MathBrush
←  All problems

P5–P6 Overlap and cancel

Area of triangle

Two identical right-angled triangles overlap. Their perpendicular sides are 12 cm and 18 cm; the shaded parts total 132 cm². Find the common unshaded area.

▶  Watch it drawn
Show the answerHide the answer

42 cm². Each triangle has area 1/2 × 12 × 18 = 108 cm². Since the identical triangles contain the same unsha…

Sent in by a reader. Send me the one you are stuck on →

The pattern behind it: Overlap and cancel

Two shapes share a region, and the shared part cancels out.

In this one. Equal whole triangles with the same overlap must leave equal shaded remainders, so split the total shaded area equally.

Counted twiceA + Bgroup Athe two in bothgroup Bthe same two againCounted oncetake the shared part offin all5 + 4 − 2 = 7

How to spot it

Shapes are joined, stacked, or have a piece cut off, and the question compares a perimeter or length before and after, rather than asking for an area from scratch.

Common mistake

Children assume perimeter changes the same way area does. Joining shapes or cutting off a corner can leave the perimeter unchanged, or change it by a different amount from the area.

Try one like it

Challenging Garden PQRS — the path and the shaded triangles A rectangular garden PQRS is divided into two rectangular beds by a straight path UV, where U is on PQ and V is on SR (so PUVS and…
All 4 with this pattern →

The 33 problem patterns

All 33 patterns, one to a page — the 19 behind Primary 5 and 6 word problems, and the skills the younger years are built on. The bar model, how to spot it in the question, and the mistake children make. A booklet you can print. Free.

A PDF, and a note when new solutions go up.