How does the cross-section test work?
Each item shows a solid, such as a cube, a pyramid or a prism, with a flat plane passing through it, drawn see-through. Picture the face the cut would leave and pick its shape from four choices. Dashed edges are at the back of the solid. After you answer, the cut face is drawn in and the reason for its shape is given.
A game is 8 cuts drawn from 24, through five solids. Each cut's shape is worked out exactly from the solid and the plane, and the site's tests check every one.
How is your Cross-Section Test result calculated?
Your result is the number of cuts you named correctly, out of 8, with your time. The wrong choices are never shapes that would also be right: a square is a rectangle too, so the two are never offered against each other. No published average exists for these items, so there is no rating.
How do you work out a cross-section?
Count the faces the plane passes through: the cut face has one side for each of them. Then look at which of those faces are parallel, because a flat cut through two parallel faces leaves two parallel sides. That alone tells a trapezoid from a parallelogram, and a rectangle from other four-sided shapes.
- A cut parallel to a cube's face, or to a prism's end, has the same shape as that face: a square from a cube, a triangle from a triangular prism.
- A cut that clips one corner of a cube crosses the three faces that meet there, so it is a triangle. Clip a square pyramid's top and you get a square, because four faces meet there.
- A slanted cut can cross more faces: up to all six of a cube's.
What shapes can a cube's cross-section be?
A triangle, a four-sided shape (square, rectangle, rhombus, parallelogram or trapezoid), a pentagon or a hexagon. Never more than six sides, because a cube has only six faces to cross. A cut through the centre at right angles to a long diagonal gives a regular hexagon, every side the same length; the site's tests check its area.
| Solid | Faces | Cut shapes in this test |
|---|---|---|
| Cube | 6 | square, rectangle, triangle, hexagon, pentagon, trapezoid, rhombus, parallelogram |
| Square pyramid | 5 | square, triangle, trapezoid, pentagon |
| Triangular prism | 5 | triangle, rectangle, trapezoid, pentagon |
| Hexagonal prism | 8 | hexagon, rectangle, pentagon |
| Tetrahedron | 4 | square, rectangle, triangle |
One surprise is the tetrahedron, a pyramid of four triangles: a cut halfway between two opposite edges leaves a perfect square.
Is this like the Santa Barbara Solids Test?
In spirit. The Santa Barbara Solids Test, published by Cohen and Hegarty in 2012 (Learning and Individual Differences), asks people to infer cross-sections of 3D objects and was built to study that skill. Its items, wording and timing are its own; this is a short practice game with no norms behind it.
For more 3D puzzles, try the Block Counting Test or the Paper Folding Test.