AJMO Geometry and Area: Find the Relationship Before the Formula

By Ace Achievers · Published 16 October 2026 · Official information checked 14 October 2026

Why can a familiar area topic feel unfamiliar?

The official samples include tasks involving diagrams, wrapping and shaded regions. Their teaching value is the decision before the calculation: which areas are connected, what does a visible surface mean, and which equal lengths can be used? Our examples below are original teaching models, not reproductions of the official sample or the 2026 paper.

Do not read a missing length from the appearance of a drawing. Identify the stated equalities or shared dimensions. A diagram can organise information, but a neat-looking picture is not a measurement. In the first model, the critical condition is that both dividing lines continue across the whole rectangle.

How do three areas determine the fourth?

A second way is to compare columns. The right column is 20/12 = 5/3 times as wide as the left, so the lower-right part is 18 × 5/3 = 30. Both methods use the same shared-dimension structure. If the division lines did not align, this relationship would not follow from the three areas alone.

For a child ready for algebra, label the widths a and b and the heights h and k. The four areas are ah, bh, ak and bk. The products of opposite areas are equal: (ah)(bk) = (bh)(ak). That explains why the missing area is 20 × 18 / 12, rather than treating it as a memorised trick.

What happens when one area changes?

Keep the upper areas 12 and 20 but change the lower-left area to 24. Now the lower row is twice as tall, so the missing area is 40 and the total is 96. Ask the child to explain which dimension changed. Simply adding six to the earlier missing area would ignore the proportional relationship.

How do we distinguish paper area from visible surface?

Notice the units. Multiplying three lengths gives a volume, while each face area uses two lengths. The difference compares two areas, not an area with a volume. Check the answer against a physical picture: overlapping folds can hide some paper even though the box is fully covered.

How should this method be practised?

Use a short sequence: label shared dimensions, choose a representation, calculate, then change one condition. Include a problem where the convenient relationship does not apply, so the child must recognise its assumptions. Continue to harder diagrams when they can explain why a method is valid, not just name it.

For more systematic practice, inspect the geometry sections in the related year-level AMC curricula. Keep the task at a level where the child can reason independently after teaching; an award certificate alone does not select that level.