Geometry & Figures

Problem

Faces are bounded by edges joining vertices

A flat net is folded into an open-top (lidless) rectangular box. Water is poured in, and the box is tilted so it rests on one of its bottom corner edges; the water settles into the lowest region and its surface forms a slanted plane. I must figure out exactly which faces (and which parts of those faces) the water touches, then show that wetted region shaded on the cross-shaped net of the same box.
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — This is a folding-and-tilting problem, so the key skill is imagining the 3-D box in my head (and, if needed, building one) to see where the water sits, then mapping each face back onto the flat net. I picture the tilt, mark the wet faces on the solid, then unfold those marks onto the cross-shaped net, lining the shading up across each fold edge.
1STEP 1

Name the 5 faces of the lidless box

The cross folds into a floor plus four walls, so the box has 5 faces and an open top.

2STEP 2

See where the water settles when the box is tilted

Tipped onto one bottom edge, the water pools there as a wedge with a flat, slanted top.

3STEP 3

Decide which faces the wedge of water touches

The wedge lies on the floor and against the two walls meeting at that low edge: 3 of the 5 faces.

4STEP 4

Find the shape of each wet region

The floor is wet all over, and each near wall is wet in a triangle standing on the fold edge.

5STEP 5

Color the regions onto the cross-shaped net

Unfolded, shade the centre square whole and one triangle on each of the two neighbouring flaps.

Answer
Shade the center square (bottom) of the cross-shaped net completely, plus a triangle on each of the two adjacent flaps (the two side walls that meet at the lowest edge), each triangle resting on the fold line it shares with the bottom; the other two flaps and the open top stay blank — 3 of the 5 faces are wet.
The box has 5 faces (no lid). Water sinking into one corner edge must cover the floor and lean on the two walls forming that corner, so 3 wet faces is exactly what we expect — not all 5 (the high walls and open top stay dry) and not just 1. The two wet walls are triangular because a flat slanted surface cuts a wall along a straight line. Everything matches the blue wedge in the picture.
Takeaway

Water always slides to the lowest corner: it covers the floor and wets the two walls that meet there — so on the unfolded net you color the bottom plus two triangles.

  • Name the 5 faces of the lidless box
  • See where the water settles when the box is tilted
  • Decide which faces the wedge of water touches
  • Find the shape of each wet region
  • Color the regions onto the cross-shaped net
Where next?