Chapter 11.1 — Introduction to Areas of Plane Figures

Five Formulas Already Familiar, Collected Together

Every single one of these five formulas was already used somewhere in earlier chapters for a different purpose — this chapter's real focus isn't re-deriving any of them, but instead proving relationships between figures that share a base or sit between the same parallel lines, using these familiar formulas as the starting toolkit rather than the main event. Notice, too, that four of these five formulas are secretly the same idea in disguise: a rectangle is base × height with two right angles built in, a square is that same rectangle formula with base and height forced equal, and a parallelogram's base × height works identically once those right angles are removed and the sides are allowed to slant instead — only the triangle and the rhombus bring in an extra factor of ½ each, for closely related reasons this chapter returns to directly and carefully later on.

A General Quadrilateral, Split by One Diagonal

For a quadrilateral with diagonal length d, and perpendiculars h₁ and h₂ dropped carefully from the two remaining vertices onto that diagonal, area = ½ × d × (h₁ + h₂) exactly. This is really and genuinely nothing more than the plain triangle formula applied twice over: the diagonal cleanly splits the quadrilateral into two separate triangles sharing the same base d, one with height h₁ and one with height h₂, and carefully adding ½dh₁ + ½dh₂ together gives exactly this combined formula — no genuinely new idea, just the triangle-area formula reapplied on both sides of a shared diagonal. This same diagonal-splitting move, applied specifically to a rhombus, is exactly where the rhombus's own ½ × d₁ × d₂ formula comes from too: a rhombus's two diagonals bisect each other at right angles, cutting it into four small right triangles whose combined area works out, after the algebra is carried through, to precisely half the product of the two full diagonals — a result this chapter's own exercises prove explicitly and carefully, rather than simply asserting it outright. It's worth sitting with why the diagonal formula above needed h₁ and h₂ as two separate perpendicular measurements, while the rhombus formula collapses down to just the two diagonals with no separate height terms at all: a rhombus's own diagonals already are the two perpendiculars needed, since they cross each other at exactly 90°, so there's no additional height measurement left to specify once the two diagonal lengths are already known — the general quadrilateral formula and the rhombus formula are really the same underlying idea, just applied to a figure special enough that one of its two required measurements collapses into the other automatically.

Building Up an Irregular Figure From Simple Pieces

An irregular shape — one with no name and no single formula of its very own — can still be measured accurately by breaking it into non-overlapping pieces that do each have known formulas: a triangle plus a rectangle plus a semicircle, for instance, all added carefully together. The total area of the whole irregular figure is simply the sum of the individual areas of whichever simple, cleanly non-overlapping pieces it happens to be genuinely built from, chosen carefully so that every part of the original figure is covered exactly once, with nothing ever counted twice and nothing left out. Splitting a single figure into pieces this way isn't limited to irregular shapes with no name of their own, either — a trapezium, for instance, can be split along either diagonal into a triangle and a smaller triangle, or split by dropping perpendiculars into a rectangle and two right triangles, and this exercise's own worked problems lean on exactly that kind of decomposition to reach an answer that no single named formula could produce directly on its own. Choosing where to make the split is itself a genuine skill worth practicing: the same irregular figure can usually be decomposed several different valid ways, and a split that lines up neatly with shapes whose formulas are already firmly known — rather than one that merely happens to look natural on the page at first glance — is what actually makes a problem solvable quickly rather than turning it into a much longer, messier calculation than necessary.

Where "Same Base, Same Parallels" Comes In

Every formula collected above computes one figure's area in isolation. Exercise 11.1 puts these formulas to direct, practical use on quadrilaterals split carefully into triangles, before Exercise 11.2 and Exercise 11.3 introduce this chapter's real centerpiece — theorems comparing two different figures' areas whenever they share a base and sit between the same pair of parallel lines, a genuinely different kind of question from anything asked so far. The phrase "between the same parallels" is worth pausing on before moving ahead, since it does real work in every one of the theorems that follow: two figures sit between the same pair of parallel lines when one entire figure fits inside the strip of space bounded by those two lines on either side, with its base running along one of the lines and its opposite vertex or vertices touching the other — not merely pointed somewhere in the same general direction, but genuinely and fully enclosed between those two specific lines, with nothing hanging outside either boundary at any point.

What makes "same base, same parallels" worth an entire pair of exercises, rather than a single quick observation, is how counterintuitive the resulting theorems can look at first glance: two parallelograms, or two separate triangles, can look genuinely dramatically different in overall shape — one short and wide, another tall and narrow — and still end up with exactly equal areas, purely because they share a base and both fit fully inside that same parallel strip. The shared base fixes one entire factor in the area formula outright and completely, and "between the same parallels" turns out to pin down the height too, in exactly the same fixed way, even though the two figures themselves can be slanted or stretched into genuinely very different overall silhouettes on the page. Height, in this context, always means the perpendicular distance from the base line to the opposite parallel line — not the length of any slanted side running along the figure itself — which is exactly why two visually different parallelograms sharing a base and sitting inside that same strip end up with genuinely identical area, even when neither one looks anything like a simple stretched copy of the other.