How to Calculate Square Root Quickly
Use the Babylonian method to approximate any non-negative square root. Each round averages an estimate with the number divided by that estimate, and the final answer is checked by squaring it.
Written and checked by TheToolNet Editorial Team · Last reviewed: September 19, 2026
Quick square-root steps
Babylonian square-root algorithm
For a number N and current estimate x, the next estimate is (x + N/x) ÷ 2. If x is too high, N/x is too low; their average lands closer to the root. Repeating the same short calculation quickly stabilizes.
For √2 starting at 1: the first estimate is (1 + 2/1)/2 = 1.5. The next is about 1.4167, then 1.4142. Squaring 1.4142 returns approximately 2.
Estimate with nearby perfect squares
Before calculating, bracket the answer. Because 7² = 49 and 8² = 64, √50 is a little more than 7. This catches decimal-placement mistakes and gives the iteration a strong starting point.
| Perfect square | Root | Perfect square | Root |
|---|---|---|---|
| 25 | 5 | 36 | 6 |
| 49 | 7 | 64 | 8 |
| 81 | 9 | 100 | 10 |
| 121 | 11 | 144 | 12 |
Frequently asked questions
Is this the same as Newton's method?
Yes. Applied to x² − N = 0, Newton's method simplifies to the Babylonian averaging formula.
How many iterations are needed?
A reasonable starting estimate often gives several accurate decimal places in three or four rounds. The tool displays six for a clear convergence trail.