# Square 1 to 30 — Table of Square Values and Patterns

[Algebra](/content/tag/algebra/index.html)

TL;DR

The squares from 1 to 30 run from 1^2 = 1 to 30^2 = 900. This article gives the complete table, the odd-number pattern that links each square to the next, the fastest memory tricks, and what makes each of these values a perfect square.

**Last updated on June 13, 2022** 7 min read

## The Squares From 1 to 30 Run From 1 to 900

The squares from 1 to 30 are the values n² for n=1 through 30 — beginning at 1² = 1 and ending at 30² = 900. Every one of these numbers is a **perfect square**, the result of an integer multiplied by itself.

Knowing them on sight is what makes mental arithmetic, factoring, and estimating square roots fast. The table below is the core reference.

## Quick Answer

### **Quick Answer:**

**Result:** the squares from 1² = 1 to 30² = 900.

**Notation:** n² = n × n.

**Method shown:** direct multiplication, the algebraic-identity shortcut, and the numbers-ending-in-5 trick.

**Pattern:** consecutive squares differ by successive odd numbers — n² to (n+1)² rises by 2n + 1.

## Quick Reference Table — Squares 1 to 30

| 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 |

The first ten (1² to 10²) appear in almost every algebra problem; the next twenty are where competitive-exam and mental-maths speed is won.

## Where The Squares 1 to 30 Appear

These thirty values are the most-used perfect squares in school maths. They sit behind every Pythagorean theorem calculation — for instance, 5² + 12² = 13² uses 25 + 144 = 169, three entries straight from this table. They surface in quadratic discriminants and in areas of square grids. Numbers in this range also anchor the odd-number pattern that this article builds on.

## What Makes A Number A Perfect Square

A **perfect square** is an integer equal to some integer multiplied by itself: n² = n × n. For positive n, n² is also the area of a square with side length n. Every value in the table above is a perfect square by construction. A number is a perfect square exactly when every prime in its factorization appears to an even power.

## The Odd-Number Pattern Behind The Squares

Consecutive squares differ by consecutive odd numbers:

1 → 4 is +3;
4 → 9 is +5;
9 → 16 is +7;
16 → 25 is +9.

Each jump from n² to (n+1)² is 2n + 1, the next odd number. This is why the sum of the first n odd numbers is always n²: 1 + 3 + 5 + 7 = 16 = 4². It also provides a quick check — for example, if you know 14² = 196, then 15² = 196 + (2×14 + 1) = 196 + 29 = 225, with no multiplication required.

## How To Compute The Squares 1 to 30

### Method 1 — Direct multiplication

n × n. For example, 23 × 23 = 529 by column multiplication.

### Method 2 — The algebraic-identity shortcut

Split n into a round number plus or minus a small one, then use (a±b)² = a² ± 2ab + b².

### Method 3 — Numbers ending in 5

For any number ending in 5, write n = 10a + 5; the square is a(a+1) followed by 25.

## Examples of Squares 1 to 30

### Example 1

**Find 17² using the table.**
Read it off: 17² = 289.

### Example 2

A student wants 16² and reaches for the pattern.

**Correct.** From 15² = 225 the jump is 2(15) + 1 = 31, giving 16² = 225 + 31 = 256.

### Example 3

**Compute 26² with the identity shortcut.**
(25 + 1)² = 625 + 2(25)(1) + 1 = 676.

### Example 4

**Use the ending-in-5 trick for 35².**
a = 3, a(a+1) = 12, so 35² = 1225.

### Example 5

**Check 19² using the odd-number pattern from 18² = 324.**
19² = 324 + (2×18 + 1) = 324 + 37 = 361. Matches the table.

## Common Mistakes With Squares 1 to 30

### Mistake 1: Treating consecutive-square gaps as constant

**Correct:** The gap from n² to (n+1)² grows each step, so 16² = 225 + 31 = 256.

### Mistake 2: Confusing n² with 2n

**Correct:** n² = n × n, so 7² = 49.

### Mistake 3: Off-by-one when recalling the table

**Correct:** The gap as a check helps in memorizing values.

## Conclusion

- The squares from 1 to 30 run from 1² = 1 to 30² = 900, all perfect squares.
- Consecutive squares differ by consecutive odd numbers.
- The ending-in-5 trick and the (a±b)² identity make the larger values quick to compute.
- The most common error is treating the gaps as constant.
- The first ten squares are worth memorizing; the odd-number pattern extends them to 30.

## A Practical Next Step

1. Cover the table and rebuild 1² through 25² using only the odd-number gaps.
2. Compute 24² two ways — direct multiplication and the (25−1)² identity — and confirm they match.
3. Use the ending-in-5 trick to write 25² from memory.

## Frequently Asked Questions

**What is the square of 30?**
30² = 900.

**What is the sum of squares from 1 to 30?**
∑k=1 to 30 k² = 9,455.

**How many perfect squares are there from 1 to 30?**
Among 1 to 30, five are perfect squares: 1, 4, 9, 16, 25.

**Are the squares of odd numbers always odd?**
Yes. Odd × odd is odd.

**What is the fastest way to memorise squares 1 to 30?**
Learn 1–10 first, then use the odd-number gap to extend.