Why shapes behave as they do

In Investigate you discovered patterns by trying examples: diagonals grow 2, 3, 4, 5 …; triangle angles always make 180°; F + V − E came out as 2 every time. But a few examples, however many, never prove that something is always true. The 1,000th case might break the rule.

In this layer you will find out why the patterns must hold, using arguments that work for every case at once: counting in two ways, cutting shapes into triangles, walking around a polygon, and squashing a solid flat. You will also meet the edge cases where the rules bend or break, and the long history, much of it Indian, of how people came to understand shape.

Claim: a polygon with n sides has n(n − 3) ÷ 2 diagonals.

Proof by counting ends. Each of the n vertices can be joined to the other n − 1 vertices. Two of those joins are sides (to its two neighbours), so each vertex is the end of n − 3 diagonals. Counting from every vertex gives n(n − 3) diagonal-ends. Every diagonal has exactly two ends, so it has been counted twice. Therefore the number of diagonals is n(n − 3) ÷ 2. ∎

A second proof, by handshakes. Joining every pair of vertices is like everyone at a party shaking hands with everyone else once: n people make n(n − 1) ÷ 2 handshakes. Among all these segments, exactly n are sides. The rest are diagonals: n(n − 1) ÷ 2 − n, which simplifies to n(n − 3) ÷ 2. Two different arguments reaching the same formula is strong confirmation.

Exterior angle property. Extend one side of a triangle past a vertex. The angle between the extension and the next side is an exterior angle. It equals the sum of the two interior angles opposite it. Why? The exterior angle and the interior angle next to it together make a straight line (180°). The three interior angles also make 180°. Take away the shared interior angle from both, and what is left must match: exterior angle = the other two interior angles.

Angle sum of any polygon. From one vertex of an n-sided (convex) polygon, draw all n − 3 diagonals. They cut it into n − 2 triangles, whose angles together make up exactly the polygon's angles. So the angle sum is (n − 2) × 180°.

Exterior angles always total 360°. Walk round a convex polygon. At each vertex you turn through the exterior angle. By the time you are back where you started, facing the same way, you have turned through one full turn: 360°, whatever the number of sides.

Claim: in any triangle, each side is shorter than the other two added together.

Why: the side BC is a straight path from B to C. Going from B to A and then on to C is a detour. The straight path between two points is the shortest of all paths, so BC is shorter than BA + AC. The same holds for every side. ∎

This gives a precise rule for the third side: if two sides are a and b (with a ≥ b), the third side must be longer than a − b and shorter than a + b. Too short and the two short sides cannot reach; too long and the other two cannot stretch across it.

Why do mathematicians insist that a square is a rectangle? Because of inheritance. Anything proved for all rectangles, such as the diagonals are equal, then automatically holds for squares, with no extra work. If squares were excluded from rectangles, every theorem would need a separate square version.

Here are three short arguments that show how the family's properties follow from the definitions:

1. Parallelogram: neighbouring angles add to 180°. Two neighbouring angles sit between a pair of parallel sides, cut by the side joining them. They are co-interior angles, which add to 180°. It follows that opposite angles are equal (each is 180° minus the same neighbour).
2. Rectangle: one right angle forces four. A rectangle is a parallelogram with one right angle. Its neighbours are 180° − 90° = 90°, and so on round the shape.
3. Rhombus: diagonals are perpendicular. In a rhombus ABCD, B and D are each the same distance from A and from C (all sides equal). Points equally far from A and C lie on the perpendicular bisector of AC. So BD is the perpendicular bisector of AC: the diagonals cross at right angles.

A wheel works because the axle is at the centre, always exactly one radius above the road. Two other facts make circles special:

Constant width. Measure a circle's width in any direction and you get the diameter. That is why a round manhole cover cannot fall into its hole whichever way you turn it: a square cover can, if you tip it so its side slips down the longer diagonal of the hole.

Finding the centre from a chord. The perpendicular bisector of any chord passes through the centre. Archaeologists use this to find the size of a whole plate or pot from one broken piece: draw two chords on the rim fragment, construct their perpendicular bisectors, and where they cross is the centre.

For every prism and pyramid. We can check Euler's formula for whole families at once using algebra.

• A prism with an n-sided base has F = n + 2, V = 2n, E = 3n. Then F + V − E = (n + 2) + 2n − 3n = 2, whatever n is.
• A pyramid with an n-sided base has F = n + 1, V = n + 1, E = 2n. Then F + V − E = (n + 1) + (n + 1) − 2n = 2.

That proves the formula for infinitely many solids in two lines. But there are polyhedra that are neither prisms nor pyramids. For those, we need a cleverer argument.

Why do some six-square shapes fold into a cube and others not? Three reasoned rules settle almost every case:

1. No 2 by 2 block. Four squares round one point would all need to meet at one vertex of the cube, but only three faces meet at each vertex of a cube. So the fourth overlaps.
2. No row of 5. A row of squares wraps round the cube's middle, and the ring round a cube has only 4 faces, so the 5th lands on the 1st.
3. Opposite faces. In a working net, two squares separated by exactly one square in a straight row (or at the ends of a line of three) become opposite faces. Each face must have exactly one opposite.

The net also tells you the surface area of a solid: it is simply the area of the net. A cube of side 5 cm has a net of 6 squares, each 25 cm², so its surface area is 6 × 25 = 150 cm².

Why a regular n-gon has exactly n lines of symmetry. Each mirror line must pass through the centre. If n is odd, each line runs from a vertex to the midpoint of the opposite side: one line per vertex, n lines. If n is even, the lines come in two kinds: n ÷ 2 through pairs of opposite vertices and n ÷ 2 through midpoints of opposite sides. Again n in total.

Rotational symmetry. A shape has rotational symmetry if it fits onto itself after a turn of less than 360° about its centre. The number of positions in one full turn where it looks the same is its order. A regular n-gon has order n (turns of 360° ÷ n). A parallelogram has order 2 but no mirror lines; the letter Z has order 2 but no mirror lines.

Mirrors make turns. Reflect a shape in one mirror line and then in a second line that crosses the first at angle θ: the result is a rotation through 2θ about the crossing point. That is why a shape with two mirror lines at right angles (like a rectangle, θ = 90°) always has half-turn (180°) symmetry too.