Problem
Use the overlap as the unit to find a figure's area
Two congruent squares overlap. The whole figure's area is known and one square is a known multiple of the overlap. We need the overlap's area.
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 × ■
Everything is now measured in overlaps.
6.G.A.1Draw A Diagram2STEP 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 × ■
The figure is 9 overlaps.
6.G.A.1Identify SubproblemsThe whole figure measures nine overlaps.
Why?
The figure is the two squares put together with the shared piece counted once, so it is square plus square minus one overlap.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each square is five overlaps, so two squares are ten of them and taking one back leaves nine.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
Solve for one unit
Those 9 units make up 36/5 cm², so divide by 9.
36/5 ÷ 9 = 4/5
The overlap is 4/5 cm².
6.EE.B.7Work BackwardsAnswer
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