Area of Rectilinear Figures

Given a polygon’s vertices, 0606 expects the shoelace method (the “array” method), fast, general, and worth full marks when two rules are respected: vertices in order, and a modulus at the end.

The array layout

Find the area of triangle $A(1, 2)$, $B(5, 3)$, $C(4, 7)$. Write the coordinates as columns, repeating the first vertex at the end:

| $x$: | $1$ | $5$ | $4$ | $1$ | | $y$: | $2$ | $3$ | $7$ | $2$ |

Down-products ($x$ of one column $\times$ $y$ of the next): $1 \cdot 3 + 5 \cdot 7 + 4 \cdot 2 = 46$ Up-products ($y \times$ next $x$): $2 \cdot 5 + 3 \cdot 4 + 7 \cdot 1 = 29$ Area $= \tfrac{1}{2}|46 - 29| =$ $8.5$

The written array is the method mark, even with one arithmetic slip, the structure scores. It generalises unchanged to quadrilaterals and beyond: more columns, same weave.

The two non-negotiables

Order the vertices around the shape (either direction, but consistently). A crossed ordering. $ABDC$ instead of $ABCD$, computes the area of a bow-tie, not your quadrilateral, and the error is silent. If the question doesn’t hand you an order, plot the points roughly first; thirty seconds of sketch prevents the most expensive mistake in the subtopic.

Take the modulus and halve. The raw difference is negative when you happened to go clockwise, area is $\tfrac{1}{2}|\text{difference}|$, always positive.

Where the vertices come from

Area questions usually sit at the end of a multi-part question: the vertices are intersection points found earlier, a foot of a perpendicular, or points constructed from parallel conditions. Exact coordinates (fractions, surds) go into the array exactly, premature rounding here corrupts the final answer beyond mark-scheme tolerance.

Alternative decompositions (rectangle minus triangles, $\tfrac{1}{2} \times \text{base} \times \text{height}$ with a convenient horizontal side) earn the same marks when valid, but the shoelace is the method that never needs an idea, which is what you want under time pressure.

Common mistakes

• Vertices unordered (the bow-tie error)
• First vertex not repeated, so one product pair goes missing
• Modulus skipped, negative “area” presented
• The $\tfrac{1}{2}$ forgotten
• Rounded coordinates fed in when exact ones existed

Full topic context: Straight-Line Graphs notes.