Methods and worked examples: Exploring Some Geometric Themes

2.1 Visualising a solid

A solid is seen through a profile: the outline it makes from a chosen viewpoint. Turn the object and the outline can change completely. A cylinder seen from the side looks like a rectangle, but from above it looks like a circle. The same solid can produce different profiles from different directions.

Three terms describe plane-faced solids. Faces are the flat surfaces that form the boundary, edges are the line segments where faces meet, and vertices are the points where edges meet. Every cube is bounded by six square faces, twelve edges and eight corners.

A prism has two congruent polygon faces at opposite ends and parallelogram side faces joining corresponding vertices. A pyramid has one polygon base and a single apex point, joined to every base vertex. A triangular pyramid is called a tetrahedron.

Worked example 1. A prism has a 10-sided polygon as its congruent faces. Then F = 10 + 2 = 12, E = 3 × 10 = 30 and V = 2 × 10 = 20. A pyramid with a 10-sided base has F = 11, E = 20 and V = 11. The same formulas with n = 4 give a cuboid-shaped prism with 6 faces, 12 edges and 8 vertices, and a square pyramid with 5 faces, 8 edges and 5 vertices.

2.2 Nets

A net is a flat shape that folds into a solid. A cube has 11 different nets when rotations and flips count as the same net. A regular tetrahedron has 2 nets. A cylinder unfolds into two circles and a rectangle whose width equals the circumference of the base and whose height equals the cylinder's height. A cone unfolds into a circle and a sector of a circle.

Not every arrangement of squares is a cube net. The folding test is the reliable method: choose one square as the base, fold the neighbours up, and continue around the edges. If two faces overlap or one face has no partner, the arrangement is not a net. The 11 valid arrangements are classified by their row patterns: 1-4-1 (six nets), 2-3-1 (three nets), 2-2-2 (one net) and 3-3 (one net).

2.3 Shortest path on a cuboid

A path on the surface of a cuboid becomes a straight line when the cuboid is unfolded into a net. The shortest surface path is therefore the straight line between the two points in the correct net, and its length is found with the Baudhayana theorem. Different unfoldings can place the points differently, so all sensible unfoldings must be checked.

Worked example 2. Two points on a cuboid unfold to a right triangle with legs 24 cm and 32 cm. The distance is d = √(24² + 32²) = √(576 + 1024) = √1600 = 40 cm. If a different unfolding gives a path that leaves the net, that unfolding does not correspond to a surface path, and it is rejected. The method is always: unfold, join the points by a straight line, check that the line stays inside the net, then apply the theorem.

2.4 Projections

A projection sends every point of an object to a plane along a line perpendicular to the plane. The projection of a line segment of length l onto a plane is p = l cos θ, where θ is the angle between the line and the plane. So p ≤ l. The projection equals the actual length when the line is parallel to the plane, and shrinks to a point when the line is perpendicular to it.

For the three mutually perpendicular planes used in drawing, the projections satisfy p(front)² + p(top)² + p(side)² = l². This is the three-dimensional statement that the squares of the projections add to the square of the true length, just as the Baudhayana theorem does in two dimensions.

The front view is the projection on the vertical plane, the top view is the projection on the horizontal plane, and the side view is the projection on the side plane. A cube in the usual orientation gives a square in each view. A cuboid gives three rectangles whose dimensions come from its three edge lengths. A cylinder gives a rectangle from the front and a circle from the top. A cone gives a triangle from the front and a circle from the top.

Projection loses information: different objects can have the same projection. A sphere and a cylinder seen along their axes both project to a circle. That is why engineering drawings give three views instead of one.

An isometric projection orients the solid so that the three axes make equal angles with the paper, and the projected lengths along all three axes are equal. This is why isometric grids use three families of parallel lines at 120°. A cube drawn isometrically looks like a regular hexagon with three visible faces.