The Baudhāyana-Pythagoras theorem
a² + b² = c²: one rule for every right triangle.
One rule for every right triangle
Baudhāyana's way of combining squares works for any right triangle.
A right triangle has perpendicular sides a and b, and a hypotenuse c. The squares on the three sides have areas a × a, b × b and c × c.
Which rule do you think is true for every right triangle?
What this lesson covers
The idea
If a right-angled triangle has sidelengths a, b and c, where c is the hypotenuse, then a² + b² = c²; so an unknown side can be found from the other two.
One rule for every right triangle
Baudhāyana's way of combining squares works for any right triangle.
A right triangle has perpendicular sides a and b, and a hypotenuse c. The squares on the three sides have areas a × a, b × b and c × c.
Which rule do you think is true for every right triangle?
- a + b = c
- a × a + b × b = c × c
- a × b = c × c
A balance for squares
Each triangle has one missing side. The left pan holds a × a + b × b. The right pan holds c × c. Change the guess until the balance is level.
The theorem
Baudhāyana-Pythagoras theorem. If a right triangle has perpendicular sides a and b and hypotenuse c, then a² + b² = c².
Find c. Sides 3 and 4: 3² + 4² = c² 9 + 16 = c² 25 = c², so c = 5.
Find a leg. Hypotenuse 13, one leg 5: 5² + b² = 13² 25 + b² = 169 b² = 144, so b = 12.
Notes
Baudhāyana-Pythagoras theorem. In a right triangle with perpendicular sides a, b and hypotenuse c: a² + b² = c². Know two sides? Put them in, then find the third.
Check yourself
A right triangle has perpendicular sides 7 and 24. How long is the hypotenuse?
Answer: 25
7² + 24² = 49 + 576 = 625. And 25 × 25 = 625, so c = 25.
A right triangle has hypotenuse c = 10 and one perpendicular side a = 6. Which equation finds the other side b?
A right triangle has hypotenuse 20 and one perpendicular side 12. How long is the other perpendicular side?
Answer: 16
144 + b² = 400, so b² = 256 and b = 16, because 16 × 16 = 256.
In Bhāskarāchārya's Līlāvatī, a lotus stem sticks 1 unit out of the water. A breeze bends it until the tip touches the water 3 units away. If the lake is x deep, the stem is x + 1 long, and the right triangle gives 3² + x² = (x + 1)², which simplifies to 9 = 2x + 1. How deep is the lake?
Answer: 4
9 = 2x + 1 gives 2x = 8, so x = 4. Check: 3² + 4² = 25 = 5², and the stem is 4 + 1 = 5.
- 6² + 10² = b². The hypotenuse 10 is on the wrong side. The hypotenuse stands alone: a² + b² = c².
- 6² + b² = 10² — correct. Yes! a² + b² = c² with a = 6 and c = 10. Then b² = 100 − 36 = 64.
- b² + 10² = 6². The hypotenuse is the longest side, 10, not 6. It must stand alone on one side of the equation.