√Invention · free download
Square root of any non-perfect square — the shortcut
Invented by Mr. Aniket Bhardwaj on 2 October 2024 and published on 2 February 2025: a one-line estimate of √F from the two perfect squares around F, accurate enough for every aptitude paper and faster than long division.
Invented 02-10-2024Published 02-02-2025Excel · PDF · PPT
Formula and calculation used
| P | previous perfect square below F |
| F | the number whose square root you want |
| N | next perfect square above F |
| G1 | F − P (gap 1) |
| G2 | N − P (gap 2) |
Formula to be used
√F ≈ √P + G1 / G2
What you require to use this formula
Only the two perfect squares that F lies between — for √3: 1 and 4; for √48: 36 and 49; for √78: 64 and 81; for √85: 81 and 100.
Excel calculation and graphs — worked examples
- √3: P = 1, N = 4 → √3 ≈ 1 + 2/3 = 1.667 (actual 1.732)
- √48: P = 36, N = 49 → √48 ≈ 6 + 12/13 = 6.923 (actual 6.928)
- √78: P = 64, N = 81 → √78 ≈ 8 + 14/17 = 8.824 (actual 8.832)
- √85: P = 81, N = 100 → √85 ≈ 9 + 4/19 = 9.211 (actual 9.220)
All you need to know is which two perfect squares F lies between. The error shrinks as F grows — it is a linear interpolation between √P and √N — so it is at its weakest for F under 10 and within 0.01 for most aptitude-paper numbers.