The square on the diagonal
It has double the area of the first square.
Use the diagonal
Doubling the side gave 4 times the area. That is too much. Baudhāyana tried something else: let the diagonal of the old square be the side of the new square.
The old square has area 9 sq. units.
What will the area of the new square on the diagonal be?
What this lesson covers
The idea
The square constructed on the diagonal of a square has double the area of the original square, because the original is made of two congruent triangles and the new square of four.
Use the diagonal
Doubling the side gave 4 times the area. That is too much. Baudhāyana tried something else: let the diagonal of the old square be the side of the new square.
The old square has area 9 sq. units.
What will the area of the new square on the diagonal be?
- 12 sq. units
- 18 sq. units
- 36 sq. units
Build it from triangles
The diagonal cuts the old square into 2 triangles. The new square stands on this diagonal. Tap the empty triangles to build it. How many triangles make the new square?
Two triangles, then four
The square constructed on the diagonal of a square has double the area of the original square. The original is made of 2 congruent triangles. The new square is made of 4 of the same triangles.
For side 3: the old square is 3 × 3 = 9 and the new square is 4 × 4.5 = 18. And 18 = 2 × 9.
Do it again on the new square. It has 4 triangles, the next one has 8. So the areas go 2, 4, 8 triangles, each double the one before.
Notes
The square on the diagonal of a square has double the area. Old square: 2 triangles. New square: 4 of the same triangles.
Check yourself
A square has area 18 sq. cm. A new square is built on its diagonal. What is the area of the new square in sq. cm?
Answer: 36
Double of 18 is 36 sq. cm. The old square is 2 triangles of 9 sq. cm each, the new one is 4 triangles of 9 sq. cm.
The square on the diagonal of a square has area 50 sq. cm. What is the area of the original square in sq. cm?
Answer: 25
Half of 50 is 25 sq. cm. Check: 2 × 25 = 50.
Why does the square on the diagonal have double the area?
A square has area 3 sq. units. Build a square on its diagonal. Then build another square on the diagonal of that new square. What is the area of the last square?
Answer: 12
3, then 6, then 12. Two doublings make 4 times: the same as doubling the side of the first square.
- Its side is twice the side of the old square. Doubling the side gives 4 times the area. The diagonal is longer than the side, but it is not twice as long.
- It is made of 4 small triangles, the old square of 2 of the same triangles — correct. Yes! 4 triangles against 2 triangles, all of the same size: double.
- A square has two diagonals. We use only one diagonal, as the side of the new square. The count of triangles tells the story: 4 against 2.