Area of the square?

A hard-level "Area" problem.Chasing the side length is a dead end at elementary level. Switching to the diagonals is the key move.

GivenThe radius of the circle is 6 cm

Hints

1. A square's area is "side × side", but this side length cannot be found with elementary arithmetic. So we switch our attention to the diagonals. Draw both of them.
2. Look closely: both diagonals pass exactly through the centre of the circle.
3. A line through the centre reaching from edge to edge is the diameter. The radius is 6 cm, so the diameter is 6 × 2 = 12 cm. Both diagonals are 12 cm.
4. What is more, the two diagonals cross at right angles. That is always true for a square.
5. Cutting along the diagonals gives four identical right triangles.
6. Now fold those four outwards. They fit perfectly into the gaps.
7. The result is a square with side 12 cm. So the red square is exactly half of this bigger square.
8. In short, "diagonal × diagonal ÷ 2". 12 × 12 ÷ 2 is the answer.

Answer 72cm²

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