Everyday objects, abstracted to their ideal shapes: a building block becomes a cube, a ball a sphere, a soup can a cylinder, an ice-cream cone a cone. Geometry is the study of shapes like these.
You have spent whole stages on numbers: the number line, expressions, equations, inequalities. Every one of those tools answers the same kind of question — how much? How long, how many, how fast. That is one great half of mathematics. Now we turn to the other half. Look up from the page: the world is full of shapes — the corner of a room, the rim of a cup, the path of a thrown ball. Geometry is the part of math that describes those shapes exactly, and this lesson is your first step into it.
13.1.1 Two questions about the world
Hold a single coin in your hand. You can ask two completely different questions about it.
The first is a question of quantity: how much is it worth, how heavy is it, how many do you have? These are the questions of algebra and arithmetic. Their natural picture is the number line — a single straight track where every position is one number, one amount.
The second is a question of shape: what does it look like? It is round, flat, with a rim and two faces. These are the questions of geometry. Their natural pictures are figures — triangles, circles, boxes — things you can draw and turn and measure.
Two halves of mathematics. Left: algebra lives on the number line — a question of how much. Right: geometry lives in figures — a question of what shape.
Key idea
Algebra measures quantity — how much. Geometry describes shape — what form. Both are exact; they simply ask different questions about the same world.
Of course the two are old friends. The number line is itself a geometric object (a line!), and soon every shape we draw will carry numbers — lengths and angles. But the starting question is what is new. From here on we ask, first of all, what shape is this?
13.1.2 Solid figures: things that take up space
Start with the shapes you can actually hold. A brick, a marble, a can of soup — each one fills a chunk of space; it has length, width, and height. Any figure that takes up space this way is called a solid figure, or simply a solid.
You already know more of them than you think. Here is a gallery of the most common solids, drawn the way an artist sketches a box — slightly turned, so you can see that they are three-dimensional.
A gallery of solids: cuboid (a box), cube, cylinder, cone, sphere, triangular prism, and square pyramid. Every one of them takes up space.
Spot them around you
A dice is a cube; a cereal box is a cuboid; a soda can is a cylinder; a party hat is a cone; a basketball is a sphere; a camping tent is often a triangular prism; the top of an obelisk is a pyramid.
13.1.3 Sorting solids: flat faces vs a curved surface
Run your finger over each solid. On some, your finger always travels along a flat face and turns a sharp corner to reach the next one — a cube is all flat faces. On others, your finger glides over a curved surface that never lies flat — think of rolling a marble or a can.
That single test sorts every solid into two families:
A polyhedron is a solid whose surface is made entirely of flat faces (the faces are polygons). Prisms and pyramids are polyhedra. The cube, cuboid, triangular prism, and square pyramid all belong here.
A solid of revolution has at least one curved surface. The beautiful fact behind the name: each of these is made by taking a flat shape and spinning it around a line. Spin a rectangle and you sweep out a cylinder; spin a right triangle and you sweep out a cone; spin a half-disk (a semicircle) and you sweep out a sphere.
Try it Step through the solids and classify each
Press + and − to cycle through seven solids. Watch the readout name each one and tell you its family.
Solid1
The two families
Polyhedron — every face is flat (a polygon). Examples: cube, cuboid, prism, pyramid. Solid of revolution — at least one curved surface, made by spinning a flat shape. Examples: cylinder, cone, sphere.
13.1.4 Flat figures: shapes with no thickness
Now press a solid flat — or just look at its shadow on the wall. A shadow has shape but no thickness at all: it lies entirely in one flat plane. A paper cut-out, a chalk circle on the sidewalk, the screen you are reading — these are plane figures (also called flat figures).
A plane figure has length and width but no height — no third dimension. The everyday plane figures are the ones you have drawn since you were small: the triangle, the rectangle and square, the circle, and many-sided shapes like the pentagon and hexagon.
A gallery of plane figures: triangle, square, rectangle, circle, pentagon, and hexagon. Each lies flat — length and width, but no thickness.
Don't mix them up
A square is flat — a plane figure. A cube takes up space — a solid. The square is one face of the cube. Plane figure = no thickness; solid = takes up space.
13.1.5 How solids and flat figures connect
Solids and plane figures are not two separate worlds — they are tied together in two natural ways.
Unfold a solid, and you get a flat figure. Cut a cardboard box along its edges and flatten it: the box opens out into a pattern of flat polygons called a net. A net is simply a solid taken apart and laid flat, so that folding it back up rebuilds the solid.
Look at a solid from one side, and you get a flat figure too. Stare straight at a cube and you see a square; that flat picture is a view of the solid. Nets and views are both bridges between the flat and the spatial.
Try it Fold a net up into a cube — and unfold it back
Toggle between the flat net and the assembled solid. The colored faces are the same six squares, just folded.
Show
The bridge
A net is a solid unfolded flat; a view is a solid seen from one side. Both turn a solid into plane figures — and folding a net back up returns the solid. We will cross this bridge again in solid geometry (Stage 28).
Looking ahead, in Lesson 13.2 we go the other direction and break any figure — flat or solid — down into its tiniest building blocks: points, lines, surfaces, and solids, each one growing out of the last by motion.
★ Recap
In one breath: algebra asks how much; geometry asks what shape. A figure that takes up space is a solid; a figure with no thickness is a plane figure. Solids split into two families — polyhedra (all flat faces) and solids of revolution (a curved surface, made by spinning). And the two worlds connect: unfold a solid into a flat net, or view it from one side as a flat picture.
Question
Branch
Pictures
How much?
Algebra
number line, amounts
What shape?
Geometry
solids & plane figures
Family of solid
Surface
Examples
Polyhedron
all faces flat
cube, cuboid, prism, pyramid
Solid of revolution
a curved surface
cylinder, cone, sphere
✎ Exercises 13.1
Sort each item into solid figure or plane figure: (a) a basketball, (b) a triangle drawn on paper, (c) a brick, (d) a circle, (e) a soup can.
Answer
Solids: (a) basketball, (c) brick, (e) soup can — each takes up space. Plane figures: (b) triangle, (d) circle — each lies flat with no thickness.
Name the solid from the description: "Two flat circular ends joined by one curved surface; you can roll it on its side."
Answer
A cylinder (a soda can). Its curved surface makes it a solid of revolution.
Sort this list into polyhedra and solids of revolution: cube, cone, triangular prism, sphere, square pyramid, cylinder.
Answer
Polyhedra (all flat faces): cube, triangular prism, square pyramid. Solids of revolution (a curved surface): cone, sphere, cylinder.
How many faces does a cube have? How many faces does a square pyramid have?
Answer
A cube has 6 square faces. A square pyramid has 5 faces — 1 square base plus 4 triangular sides.
A flat pattern of six squares is folded up. If it closes perfectly with no gaps or overlaps, what solid does it form? What is this flat pattern called?
Answer
It forms a cube. The flat pattern is a net of the cube — the cube unfolded and laid flat.
For each question, say whether it belongs to algebra (how much) or geometry (what shape): (a) "How many sides does this figure have?" (b) "What is 7 + 5?" (c) "Is this surface flat or curved?" (d) "How fast is the car going?"
Answer
Geometry: (a) and (c) — both ask about shape. Algebra: (b) and (d) — both ask how much.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
The big idea. This lesson is the hinge between the arithmetic/algebra strand (quantity) and the geometry strand (shape). The single most useful sentence a student can carry forward is "algebra asks how much, geometry asks what shape." Everything else — solids vs. plane figures, polyhedra vs. solids of revolution, nets and views — hangs off that distinction.
The misconception to watch. Beginners blur plane figures and solids: they call a cube a "square" or a sphere a "circle." Anchor the difference physically — a square is a flat face you could cut out; a cube takes up space and has six such faces. The hands-on cure is to unfold a real cereal box into its net and fold it back; the net widget mirrors that experience on screen.
Common Core. This sits alongside Grade 1–2 work distinguishing two- and three-dimensional shapes (1.G.A.1–2, 2.G.A.1) and Grade 6 surface-area-from-nets (6.G.A.4), where students "represent three-dimensional figures using nets." The polyhedron / solid-of-revolution sort previews high-school G-GMD.B.4 (relating 2-D cross-sections and rotations to 3-D objects).
eastmath.com · Stage 13 · 13.1 Stepping into Geometry · Intuition before notation
eastmath.com · 13.1 From Numbers to Shapes: Stepping into Geometry · 13.1.1 Two questions about the world