10.3 Three-Dimensional Objects & Nets

From Paper A onwards, ICAS asks you to describe and compare 3D objects; by the middle papers you are counting faces, edges and vertices, and by Papers D–F you are matching nets to solids and reading views. These are visual questions, but they reward systematic counting and a few memorised facts far more than artistic talent.

The Solid Family

Prisms have two identical, parallel ends (the bases) joined by rectangular faces. The name comes from the base shape: a triangular prism has triangle ends, a rectangular prism (cuboid) has rectangle ends, and a cube is the special case where every face is a square.

Pyramids have one base and triangular faces that meet at a single point (the apex). A square-based pyramid has a square base and four triangular faces.

Curved solids:

• Cylinder — like a prism with circular ends (a can)
• Cone — like a pyramid with a circular base (an ice-cream cone)
• Sphere — a perfectly round ball; it has no faces, edges or vertices in the flat sense

Faces, Edges, Vertices and Euler's Rule

• A face is a flat surface.
• An edge is where two faces meet.
• A vertex is a corner where edges meet (plural: vertices).

Counting method: count the bases first, then the side faces. A triangular prism has 2 triangle faces + 3 rectangle faces = 5 faces. For edges: 3 around each triangular end plus 3 joining the ends = 9. Counting in groups stops you missing or double-counting.

Every convex solid obeys Euler's rule:

F + V − E = 2

Check the cube: 6 + 8 − 12 = 2. Use it two ways in the exam — to find a missing value (a solid with 6 faces and 8 vertices must have 6 + 8 − 2 = 12 edges) and to check your counting (if your numbers do not give 2, recount).

Nets: Folding Flat Shapes into Solids

A net is a flat arrangement of polygons that folds up into a solid. ICAS shows a net and asks which solid it makes — or shows four nets and asks which one fails.

Key nets to recognise:

• Cube: six squares. There are exactly 11 valid nets. A reliable test: four squares in a row form the ring of side faces; the other two squares must be attached to that ring on opposite roles (one to become the top, one the bottom). If two squares would fold onto the same face, the net fails — a 2 × 3 rectangle of squares, for example, cannot fold into a cube.
• Rectangular prism: six rectangles in three matching pairs.
• Cylinder: two circles plus one rectangle. The rectangle's length must equal the circle's circumference.
• Cone: one circle plus one sector (a 'pizza slice').
• Square-based pyramid: one square with four triangles attached to its sides.
• Triangular prism: two triangles plus three rectangles.

Strategy for 'which net' questions: count faces first (a net with 5 faces cannot make a cube), then check shapes (a cone needs a curved sector, not triangles). Elimination usually removes three options quickly.

Cross-Sections

A cross-section is the shape revealed when you slice straight through a solid.

• Slicing a prism parallel to its base always gives the base shape — a cylinder cut parallel to its base shows a circle, no matter where you cut.
• Slicing a cylinder vertically through its centre gives a rectangle.
• Slicing a cone parallel to its base gives a smaller circle; slicing it vertically through the apex gives a triangle.
• Slicing a sphere anywhere gives a circle — the largest when the cut passes through the centre.

Views: Front, Side and Top

ICAS shows a 3D object (often cubes stacked together, drawn in isometric style on triangular dot paper) and asks for the front view, side view or top (plan) view — flat 2D outlines of what you would see from that direction.

Worked example. Three cubes sit in an L-shape on a table: two cubes side by side, with a third cube stacked on top of the left one.

• Front view (looking at the long side): an L-shape — two squares along the bottom, one square above the left square. Heights column by column: 2 then 1.
• Side view (looking from the left end): a single column 2 squares tall, because the back cube hides directly behind the front one.
• Top view: two squares side by side — the stacked cube does not add a new footprint.

Method for stacked-cube views: for each column in the viewing direction, record the maximum height you would see. The view is just those column heights drawn as squares.

Isometric drawings show three faces of each cube (top, left, right) using slanted edges at 30°. When ICAS gives an isometric drawing and asks which flat view matches, pick one direction at a time and rebuild the view column by column. The trap is counting cubes you cannot see — always check whether a hidden cube must exist to support a floating one.