Square 1 to 50 — List of Perfect Square Values

The Answer in One Line

The squares from 1 to 50 are $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500$.

The first ten ($1^2$ to $10^2$) appear in nearly every algebra problem.

Quick Answer

Result: the squares from $1^2 = 1$ to $50^2 = 2500$.

Notation: $n^2 = n \times n$.

Method shown: direct multiplication and the squares-near-50 shortcut.

Pattern: the last digit of $n^2$ depends only on the last digit of $n$ (specifically the pattern $0, 1, 4, 9, 6, 5, 6, 9, 4, 1$).

Quick Reference Table — Squares 1 to 50

Where the Squares 1 to 50 Appear

The squares from 1 to 50 are the values most students see most often. They appear in the Pythagorean theorem (a leg of length 7 and a leg of length 24 produce a hypotenuse $\sqrt{49 + 576} = \sqrt{625} = 25$, the classic $7$–$24$–$25$ triple), in quadratic-equation solutions (every perfect-square discriminant in the table makes a quadratic factor over the integers), and in any SAT or ACT mental-maths section where speed depends on knowing the table by heart.

Concept Definition — Perfect Squares

A perfect square is an integer that equals some integer multiplied by itself. The square of $n$ is $n^2 = n \cdot n$.

For positive integers, $n^2$ is also the area of a square with side length $n$. The geometric interpretation is built into the name.

How to Compute Squares Quickly

Method 1 — Direct multiplication

$n \times n$.

For two-digit $n$: $34^2 = 34 \times 34 = 1156$ (use long multiplication or column form).

Method 2 — The squares-near-50 trick

For $n = 50 \pm k$:

$$n^2 = 2500 + 100k \cdot (\text{sign}) + k^2.$$

More precisely, $(50 + k)^2 = 2500 + 100k + k^2$ and $(50 - k)^2 = 2500 - 100k + k^2$.

Example. $47^2 = (50 - 3)^2 = 2500 - 300 + 9 = 2209$. Faster than long multiplication.

Method 3 — Squares of multiples of 5

$(5n)^2 = 25 n^2$. So $45^2 = 25 \cdot 81 = 2025$.

Method 4 — Last-digit pattern

The last digit of $n^2$ follows the cycle $0, 1, 4, 9, 6, 5, 6, 9, 4, 1$ as the last digit of $n$ goes $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. Useful as a sanity check.

Common Mistakes With Square 1 to 50

1. Confusing $n^2$ with $2n$.

Where it slips in: Student writes $7^2 = 14$.

Don't do this: $n^2 = 2n$.

The correct way: $n^2 = n \times n$. $7^2 = 49$.

2. Off-by-one in the table.

Where it slips in: Student writes $13^2 = 144$ (mixing with $12^2$).

Don't do this: Trust memory without checking neighbours.

The correct way: $13^2 = 169$. $12^2 = 144$. Adjacent squares differ by $2n + 1$ — useful sanity check.