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Mathematics
Solid Sections and Nets
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An interactive learning material to observe 3D solids, cut them with a plane to see the cross-section, and unfold the solid into a net to understand the relationships between faces.
Drag to rotate; hold Ctrl and scroll or use the buttons to zoom. Names, measurements and intersection lines are listed in the information panel.
Preparing the 3D scene…
ABCDA′B′C′D′
Cube ABCD.A′B′C′D′
- Vertices
- 8
- Edges
- 12
- Faces
- 6
- Lateral area
- 16 cm²
- Total surface area
- 24 cm²
- Volume
- 8 cm³
8 − 12 + 6 = 2
Drag to rotate the solid and choose a face to see its measurements; then go to Cross-section or Net.
Model notes
- The solids and cutting planes are ideal geometric models; colours, line weights and lighting only support observation. Hidden edges are dashed as in textbooks.
- Dimensions are in cm; the solids are ideal models, so only the ratios between quantities are meaningful.
- A polygonal cross-section is given its most specific name from its measurements, with a tolerance of 0.05% for lengths and 0.05° for angles; special positions are reached exactly through the suggested positions and slider snapping.
- Cylinder cross-sections are computed analytically: a plane perpendicular to the axis gives a circle; a tilted plane cutting only the lateral surface gives an ellipse; a tilted plane cutting one or both bases gives part of an ellipse cut off by one or two chords; a plane parallel to the axis gives a rectangle (through the axis: a 2r × h rectangle). A sphere's cross-section is always a circle of radius √(R² − d²).
- Cone cross-sections are computed analytically: a plane perpendicular to the axis gives a circle; a plane through the apex S gives an isosceles triangle; a plane not through the apex cuts the lateral surface along an ellipse, a parabola or a hyperbola when its angle with the base is smaller than, equal to or greater than the angle between a generatrix and the base (tolerance 0.05°); where the base cuts it off, the section is bounded by a chord.
- The unfolding animation shows how faces correspond: polyhedron faces turn about their hinge edges; the lateral surfaces of the cylinder and the cone unroll while keeping their lengths (cylinder: base circumference 2πr; cone: slant height l and base arc 2πr, flattening into a sector with central angle 360°·r/l), and the bases turn as rigid bodies. Material, paper thickness and collisions are not simulated.
- A sphere cannot be flattened without stretching, so it has no net.
- The perspective view can make lengths and angles look different from their true values; the measurements in the information panel are the true geometric values.