MISSION · Geometry
Nets and Cross-Sections
Two ways of turning a 3D object into 2D shapes you can measure
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LEVEL 2 · LEARN
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The idea in one sentence
A net is a solid unfolded flat, and a cross-section is the flat shape you see when you slice a solid through — one shows the outside, the other shows the inside.
Nets: cutting along the edges
Take a cardboard box, cut along enough edges, and press it flat. What you get is a net: every face of the solid, laid out and still joined.
A cube unfolds into six squares. There are eleven genuinely different ways to arrange them, and plenty of six-square arrangements that don’t fold back into a cube — four in a row with two stuck on the same side will overlap. The test is whether every square ends up on its own face with nothing doubled and nothing missing.
The reason nets matter isn’t craft. It’s that surface area becomes easy: the surface area of a solid is just the total area of its net, and a net you can see is much harder to count wrongly than faces hiding round the back.
| Solid | Net |
|---|---|
| Cube | 6 squares |
| Rectangular prism | 3 pairs of matching rectangles |
| Triangular prism | 2 triangles + 3 rectangles |
| Square pyramid | 1 square + 4 triangles |
| Cylinder | 2 circles + 1 rectangle |
That last one is the interesting one. Peel the label off a tin and it’s a rectangle whose length is the circumference of the circle, \(2\pi r\). That single fact explains the whole surface area formula:
\[A = 2\pi r^2 + 2\pi r h\]Cross-sections: cutting through
Now slice the solid instead of unfolding it. The exposed face is the cross-section, and it depends entirely on where and at what angle you cut.
A cylinder sliced parallel to its base gives a circle — the same circle, wherever you cut, which is what makes it a prism-like solid. Slice the same cylinder straight down through its axis and you get a rectangle. Slice it at a slant and you get an ellipse.
For a prism, any cut parallel to the base gives the identical base shape. That is the definition of a prism, and it’s also why the volume formula is base area × height: the solid is that cross-section, stacked.
For a pyramid or cone, parallel slices give the same shape shrinking towards the apex, which is why the volume comes out as a third of the prism with the same base.
A cube can produce a square, a rectangle, a triangle, a pentagon, or a regular hexagon, depending on the angle. The hexagon — cut through the middle at the right slant — is worth seeing once, because it kills the assumption that the cross-section looks like the solid.
Common mix-ups
- Assuming any six squares fold into a cube. Some overlap when folded.
- Forgetting a face in a net, usually the bottom one.
- Drawing the cylinder’s rectangle any old length. It must equal the circumference.
- Assuming every slice of a prism is the base shape. Only the parallel ones are.
Check yourself
How many faces are in the net of a cube? And what shape is a cylinder’s cross-section when you slice parallel to the base?
Six squares — one per face, which is why the cube’s surface area is 6s². And the parallel slice through a cylinder is a circle, the same circle at every height.
LEVEL 3 · REFLECT
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