some are bounded entirely by flat faces — every surface is a polygon. Those are the polyhedra: prisms, pyramids, frustums. The others have at least one curved surface, and each is made by spinning a flat figure about a straight axis. Those are the solids of revolution: cylinders, cones, truncated cones, and spheres. Knowing which family a solid belongs to tells you immediately whether its skin will unroll into flat polygons (28.3) or into circles and sectors.
A box → polyhedron (6 flat faces). A triangular tent → polyhedron (a prism). A pencil sharpened to a point → cone-ish, a solid of revolution. A bucket → truncated cone, revolution. A die → polyhedron (a cube). A globe → sphere, revolution. The test is simply: is any surface curved?
Pick a solid, then dial the azimuth (spin) and elevation (tilt). The edges tucked behind the body stay correctly dashed no matter how you turn it.
A prism is what you get by stacking identical flat cards. Take any polygon — the base — and copy it straight up to a parallel, congruent top; the sides that connect them are parallelograms. When the stack rises straight up (each card directly over the one below), every side is a rectangle and we call it a right prism; if the stack leans to one side, the sides slant and it is an oblique prism. A cube and a shoebox are right rectangular prisms; a triangular prism has two triangle bases and three rectangle sides.
Now do the same idea but with a round sweep. Take a rectangle and spin it about one of its sides as an axis: the opposite side traces a circle, and the rectangle sweeps out a cylinder. A cylinder is a "round prism" — two parallel congruent circle bases, joined by one curved lateral surface. The straight side you spun around becomes the cylinder's axis, and the rectangle's other dimension becomes the radius.
A prism = two parallel congruent bases + parallelogram sides; a right prism has rectangular sides. A cylinder = the round version, swept by spinning a rectangle about one side: two circle bases + one curved surface.
Where a prism keeps the same cross-section all the way up, a pyramid shrinks it to nothing. Start from a polygon base and join every corner to a single point above — the apex; the sides are triangles meeting at that apex. The Great Pyramid is a square pyramid: a square base and four triangular faces. Slice the pointed tip off with a cut parallel to the base and you are left with a frustum — two parallel, similar polygon bases (a big one below, a small one above) joined by trapezoids.
The curved cousins again come from spinning a flat figure about an axis:
So a cone is to a pyramid as a cylinder is to a prism: the round version. And a frustum (polyhedral) and a truncated cone (round) are the same idea — a cut-off taper — done with flat faces or with a curved one.
A cone tapers to a point; a truncated cone keeps a smaller circle on top. A pyramid is the flat-faced taper; a cone is the round one. Don't confuse a cylinder (two circles, straight sides) with a truncated cone (two different circles, slanted sides). One spinning figure decides which solid you get.
How do you draw a three-dimensional solid believably on a flat page? You need a projection — a rule that turns each space point into a page point. The textbook workhorse is the oblique or cabinet sketch (in Chinese, 斜二测, "slant two-measure"). Its rules are short and you can follow them with a ruler:
The "half the depth at 45°" rule is the one everybody gets wrong. It is there for a good reason: a full-length depth makes the drawing look stretched and unconvincing; halving it gives the familiar, readable "textbook box." The price is that a cube does not draw as three equal squares — its front is a true square, but its depth edges are drawn shorter and slanted, so the top and side look like squashed parallelograms. That is correct, not a mistake. The widget below lets you dial the true depth d and watch the rule at work; the readout always reports the drawn depth as exactly half of d.
Dial the true depth d of the box. Width and height are drawn true; depth is drawn at 45° and half its real length. Hidden edges stay dashed.
① Depth is halved and slanted: a cube of edge a draws with depth edges of length a/2 at 45°, not a. ② Hidden edges are dashed, never omitted and never drawn solid — they belong to the figure. ③ A right prism stands straight (rectangular sides); an oblique prism leans. Mixing these up is the most common way a solid sketch lies.
Two things: the family tree of solids, and the cabinet rule for drawing them.
| Idea | Polyhedra (flat faces) | Solids of revolution (a curved surface) |
|---|---|---|
| Same cross-section up | prism (right or oblique) | cylinder (spin a rectangle) |
| Taper to a point | pyramid (apex) | cone (spin a right triangle) |
| Cut-off taper | frustum (two ∼ bases) | truncated cone (spin a trapezoid) |
| All-round | — | sphere (spin a semicircle) |
| Cabinet (斜二测) rule | How to draw it |
|---|---|
| Front face | true shape — real lengths, right angles kept |
| Depth axis | at 45° |
| Depth lengths | drawn at half their true size |
| Verticals | vertical and true length |
| Hidden edges | dashed — never erased, never solid |
Next, in 28.2 · Three views of a solid, we pin a solid down exactly by photographing it straight-on from the front, the side, and the top — and we will see those three flat pictures lock together by the rule length lines up, height stays level, width matches.
Sort each solid into polyhedron or solid of revolution: (a) a triangular prism, (b) a basketball, (c) a square pyramid, (d) a soup can, (e) a flowerpot (wider at the top), (f) a frustum of a hexagonal pyramid.
Polyhedra (all flat faces): (a) triangular prism, (c) square pyramid, (f) hexagonal frustum. Solids of revolution (a curved surface): (b) sphere, (d) cylinder, (e) truncated cone.
Which flat figure, spun about which line, makes each curved solid? (a) a cylinder, (b) a cone, (c) a sphere.
(a) A rectangle spun about one of its sides. (b) A right triangle spun about one of its legs. (c) A semicircle spun about its diameter. In each case the spun-about line becomes the solid's axis.
A student sketches a cube of edge 4 in cabinet projection. They draw the front face as a 4×4 square (correct), then draw each depth edge 4 long at 45°. What rule did they break, and what should the depth edges measure on the drawing?
They broke the half-depth rule. In cabinet projection depth lengths are drawn at half their true size, so each depth edge should be drawn 2 long (= 4 ÷ 2) at 45°, not 4. (The width and height stay true at 4.)
In a cabinet sketch of a box, which edges are drawn dashed, and why are they not simply erased?
The three edges meeting at the far-back-bottom corner — the corner tucked behind the body — are hidden, so they are drawn dashed. They are real edges of the solid; erasing them would hide information the reader needs (where the back corner is). A correct sketch shows hidden edges, just with a dashed line.
Explain the difference between a right prism and an oblique prism, and between a cone and a truncated cone.
A right prism stands straight up, so its side faces are rectangles, perpendicular to the bases; an oblique prism leans, so its sides are slanted parallelograms. A cone tapers all the way to a single apex (one circle base); a truncated cone is a cone with its tip cut off parallel to the base, leaving a smaller circle on top (two circle bases).
A cube has edge a. Without coordinates, use the Pythagorean theorem to argue that a face diagonal has length a√2 and the space diagonal (corner to opposite corner, e.g. A to C₁) has length a√3.
A face is an a×a square, so a face diagonal is √(a²+a²) = a√2. The space diagonal is the hypotenuse of a right triangle whose legs are a face diagonal (a√2) and one vertical edge (a): √((a√2)² + a²) = √(2a²+a²) = √(3a²) = a√3. (We will redo this in seconds with coordinates in Stage 29.)
Six questions to lock it in. Tap the answer you think is right.
This opening lesson of Stage 28 builds the bridge from plane to solid geometry and aligns with the Common Core cluster G-GMD (visualize relationships between two- and three-dimensional objects) and G-MG (model with geometric solids). Naming and classifying the basic solids — distinguishing polyhedra from solids of revolution, and identifying each as a figure swept or spun — supports G-GMD.4 (cross-sections and rotations of 2-D shapes). The oblique/cabinet sketching convention is the standard way to represent solids on paper before coordinates arrive; the recurring emphasis on honest hidden (dashed) edges and the half-depth rule trains the careful, truthful diagramming that the three views (28.2), nets (28.3), and the coordinate methods of Stage 29 will all rely on.