How to Calculate Square Root Manually

What You'll Need

• Paper and pencil
• Eraser (optional but helpful)
• Basic multiplication skills

Step-by-Step Guide to Calculate Square Root Manually

This method, known as the long division method, works for both perfect squares (like 16 or 25) and non-perfect squares (like 2 or 7). For a refresher on square root basics, see our What Is a Square Root? page.

1. Pair the digits from right to left. For a whole number, group the digits in pairs starting from the decimal point (or the rightmost digit). For example, 2025 becomes (20)(25). For 2, write it as 2.00 00 00... (add decimal zeros as needed).
2. Find the largest number whose square is ≤ the first pair. Write that number above the first pair. This is the first digit of the answer. For 2025, the first pair is 20. The largest integer with square ≤ 20 is 4 (since 4²=16). Write 4 above the 20.
3. Subtract and bring down the next pair. Subtract the square (16 from 20) to get 4. Bring down the next pair (25) to get 425.
4. Double the current result (ignoring any decimal) and write it as a divisor with a blank digit. For 2025, current result is 4, double is 8. Write 8_ (with a blank space) as the divisor.
5. Find the next digit. Pick the largest digit (0-9) such that when placed in the blank, the product of that digit and the new divisor is ≤ the current remainder. For 425, try 5: 85 × 5 = 425 exactly. Write 5 above the next pair. Now the result is 45.
6. Repeat for more digits (if needed). For perfect squares, you may stop. For non-perfect squares, continue by bringing down pairs of zeros and repeating steps 4-5.
7. Place the decimal point. The decimal point in the result goes directly above the decimal point in the original number (if any). For whole numbers, it's after the last digit.

Worked Example 1: Perfect Square (2025)

Let’s find √2025 manually.

Step 1: Pair digits: (20)(25).
Step 2: Largest number with square ≤ 20 is 4. Write 4 above. Subtract 16 from 20 → remainder 4.
Step 3: Bring down next pair: 425.
Step 4: Double current result (4) → 8. Write divisor as 8_.
Step 5: Find digit: 85 × 5 = 425. So digit is 5. Write 5 above. Result so far: 45.
Step 6: Remainder 0. Since 2025 is a perfect square, we stop. √2025 = 45. Check: 45² = 2025.

Worked Example 2: Non-Perfect Square (2)

Now let’s calculate √2 to three decimal places.

Step 1: Write 2 as 2.00 00 00. Pairs: (2)(00)(00)(00).
Step 2: Largest number with square ≤ 2 is 1 (1²=1). Write 1 above. Subtract 1 → remainder 1.
Step 3: Bring down first pair of zeros: 100.
Step 4: Double current result (1) → 2. Divisor: 2_.
Step 5: Find digit: 24 × 4 = 96 ≤ 100. So digit is 4. Write 4 above. Remainder: 100 - 96 = 4.
Step 6: Bring down next pair: 400. Double current result (14) → 28. Divisor: 28_.
Step 7: Find digit: 281 × 1 = 281 ≤ 400. Digit is 1. Write 1 above. Remainder: 119.
Step 8: Bring down next pair: 11900. Double result (141) → 282. Divisor: 282_.
Step 9: Find digit: 2824 × 4 = 11296 ≤ 11900. Digit is 4. Write 4 above. Remainder: 604.
Thus √2 ≈ 1.414 (three decimal places). For more on interpreting such values, see our Understanding Square Root Values page.

Common Pitfalls to Avoid

• Incorrect pairing: Always pair from the decimal point (or rightmost digit for whole numbers). For 2025, correct pairing is (20)(25), not (2)(02)(5).
• Forgetting to double the result before finding the next digit.
• Misplacing the decimal point: The decimal in the answer goes directly above the decimal in the original number. For whole numbers, it's after the last digit.
• Errors in multiplication: Double-check your products, especially when the divisor has multiple digits.
• Thinking only perfect squares have rational square roots: Many numbers, like 2 and 3, are non-perfect squares and require decimal approximations. Visit our Square Root Formula page for more properties.