Perfect squares are the backbone of square root problems. If you can recognize them instantly, you save time on every question — from simplifying radicals to solving quadratics.
Here's the complete chart plus tricks to memorize and spot them fast.
Perfect Squares Chart: 1² to 30²
| n | n² | n | n² | n | n² | |---|----|---|----|---|----| | 1 | 1 | 11 | 121 | 21 | 441 | | 2 | 4 | 12 | 144 | 22 | 484 | | 3 | 9 | 13 | 169 | 23 | 529 | | 4 | 16 | 14 | 196 | 24 | 576 | | 5 | 25 | 15 | 225 | 25 | 625 | | 6 | 36 | 16 | 256 | 26 | 676 | | 7 | 49 | 17 | 289 | 27 | 729 | | 8 | 64 | 18 | 324 | 28 | 784 | | 9 | 81 | 19 | 361 | 29 | 841 | | 10 | 100 | 20 | 400 | 30 | 900 |
Memorization Tricks
The "Last Digit" Pattern
Perfect squares can only end in: 0, 1, 4, 5, 6, 9
They NEVER end in: 2, 3, 7, 8
If a number ends in 2, 3, 7, or 8 → it's NOT a perfect square. Eliminate it immediately.
The "Digital Root" Test
Add the digits repeatedly until you get a single digit:
| Digital Root | Perfect Square? | |-------------|----------------| | 1 | ✅ (1, 64, 100...) | | 4 | ✅ (4, 49, 121...) | | 7 | ✅ (16, 25, 169...) | | 9 | ✅ (9, 81, 225...) | | 2, 3, 5, 6, 8 | ❌ Never |
Example: Is 247 a perfect square? 2+4+7 = 13 → 1+3 = 4. Maybe. But 15² = 225 and 16² = 260... not quite. 247 is not a perfect square.
The "Difference" Pattern
Consecutive perfect squares differ by consecutive odd numbers:
| n² | Difference to next | |----|-------------------| | 1 → 4 | +3 | | 4 → 9 | +5 | | 9 → 16 | +7 | | 16 → 25 | +9 | | 25 → 36 | +11 |
The difference increases by 2 each time. This means: (n+1)² = n² + 2n + 1
Spotting Perfect Square Factors
To simplify radicals, you need to spot perfect squares that divide evenly into the number under the radical.
Quick test: Is 72 divisible by a perfect square > 1?
- ÷4 = 18 ✅ → √72 = √(4×18) = 2√18 (but can go further)
- ÷9 = 8 ✅ → √72 = √(9×8) = 3√8 (but can go further)
- ÷36 = 2 ✅ → √72 = √(36×2) = 6√2 (simplest!)
Always find the largest perfect square factor for the simplest result.
Estimation Using Nearest Perfect Squares
When you need √n without a calculator:
- Find the perfect squares on either side
- Estimate where n falls between them
√50:
- 7² = 49, 8² = 64
- 50 is very close to 49 → √50 ≈ 7.07
√90:
- 9² = 81, 10² = 100
- 90 is 9/19 of the way from 81 to 100 → √90 ≈ 9.49
The Trench Truth: On standardized tests, you rarely need exact decimal values. If the answer choices are far apart, just knowing which two perfect squares a number falls between is enough. √90 is between 9 and 10, closer to 10. If the choices are 9.2, 9.5, 9.8, 10.1 — pick 9.5.
Try our square root calculator to check any number — it shows the nearest perfect squares automatically.
Related: Quadratic Formula Calculator · Derivative Calculator · Area Converter
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