Chapter 9: Boxes and Sketches

Nets & 3D Shapes:
• (a) Net definition: A 2-dimensional flat pattern of polygons that can be cut and folded along edges to form a 3-dimensional solid shape.
• (b) 3D Shape formed: An open cuboid box (or cube if sides are equal).
• (c) Cuboid properties: A closed cuboid box has 6 faces, 12 edges, and 8 vertices.

Cube Nets Analysis:
• (a) Faces: A cube has exactly 6 square faces.
• (b) Single row test: No, 6 squares in a single straight line cannot fold into a cube. Folding a strip of 6 squares only creates a hollow cylinder with 4 sides, overlapping the remaining 2 squares, leaving the top and bottom open with no lids.

Opposite Face Rule (Sum = 7):
• (a) Opposite to 1: 7 - 1 = 6.
• (b) Opposite to 3: 7 - 3 = 4.
• (c) Opposite to 5: 7 - 5 = 2.

Architectural Drawings:
• (a) Floor Map: A 2D top-down plan showing the layout of rooms, doors, and walls as flat outlines.
• (b) Deep Drawing: A 3-dimensional perspective drawing that shows length, width, and height, making the house look realistic like a picture.
• (c) Reason: A floor map only views from directly above (bird's-eye view), so vertical height is invisible. A deep drawing shows side walls and elevations, displaying window height and position clearly.

Verification Checks:
1. Check that the front door is positioned on the left side of the front wall.
2. Check that the window appears on the right side of the front wall.
3. Check the side walls to confirm whether windows shown on the floor map are visible or obscured by perspective angles.

Views of a Bridge:
• (a) Top View: Shows two parallel straight tracks or road surface with crossing railway sleepers, appearing as a long rectangle.
• (b) Front View: Shows the cross-section of the roadbed supported by vertical pillars with empty space below.
• (c) Side View: Shows a row of vertical pillars supporting a horizontal deck across the span, resembling a colonnade.

2D Nets for Curved Shapes:
• (a) Cylinder curved surface: A Rectangle (when rolled, the length becomes the circumference of the circle).
• (b) Cone net: A Sector of a circle (pie-slice shape).

Cube Visibility:
In a realistic 3D perspective drawing of an opaque cube, you can see at most 3 faces at one time (the top face and two adjacent side faces). The other 3 faces remain hidden at the back and bottom.

Open Box Net:
• An open box without a lid has 5 square faces (1 bottom and 4 vertical sides).
• A common net shape is a cross shape: 1 central base square with 4 squares attached to its four edges.

Cube Calculations:
• (a) Total cubes: Volume = 3 × 3 × 3 = 27 unit cubes.
• (b) Cubes painted on 3 faces: Only the 8 corner cubes have three exposed faces painted red.

Triangular Prism Properties:
• Faces: 2 triangles + 3 rectangles = 5 faces.
• Edges: 3 on top triangle + 3 on bottom triangle + 3 connecting vertical edges = 9 edges.
• Vertices: 3 corners on each triangle = 6 vertices.

Matchbox Addition:
• Bridge span: 6 matchboxes.
• Supports: 2 piles of 3 matchboxes = 2 × 3 = 6 matchboxes.
• Total matchboxes: 6 + 6 = 12 matchboxes.

Isometric Dot Paper:
• Isometric dot paper has dots arranged in equilateral triangular grids.
• It allows artists to draw receding parallel lines at 30° angles to show true 3D depth and proportion without distortion, making edges of equal length look natural in 3D.

Square Pyramid:
• (a) Net description: A central square with 4 identical triangles attached to each of its four edges (resembling a 4-pointed star).
• (b) Edges: 4 edges on the square base + 4 slanting edges meeting at the top vertex = 8 edges.

Tumbler Views:
• (a) Top View: You see concentric circles (the circular rim and base).
• (b) Side View: You see a rectangle (or trapezoid if the glass is tapered).

Cube Paper Area:
• Area of one square face = 4 cm × 4 cm = 16 sq cm.
• A cube has 6 faces.
• Total paper area needed: 6 × 16 sq cm = 96 sq cm.

Internal Cube:
• Only the 1 central core cube (at row 2, column 2, layer 2) has no exposed faces to the outside.
• Therefore, exactly 1 cube is completely hidden.

Sphere Properties:
• Faces: 1 continuous curved surface (no flat faces).
• Edges: 0 edges (no straight or sharp boundaries).
• Vertices: 0 vertices (no corners).

Staircase Views:
• (a) Side View: A stepped zigzag outline (staircase profile showing 4 risers and 4 treads).
• (b) Top View: A single large rectangle divided into 4 equal parallel horizontal strips (the steps).

Folding Process:
1. The second square acts as the base.
2. The left and right squares fold upwards at 90° to form the left and right walls.
3. The top square folds down to form the front wall.
4. The two bottom squares fold up: the third square forms the back wall and the fourth square folds over the top to form the lid. This perfectly closes all 6 faces of the cube.

Differences:
1. Dimensions: 2D shapes have only 2 dimensions (length and width). 3D shapes have 3 dimensions (length, width, and height/depth).
2. Flatness: 2D shapes are completely flat and lie entirely within a plane. 3D shapes occupy physical space in the real world.
3. Measurement: 2D shapes are measured by Area (sq units). 3D shapes are measured by Volume (cubic units).

Euler's Formula Verification (F + V - E = 2):
• (a) Cube: F = 6, V = 8, E = 12.
Calculation: 6 + 8 - 12 = 14 - 12 = 2 (Verified!).
• (b) Triangular Prism: F = 5, V = 6, E = 9.
Calculation: 5 + 6 - 9 = 11 - 9 = 2 (Verified!).

Total Surface Area of Cuboid = 2(lb + bh + hl):
• lb = 30 × 18 = 540 sq cm.
• bh = 18 × 12 = 216 sq cm.
• hl = 12 × 30 = 360 sq cm.
• Sum = 540 + 216 + 360 = 1,116 sq cm.
• Total Surface Area = 2 × 1,116 = 2,232 sq cm.

Base Shapes:
• (a) Cone base: A flat Circle.
• (b) Cylinder flat faces: Exactly 2 flat circular faces (one top, one bottom).