Patterns & Reasoning

Depth · one concept from first principles · step 2 of 4

Use the overlap as the unit to find a figure's area

The figure at the right is made by overlapping 22 congruent squares, and the whole figure covers 715 cm27\frac{1}{5}\ \text{cm}^2. If one square's area is 55 times the area of the overlap, what is the area of the overlap, in cm2\text{cm}^2?

GeometryExpressionsFractions
Your answer
How to solve
Strategy Identify Subproblems — Measure everything in one unit: the overlap. Each square is 5 of that unit, and the figure is the two squares with the shared unit counted once instead of twice, which turns the whole picture into a single count.
1STEP 1

Call the overlap one unit

Let the overlap's area be ■. Then each square, being 5 times it, is 5 × ■.

(square) = 5 × ■
2STEP 2

Write the whole figure in that unit

Adding both squares gives 10 units but counts the shared one twice, so take one unit back off.

5 × ■ + 5 × ■ - ■ = 9 × ■
3STEP 3

Solve for one unit

Those 9 units make up 36/5 cm², so divide by 9.

36/5 ÷ 9 = 4/5
Answer
45\frac{4}{5} cm²
Rebuild it: each square is 5 × 4/5 = 4 cm², and 4 + 4 - 4/5 = 36/5, the given total.
Takeaway

Measure a picture in whatever piece repeats -- even the overlap can be the unit.

  • Call the overlap one unit
  • Write the whole figure in that unit
  • Solve for one unit
Where next?
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