Perfect Squares and Cubes Chart- Quick Reference Guide
What Are Perfect Squares and Cubes?
A perfect square is a number you get when you multiply an integer by itself. So 9 is a perfect square because 3 × 3 = 9. Simple enough.
A perfect cube is what you get when you multiply an integer by itself three times. So 27 is a perfect cube because 3 × 3 × 3 = 27.
These come up constantly in algebra, geometry, and standardized tests. Having a quick reference saves you from wasting time on basic calculations when you should be solving actual problems.
Perfect Squares Chart (1–30)
| Number (n) | n² (Square) |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| 16 | 256 |
| 17 | 289 |
| 18 | 324 |
| 19 | 361 |
| 20 | 400 |
| 21 | 441 |
| 22 | 484 |
| 23 | 529 |
| 24 | 576 |
| 25 | 625 |
| 26 | 676 |
| 27 | 729 |
| 28 | 784 |
| 29 | 841 |
| 30 | 900 |
Perfect Cubes Chart (1–20)
| Number (n) | n³ (Cube) |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1,000 |
| 11 | 1,331 |
| 12 | 1,728 |
| 13 | 2,197 |
| 14 | 2,744 |
| 15 | 3,375 |
| 16 | 4,096 |
| 17 | 4,913 |
| 18 | 5,832 |
| 19 | 6,859 |
| 20 | 8,000 |
Squares vs. Cubes Side-by-Side
Sometimes you need both values at once. Here's a compact reference for numbers 1–15:
| n | n² | n³ |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 729 |
| 10 | 100 | 1,000 |
| 11 | 121 | 1,331 |
| 12 | 144 | 1,728 |
| 13 | 169 | 2,197 |
| 14 | 196 | 2,744 |
| 15 | 225 | 3,375 |
How to Use These Charts
Here's what most people actually need this for:
- SAT/GRE/GMAT prep — Factoring problems show up constantly. If you know 144 is 12² without thinking, you save precious test minutes.
- Factoring equations — Recognizing that 169 = 13² helps you spot difference of squares and perfect square trinomials instantly.
- Volume calculations — Cube roots come up in geometry when you're working with cubic volumes.
- Simplifying radicals — Pulling factors out of √72 is way easier when you know 36 is a perfect square.
Quick Memorization Tips
You don't need to memorize all 900 values. Focus on these:
- The squares from 1² through 15² — these appear most often in textbook problems
- The cubes of 2, 3, 4, 5, and 10 — 8, 27, 64, 125, 1000 are the workhorses
- Ending digits that signal perfect squares: 00, 1, 4, 5, 6, 9 — no square ever ends in 2, 3, 7, or 8
Common Patterns to Know
Squares follow predictable patterns:
- n² for odd n is always odd. n² for even n is always even.
- The last two digits of squares follow cycles — useful for mental math checks.
Cubes are less predictable, but notice that 10³ = 1000, 11³ = 1331, 12³ = 1728. These are the ones that show up in cube root problems.
When You're Actually Solving Problems
Don't just stare at the charts. Practice converting between forms:
- If you see 289, instantly recognize it as 17²
- If you see 2197, break it down: 13³ = 2197
- For √1156, find the pair: 34² = 1156
This recognition skill is what separates fast problem-solvers from slow ones. The charts above are your reference. The speed comes from actually using them until the common values stick in your head.