Matemáticas
Net of a Solid: The Flat Shape That Folds Into a 3D One
A net is a solid unfolded flat. A cube has eleven of them. See how to test whether a net folds, and how to read surface area straight off one.
Net
A net is a flat arrangement of a solid's faces, joined at their edges, that folds along those edges to form the solid without gaps or overlaps.
- También llamado
- Net of a 3D shape
- Dónde lo encuentra el alumnado
- Grade 4 mathematics, when cubes and cuboids are built from card, and again at GCSE where surface area is often set on the net rather than the solid.
La respuesta corta
A net is a flat arrangement of a solid's faces, joined along their edges, that folds up to make the solid with no gaps and no overlapping faces. A cube has eleven distinct nets. Because a net shows every face at its true size, adding the face areas gives the solid's surface area directly.
Un ejemplo
Surface area = 2(5×3) + 2(5×2) + 2(3×2) = 30 + 20 + 12 = 62 cm²
A cuboid measuring 5 cm by 3 cm by 2 cm unfolds into six rectangles in three matching pairs. Laid flat they are two 5 by 3 rectangles, two 5 by 2 and two 3 by 2, giving 62 cm² in total. Drawing the net first turns a question about a solid into one about six flat rectangles, which is where the pairing becomes impossible to miss.
How to tell whether a drawing actually folds
A net is not just the right number of faces in a row. Six squares laid out in a straight line will not fold into a cube — the strip wraps round on itself and two of the squares end up in the same place while another position is left open.
The mental test is quick. Choose one square as the base, fold the squares attached to it up to become walls, and see whether exactly one square is left to drop on top. If two want the same wall, or the lid is missing, the drawing is not a net.
A face count is the cheap first check. A cube net has six squares and no more; a square-based pyramid has one square and four triangles; a triangular prism has two triangles and three rectangles. A drawing with the wrong tally can be rejected before any folding is imagined.
Eleven nets, not one
There are eleven distinct nets of a cube, counting two as the same if one can be rotated or reflected onto the other. Six of them have a row of four squares with one square attached on each side of that row, which is why the familiar cross shape feels like the standard answer when it is only one option out of eleven.
Questions exploit that. A paper will show four unusual-looking arrangements and ask which one folds into a cube, and a student who has only ever seen the cross has no method to fall back on. Practising the folding test on odd-looking layouts is the point of the topic.
A net is a surface-area machine
Every face appears on a net at full size and undistorted, unlike in a drawing of the solid where faces are foreshortened. Surface area is therefore nothing more than the total area of the flat shape.
The same reasoning covers curved solids. Unroll a cylinder and the curved surface becomes a rectangle whose width is the circumference of the circular end, so its area is 2πr multiplied by the height, with two circles added for the ends.
- Distinct nets of a cube, once rotations and reflections are treated as the same
- 11Distinct nets of a cube, once rotations and reflections are treated as the same
Preguntas frecuentes
How many nets does a cube have?
Eleven, once rotations and reflections of the same layout are counted as one. Students often expect a single correct net because the cross-shaped one appears in every textbook, but any of the eleven folds up perfectly well. Exam questions rely on that gap between expectation and fact.
Why do six squares in a row not fold into a cube?
Because a cube only has four faces around its middle. Folding a strip of six squares wraps four of them into a band and sends the fifth and sixth back onto faces already used, leaving the top open and doubling up elsewhere. A working net needs the extra two squares attached off to the sides.
What is the net of a triangular prism?
Two identical triangles for the ends, and three rectangles for the sides, usually drawn as a row of three rectangles with a triangle attached above and below the middle one. Each rectangle is as long as the prism and as wide as one side of the triangle.
Do you need to draw a net to find surface area?
Not always, but it is the safest method when a solid is unfamiliar. Once a shape is routine, listing the faces and their areas is faster. The moment a solid has faces of several different sizes, sketching the net stops you from counting a face twice or forgetting one entirely.
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