Reference Table
Perfect Squares List 1–400 with Square Roots
Perfect squares — whole-number roots
What Is a Perfect Square?
A perfect square (also called a square number) is a non-negative integer that results from multiplying another integer by itself. Formally: n is a perfect square if n = k² for some integer k ≥ 0. The defining property is that its square root is a whole number — no decimals, no surds. √25 = 5 exactly; √26 ≈ 5.099 and is not a whole number, so 26 is not a perfect square.
Understanding perfect squares is foundational to simplifying radicals, factorising expressions, solving quadratic equations and recognising Pythagorean triples.
Perfect Squares 1² to 10² (The Core Set)
| n | n² | √(n²) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
Perfect Squares 11² to 20²
| n | n² | √(n²) |
|---|---|---|
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
| 16 | 256 | 16 |
| 17 | 289 | 17 |
| 18 | 324 | 18 |
| 19 | 361 | 19 |
| 20 | 400 | 20 |
How to Tell if a Number Is a Perfect Square
- Integer square root test: Take the square root on a calculator. If the result is a whole number, it’s a perfect square.
- Last digit rule: A perfect square can only end in 0, 1, 4, 5, 6 or 9. Any number ending in 2, 3, 7 or 8 is never a perfect square.
- Prime factorisation: Every prime factor in a perfect square must appear an even number of times. 36 = 2² × 3² → perfect square. 72 = 2³ × 3² → not (factor 2 appears an odd number of times).
Why Perfect Squares Matter: Simplifying Radicals
Recognising perfect squares lets you simplify expressions like √72. Because 72 = 36 × 2, and 36 is a perfect square:
√72 = √(36 × 2) = √36 × √2 = 6√2
This is simplified form — no perfect square factor remains under the radical. The strategy: find the largest perfect square that divides the radicand, extract its root, and write it outside the √ symbol.
Perfect Squares in Pythagorean Triples
| a | b | c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 ✓ |
| 5 | 12 | 13 | 25 + 144 = 169 ✓ |
| 8 | 15 | 17 | 64 + 225 = 289 ✓ |
| 7 | 24 | 25 | 49 + 576 = 625 ✓ |
Frequently Asked Questions
Is 0 a perfect square?
Yes. 0 = 0², and √0 = 0, which is a whole number. By the standard definition, 0 qualifies as a perfect square. Some curricula start their lists at 1 for practical reasons, but mathematically 0 is included.
Are negative numbers perfect squares?
No — not within the integers. Squaring any integer always gives a non-negative result. For the concept of square roots of negatives, see our guide to imaginary numbers.
What is the largest perfect square below 200?
196 = 14². The next is 225 = 15², which exceeds 200. So 196 is the largest perfect square below 200.
How many perfect squares are there between 1 and 1000?
31: the squares of 1 through 31 (since 31² = 961 ≤ 1000, and 32² = 1024 > 1000). Generally, the count from 1 to N is ⌊√N⌋ (floor of the square root of N).
What is the difference between consecutive perfect squares?
The differences form the sequence of odd numbers: 4−1=3, 9−4=5, 16−9=7, 25−16=9, and so on. In general, (n+1)² − n² = 2n+1, which is always an odd number.