What area of the table is covered by paper?
A hard-level "Area" problem.Overlap problems are really about how many times each region got counted. Subtract once for the doubles and twice for the triples, and everything lands on exactly one.
GivenThree identical sheets of paper (20 cm² each) overlap as shown. The regions covered by exactly two sheets total 6 cm²; the region covered by all three is 2 cm².
Hints
- Simply adding the three sheets gives 20 × 3 = 60 cm². But that is not the covered area — the overlaps have been counted more than once.
- The parts under exactly two sheets (6 cm² in all) were counted twice in the 60. They should count once, so subtract them once.
- The middle, under all three sheets (2 cm²), was counted three times. It should count once, so subtract it twice.
- Altogether 60 − 6 − 2 × 2 = 50 cm². Subtract just enough to bring every part back down to being counted once.
- With two sheets it is just "total − overlap". With three you must be careful not to over-subtract — that is why the middle is subtracted twice, not three times.
Answer 50cm²
More "Area" problems
- Area of the inner triangle?
- Area of triangle DOA?
- Area of quadrilateral ABCD?
- Area of the overlap?
- Total area of the two crescents?
- Area swept by the square?