Cross-Section Worksheet Generator
Slice a cube through three points on its edges and the cut face can be a triangle, a square, a rectangle, a trapezium, a pentagon or a regular hexagon. This draws a cube, a cuboid or a triangular prism with three points marked, and asks for the shape of the cut, a sketch of its true shape, or its area. The answers shade the section on the solid and draw it flat with every side measured.
What this generator does
Finds each section from the solid's edges (one corner wherever the cutting plane crosses an edge, a solid's corner on the plane kept once), orders the corners round the polygon, and checks it: every corner on the plane and on the solid's surface, the marked points on its edge, the polygon convex, its area positive. The name is read from the polygon's own measurements.
How to use this tool
- Choose a level and how many cuts.
- Press Make the sheet. Each solid is drawn with hidden edges dashed and P, Q and R marked.
- Show the answers to see each section shaded, its true shape with side lengths, its name and its area.
- Print the sheet and answers, or type a sheet code to make the same sheet again.
Understanding the controls
- Level
- Gentle: cubes, points at corners or halfway, only triangles, squares and rectangles. Standard: cubes and cuboids, any section. Stretch: adds the triangular prism and points a third of the way along, and asks for the area.
- How many cuts
- 4 or 6. Three print to a page, two at large print, with the answers after.
- Sheet code
- Leave empty for a fresh sheet from the browser's secure generator; with a code, the same code and settings make the same sheet again while the generator is unchanged.
Worked examples
- A regular hexagon
- In a cube of side 6 cm, the cut through the midpoints of three edges that do not meet, square to a long diagonal, is a regular hexagon of side 3√2 ≈ 4.2 cm and area 46.8 cm².
- Through two opposite edges
- The cut through a bottom-front edge and the top-back edge of the same cube is a rectangle 6 cm by 6√2 ≈ 8.5 cm, area 50.9 cm².
Common use cases
- Practice visualising plane sections before cutting real models
- A starter on why a cube can have a hexagonal cross-section
- Area of a section as an application of Pythagoras in three dimensions
How this generator works
The cutting plane is the one through P, Q and R. Each corner of the solid is tested for which side of it it lies on; an edge whose ends are on opposite sides is cut at the point found by proportion. Cuts that only touch the solid at a corner or along an edge, or run along a face, are refused, since none of them is a section through the inside. The area is the vector sum over the polygon's sides, and the name comes from equal sides, parallel sides and right angles, measured to a millionth.
Randomness and fairness
Cuts come from the browser's secure generator, or a seeded one when you type a code. Codes are deterministic, not cryptographic, and not a secret.
For how randomness is produced across the whole site, see how Generate Random works.
Assumptions this tool makes
- Solids are exact and the cut is a flat plane; sizes are in centimetres and drawings are not to scale.
Limitations and good to know
- Three solids only: the cube, one cuboid and one triangular prism, in fixed sizes.
- The drawings are oblique projections, so lengths and angles on the drawing are not true; the true shape is drawn separately in the answers.
- Cuts are through points at corners, halves or thirds of edges, so every section is a polygon with 3 to 6 sides.
Common mistakes
- Joining P, Q and R and stopping at a triangle
- The cut goes on across every face it meets. Extend it until it comes back round; it may cross four, five or six faces.
- Measuring the section on the drawing
- The drawing is a projection. Work the true lengths from the solid's edges, using Pythagoras where a side runs across a face.
Privacy and your data
Sheets are built in your browser. With analytics consent, a count records how many cuts a sheet had; the sheet and any code are not sent.
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