Squaring a rectangle
Baudhāyana's construction (800 BCE)
Same area, new shape
In ancient times, to square a shape meant to construct a square with the same area as that shape. The Indian mathematician Baudhāyana showed how to square a rectangle in his Śhulbasūtra, about 800 BCE. He used only a rope, pegs and arcs.
A rectangle is 8 units by 2 units, so its area is 16 square units. What side must a square of the same area have?
What this lesson covers
The idea
To square an a by b rectangle, that is, construct a square of equal area, build a right triangle with hypotenuse (a + b)/2 and one side (a – b)/2; its third side is the square's side.
Same area, new shape
In ancient times, to square a shape meant to construct a square with the same area as that shape. The Indian mathematician Baudhāyana showed how to square a rectangle in his Śhulbasūtra, about 800 BCE. He used only a rope, pegs and arcs.
A rectangle is 8 units by 2 units, so its area is 16 square units. What side must a square of the same area have?
- 4 units
- 5 units
- 10 units
Build the square
Follow the construction step by step. Tap Start, then Next step. Finish it for two rectangles.
A square from a triangle
To square a rectangle with sides a and b (a > b), build a right-angled triangle with hypotenuse (a + b) ÷ 2 and one side (a − b) ÷ 2. Its third side is the side of the square, and the square has the same area as the rectangle.
Why does it work? In the triangle HKP, HP² = HK² − BH², where HK = (a + b) ÷ 2 and BH = (a − b) ÷ 2. When you expand the two squares, the a² and b² parts cancel and what is left is 2ab ÷ 4 + 2ab ÷ 4 = ab. For 8 by 2: HK = 5 and BH = 3, so HP² = 25 − 9 = 16, which is 8 × 2.
Notes
To square a rectangle a by b (a > b), build a right-angled triangle with hypotenuse (a + b) ÷ 2 and one side (a − b) ÷ 2. Its third side is the side of a square with the same area.
Check yourself
A rectangle has AD = 8 and AB = 2. E is on AD with AE = AB, and F is the midpoint of ED. What is AF?
Answer: 5
AF = (8 + 2) ÷ 2 = 5. Check: ED = 6, so EF = 3, and AF = AE + EF = 2 + 3 = 5.
AH = AF = 5 and AB = 2. H is on AB produced. What is BH?
Answer: 3
BH = AH − AB = 5 − 2 = 3, which is (a − b) ÷ 2 = (8 − 2) ÷ 2.
In the right-angled triangle HKP, the hypotenuse HK is 5 and one side is 3. What is HP, the side of the square?
Answer: 4
HP² = 5² − 3² = 25 − 9 = 16, so HP = 4. The square has area 16, the same as 8 × 2.
Why does the square HPQS have the same area as the rectangle?
- Because HP is equal to AD. HP is shorter than AD. For 8 by 2, HP = 4 and AD = 8.
- Because HP is equal to (a + b) ÷ 2. That length is AF and HK. For 8 by 2 it is 5, while HP = 4.
- Because HP² works out to a × b, the area of the rectangle — correct. Yes. HP² = HK² − BH² works out to ab.