1.2.15 · D5Basic Geometry
Question bank — Nets of 3D shapes
Prerequisite ideas you may want open in another tab: Surface Area of 3D Shapes, Circles and Circumference, Pythagoras Theorem, Triangles and Area, Platonic Solids, Unfolding and Folding Problems.
True or false — justify
A net is a flat 2D pattern that folds along its edges into a 3D shape. A quick reminder of the key words before we test them:
Recall What must every valid net satisfy?
It must contain every face of the solid, the faces must be connected along shared fold-edges, and when folded the edges must meet with no gaps and no overlaps.
Each 3D shape has exactly one net.
False — a cube has 11 distinct nets, and most solids have many. Different nets are just different ways of "peeling open" the same solid, like peeling an orange along different cuts.
Every arrangement of 6 squares is a valid cube net.
False — you also need the right connectivity. The 6-in-a-row strip has 6 squares but the far ends can never meet when folded, so it tears or overlaps.
A cube and a rectangular box (cuboid) can share the same net shape.
False in general — a cube's net is made of 6 identical squares, but a cuboid's net has three pairs of different-sized rectangles, so the piece shapes differ unless the cuboid happens to be a cube.
The two circles in a cylinder's net can be any size, as long as the rectangle is present.
False — the rectangle's width must equal the circle's ==circumference ==. If the circle's radius doesn't match, the rectangle won't wrap around it and the seam won't close.
A cylinder's curved surface unrolls into a rectangle.
True — cutting the tube straight down its side and flattening it gives a rectangle whose height is the cylinder's height and whose width is the base circumference .
A cone's curved surface unrolls into a rectangle too.
False — it unrolls into a sector (a pie slice), because every point on the curved surface is the same slant distance from the apex, so flattening keeps them on a circle of radius .
The triangular faces of a square pyramid have height equal to the pyramid's vertical height.
False — the triangle's height in the net is the slant height (apex-to-base-midpoint along the sloping face), which is longer than the vertical height because it runs along the tilted surface.
The number of faces on the solid always equals the number of separate pieces in its net.
False — the net is one connected piece; the number of faces equals the number of polygon regions inside that single piece, all joined at fold lines.
Surface area equals the total area of the net.
True — the net is the solid "peeled flat," so summing the areas of all its faces (its total ink-on-paper area) is exactly the surface area. See Surface Area of 3D Shapes.
Opposite faces of a cube are always directly joined edge-to-edge in a net.
False — opposite faces must never touch along an edge in the net, because in the folded cube they sit on opposite sides and must meet only via the other faces.
Spot the error
Someone made a claim below. Find the flaw and correct it.
"The cylinder net rectangle is because it wraps the circle of radius ."
The width must be the ==circumference ==, not the radius . The rectangle wraps around the rim, and going around a circle covers a distance of , not .
"A cone's sector uses the base radius as its radius, since it comes from the base."
The sector's radius is the ==slant height ==, not . Every unrolled surface point lies a slant-distance from the apex; the base radius only sets the arc length .
"Six squares in a T-shape can't fold into a cube because a T isn't a cross."
Being a "cross" is not required. Many non-cross arrangements (including several T-like and Z-like layouts) are among the 11 valid cube nets — what matters is that opposite faces stay non-adjacent and everything folds without overlap.
"The pyramid's four triangles each have area where is the pyramid height."
Wrong height — the triangle uses the slant height , giving . Using the vertical height underestimates the true face area. See Triangles and Area.
"A cuboid net needs 6 congruent rectangles."
It needs 6 rectangles forming three congruent pairs, not 6 identical ones — top/bottom (), front/back (), left/right ().
"If a flat pattern has the right total area, it must fold into the target solid."
Area is necessary but not sufficient — the shape and connectivity of the pieces must also allow folding with matching edges. A right-total-area pattern can still be impossible to fold.
"When we unfold a cube, the fold edges disappear."
They don't disappear; the cut edges separate and the fold edges remain as the internal lines where faces stay joined. Each edge of the solid is either cut open or kept as a fold.
Why questions
Explain the reasoning, not just the fact.
Why does the same solid have many different nets?
Because a net depends on which edges you cut versus keep as folds. Choosing a different set of edges to cut peels the solid open differently, giving a different flat layout that still folds back.
Why must the cylinder rectangle's width equal and not the diameter ?
The rectangle wraps around the circular rim's edge, and the length of that curved edge is its circumference . The diameter is a straight chord through the middle, which is irrelevant to wrapping. See Circles and Circumference.
Why is the cone's lateral surface a sector rather than a full circle?
Because its arc length must equal only the base circumference , which is less than the full circle of radius (that would be ). So the pie slice covers an angle .
Why can we compute surface area straight from the net?
Folding doesn't stretch, shrink, or overlap any face — it only rotates flat pieces about fold lines. So the total flat area is preserved and equals the surface area.
Why does using slant height (not vertical height) matter so much for pyramids and cones?
The actual sloping face is what you cover with paper, and it lies along the slant, which is longer than the vertical drop. Vertical height would measure a shorter interior line, undercounting the material. Pythagoras links them: — see Pythagoras Theorem.
Why do only the Platonic solids have especially "clean" symmetric nets?
Because all their faces are congruent regular polygons meeting identically at every vertex, so unfoldings are highly symmetric. See Platonic Solids and Unfolding and Folding Problems.
Why can flat regular polygons tile a plane but a net usually can't be repeated to tile?
A net is designed to fold up and close a gap, so its outline is deliberately non-repeating; tiling (see Tessellations) instead needs shapes whose copies fill the plane with no folding at all.
Edge cases
Boundary and degenerate situations the topic quietly invites.
What is the "net" of a sphere?
A sphere has no true flat net — it can't be unfolded onto a plane without stretching or tearing, because its surface has curvature in two directions. Map-makers approximate it with gores (petal strips), which are not exact.
As a cone's slant height grows huge compared to , what happens to its unrolled sector angle?
The angle shrinks toward — the sector becomes a thin sliver, matching a tall, needle-like cone whose curved surface is nearly a narrow triangle.
As a cone's height shrinks to (so ), what does the net approach?
, a full circle — the flattened cone becomes a flat disc laid on its base, which is the degenerate "flat" cone.
If a cuboid's height , what happens to its net?
The four side rectangles ( and ) shrink to zero-width slivers, and the net collapses toward just the two faces — the "box" flattens into a flat sheet with no thickness.
Can a valid net have a face touching another face at only a single corner (not a full edge)?
No — fold lines must be shared edges, not points. A corner-only contact gives nothing to fold along, so that face would be effectively disconnected and the folding test fails.
If two squares in a candidate cube net overlap when folded, is it still a net?
No — a valid net folds with no overlaps. Overlap means two faces would occupy the same place, which is physically impossible for a real solid, so the pattern is rejected.
Does a cube net always have to be "in one connected piece"?
Yes — if any face is detached, you can't fold a single sheet into the closed solid; every face must reach the others through a chain of shared fold edges.
Recall One-line self-test before you move on
Give the surface-area role of the net in a single sentence. Answer ::: The net is the solid unfolded flat, so its total area is the surface area, computed by adding every face's area.