Question 5
Net of a sphere? Experiment and see if you can make a paper cutout that can perfectly wrap around a ball without leaving any wrinkles, gaps or overlaps.
- A net is a flat 2D pattern that can be folded to perfectly form a 3D solid without any wrinkles, gaps, or overlaps.
- Polyhedra (like cubes and prisms) and developable surfaces (like cylinders and cones) have flat 2D nets.
- A sphere has a surface that curves in all directions simultaneously, meaning it cannot be unrolled or flattened into a single plane without distortion.
Step 1 · Concept of a Net
A net is a two-dimensional shape that can be folded into a three-dimensional solid such that it covers the solid completely without any wrinkles, gaps, or overlaps.
For example, unfolding a cube yields a flat 2D net of squares.
Step 2 · Curvature of a Sphere
A sphere is curved uniformly in every direction. Unlike a cylinder whose curved face unrolls into a flat rectangle, a spherical surface cannot be flattened onto a 2D plane without tearing or wrinkling (just like peeling an orange and attempting to press the peel flat).
Step 3 · Experiment and Conclusion
When attempting to wrap a flat piece of paper around a ball:
- The paper wrinkles and overlaps along curved sections, or
- It leaves uncovered gaps.
Therefore, a sphere does not have a true flat net, and it is impossible to make a perfect cutout.
No, it is not possible to make a paper cutout that can perfectly wrap around a ball without leaving wrinkles, gaps, or overlaps.
- Assuming all 3D shapes have flat nets: Polyhedra (cubes, prisms) and single-curvature surfaces (cylinders, cones) have flat nets, but double-curved surfaces like spheres do not.
- Confusing map gores with nets: Segmented cutouts (like strips used to make globes) only approximate a sphere and still require stretching or leave tiny gaps.
More questions in A
Build it in Your Imagination
We will start this section by practising visualisation. For each prompt, feel free to talk to your partner, gesture, draw it in the air — but do not actually draw on paper!
- Picture your name, then read off the letters backwards. Make sure to do this by sight, not by sound — really see your name! Now try with your friend's name.
Cut off the four corners of an imaginary square, with each cut going between midpoints of adjacent edges. What shape is left over? How can you reassemble the four corners to make another square?
Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What shape do you get?
Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left?
Net of a sphere? Experiment and see if you can make a paper cutout that can perfectly wrap around a ball without leaving any wrinkles, gaps or overlaps.
Place an object in front of a plane, such as a wall of your room. Shine a torch light on the object in a direction perpendicular to the wall.
What do you see?
Observe what happens to the size of the shadow as you vary the distance between your torch and your object.
Context: Observe what happens to the size of the shadow as you vary the distance between your torch and your object.
Q. Why does this happen?
Construct a model of a cube and use your hands to keep it balanced on one corner vertex. Can you try to understand why all the projected edges have equal length?