There is an ancient poem in Tamil which tells you how to calculate Hypotenuse for a triangle. The time period in which the poem was written is unknown, but it surely provides an (approximate) alternative to Pythogoras.

Here the poem is (in Unicode Tamil):

ஓடிய நீளந்தன்னை ஓரெட்டு கூறதாக்கி கூற்றிலே ஒன்று தள்ளி
குன்றத்தில் பாதி சேர்த்தால் வருவது கர்ணம் தானே.

Poem in Romanized Tamil:

Odiya n-ILan-thannai Orettu kURathAkki kURRilE onRu thaLLi
kunRaththil pAthi sErththAl varuvathu karNam thAnE.

Divide the distance you've run (longer side) into 8 equal parts and discard one part from it. Then if you add the half the height to that, you'll get Hypotenuse.

example 1 Take a Right angled triangle with sides 3 and 4 units

Now as per the poem, Hypotenuse = (7/8)*4 + 3/2 = 5.

example 2, take a Right angled triangle with sides 3 and 6

Hypotenuse = (7/8)*6 + 3/2 = 6.75

By the actual pythagorian theorem, Hypotenuse = SQRT(6^2 + 3^2) = 6.71