Problem
Equal after removing the shared part means equal to begin with
A rectangle and a quarter circle overlap, and the two parts sticking out are equal in area. We need one side of the rectangle.
GeometryExpressions
Your answer
How to solve
Strategy Change Focus / Count the Complement — The two shaded pieces are each a whole shape minus the overlap, and the overlap is the same for both. So if the pieces are equal, the whole shapes are equal too -- and the circle's area is known, which hands over the rectangle's missing side.
1STEP 1
Add the overlap back to both
Each shaded piece is its own shape with the overlap taken away. Putting the same overlap back on both keeps them equal.
The rectangle and the quarter circle have the same area.
6.G.A.1Change Focus Count The ComplementThe rectangle and the quarter circle have the same area.
Why?
Each shaded piece is its own shape with the shared overlap taken away, and the overlap taken away is the same region both times.
🧱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?
Putting the same amount back onto two equal amounts leaves them equal, so the two whole shapes match.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Find the quarter circle's area
Its radius is the 10 cm side, and a quarter circle is a fourth of the whole circle.
10 × 10 × 3.14 ÷ 4 = 78.5
The quarter circle is 78.5 cm².
7.G.B.4Draw A Diagram3STEP 3
Run the rectangle rule backwards
The rectangle has the same area, and one of its sides is 10 cm, so divide.
78.5 ÷ 10 = 7.85
AB is 7.85 cm.
6.G.A.1Work BackwardsAnswer
7.85 cm
Check the areas: 7.85 × 10 = 78.5 cm², the same as the quarter circle, so whatever the overlap is, the two leftovers match.
Takeaway
If two shapes lose the same overlap and what is left is equal, the whole shapes were equal.
- Add the overlap back to both
- Find the quarter circle's area
- Run the rectangle rule backwards