3D Shapes & Nets

<p>Three-dimensional shapes have length, width, and height. In the 11+ exam, you may need to name 3D shapes, count their faces, edges, and vertices, or identify which net folds into a given shape.</p>

<h2>Common 3D Shapes</h2>

<ul>

<li><strong>Cube:</strong> 6 square faces, 12 edges, 8 vertices</li>

<li><strong>Cuboid:</strong> 6 rectangular faces, 12 edges, 8 vertices</li>

<li><strong>Cylinder:</strong> 2 circular faces and 1 curved surface, 2 edges, 0 vertices</li>

<li><strong>Cone:</strong> 1 circular base and 1 curved surface, 1 edge, 1 vertex (the apex)</li>

<li><strong>Sphere:</strong> 1 curved surface, 0 edges, 0 vertices</li>

<li><strong>Triangular prism:</strong> 2 triangular faces + 3 rectangular faces = 5 faces, 9 edges, 6 vertices</li>

<li><strong>Square-based pyramid:</strong> 1 square base + 4 triangular faces = 5 faces, 8 edges, 5 vertices</li>

</ul>

<h2>Euler's Formula</h2>

<p>For any polyhedron (a 3D shape with flat faces): <strong>Faces + Vertices − Edges = 2</strong>. This is a great way to check your counting. A cube: 6 + 8 − 12 = 2. ✓</p>

<h2>What Is a Net?</h2>

<p>A net is a flat pattern that folds up to make a 3D shape. Think of cutting along some edges of a cardboard box and laying it flat — that flat shape is the net of the box.</p>

<h2>Nets of Common Shapes</h2>

<p><strong>Cube:</strong> A cube has 11 different possible nets. They all consist of 6 connected squares. A common one looks like a cross (one row of 4 squares with one extra square on each side of the second square).</p>

<p><strong>Cuboid:</strong> 3 pairs of identical rectangles arranged so the shape folds into a box.</p>

<p><strong>Triangular prism:</strong> 2 triangles and 3 rectangles. The triangles go on opposite ends.</p>

<p><strong>Square-based pyramid:</strong> 1 square with 4 triangles attached to its sides.</p>

<p><strong>Cylinder:</strong> 2 circles and 1 rectangle. The rectangle wraps around to form the curved surface, so its width equals the circumference of the circle.</p>

<h2>Deciding If a Net Will Work</h2>

<p><strong>Worked Example:</strong> You are shown a cross-shaped arrangement of 6 squares and asked if it is a valid net of a cube.</p>

<p>Step 1: Count the squares — there must be exactly 6.<br>

Step 2: Check that no two squares would overlap when folded.<br>

Step 3: Check that every face is covered — no gaps.<br>

Step 4: Mentally fold (or use the trick of marking opposite faces: in a cross net, the top and bottom squares become opposite faces).</p>

<h2>Identifying Which Face Is Opposite Which</h2>

<p>In a cube net, two faces are opposite if there is exactly one face between them in a straight line. Mark the pairs and you can answer questions like "which letter is on the face opposite to B?"</p>

<h2>Views of 3D Shapes</h2>

<p>You may be asked to draw or identify the <strong>front view</strong>, <strong>side view</strong>, or <strong>plan view</strong> (bird's-eye view from above) of a 3D shape made from cubes. Count carefully how many cubes are visible from each direction.</p>

<h2>Practice Tips</h2>

<ul>

<li>Use Euler's formula to double-check your face/edge/vertex counts</li>

<li>Print nets, cut them out, and fold them — this builds your spatial skills faster than just imagining</li>

<li>For cube net questions, mark pairs of opposite faces and check no squares overlap</li>

<li>When viewing 3D shapes from different angles, use real blocks or cubes if you can</li>

</ul>