Book A Free Math Class

Square 1 to 50 — List of Perfect Square Values

#Algebra

TL;DR

The squares from 1 to 50 range from 1^2 = 1 to 50^2 = 2500. This article gives the complete list, the patterns that make memorisation faster (the last-digit pattern, the squares-near-50 trick), and the most common contexts where these values appear in school maths.

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 0,1,4,9,6,5,6,9,4,0).

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)), 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-math 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).

Method 2 — The squares-near-50 trick

For n=50±k: n^2=2500+100k\cdot(sign)+k^2.

Example. (47^2=(50−3)^2=2500−300+9=2209).

Method 3 — Squares of multiples of 5

((5n)^2=25n^2).

Method 4 — Last-digit pattern

The last digit of n^2 follows the cycle 0,1,4,9,6,5,6,9,4,0.

Common Mistakes With Square 1 to 50

1. Confusing n^2 with 2n

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

The correct way: n^2 = n \times n.

2. Off-by-one in the table.

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

The correct way: 132 = 169. Adjacent squares differ by 2n + 1 — useful sanity check.

Frequently Asked Questions

What are the perfect squares between 1 and 50?

The integers between 1 and 50 that are perfect squares: 1,4,9,16,25,36,49 — that's 1^2 through 7^2.

What is the sum of squares from 1 to 50?

(\sum_{k=1}^{50} k^2 = \frac{50 \cdot 51 \cdot 101}{6} = 42{,}925).

What is the largest perfect square below 2500?

49^2 = 2401. The next one (50^2 = 2500) reaches exactly the boundary.

Is the square of an odd number always odd?

Yes. Odd × odd = odd.

What is 50^2?

50^2 = 2500.