You have a square drawn on a sheet. Someone challenges you to draw another square that has exactly twice the area, but with one condition: you cannot measure lengths or use formulas. How can you build it using only the square itself as a guide?
Home > Riddles > The square doble
The square doble
From Plato's Meno, this small construction shows a very clean geometric idea: doubling an area does not consist of doubling a side.
Hints
Show hints
- Don't look for a “slightly longer” length: the key is already inside the square.
- Look at the segment that joins two opposite vertices.
- That segment can become the side of the new square.
Solution
Show full solution
Answer: Use the diagonal of the original square as the side of the new square. Draw a diagonal in the initial square, joining two opposite vertices. Then build a new square taking that diagonal as one of its sides. Why does it work? The diagonal divides the original square into two equal right triangles. The square built on that diagonal can be divided into four triangles equal to those. The original square contains two such triangles; the new one contains four. Therefore, the new square has exactly twice the area.
Related riddles
Keep practicing
If you enjoyed this one, try more pure-logic riddles, explore this theme, browse the full archive, or read the riddle-solving guide.