Squares of numbers from 1 to 50 range from 1² = 1 to 50² = 2500. A square of a number is the result of multiplying it by itself: n² = n × n. Memorising squares from 1 to 50 is essential for students preparing for competitive exams, board exams, and aptitude tests — it speeds up calculations in algebra, geometry, and arithmetic. This guide provides the complete squares of numbers from 1 to 50 table, shortcut tricks to calculate squares mentally, interesting patterns, a list of perfect squares, and solved examples.
Squares of numbers from 1 to 50 range from 1² = 1 to 50² = 2500.
A square of a number n is n × n, written as n².
Squares can only end in digits 0, 1, 4, 5, 6, or 9 — never 2, 3, 7, or 8.
Difference between consecutive squares: n² − (n−1)² = 2n − 1.
Sum of first n odd numbers = n² (e.g., 1+3+5+7 = 16 = 4²).
Shortcut for numbers ending in 5: multiply tens digit by (tens+1), append 25.
There are exactly 50 perfect squares between 1 and 2500.
Key squares: 10² = 100, 20² = 400, 25² = 625, 30² = 900, 50² = 2500.
Here is the complete table of squares from 1 to 50:
1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64 9² = 81 10² = 100 11² = 121 12² = 144 13² = 169 14² = 196 15² = 225 16² = 256 17² = 289 18² = 324 19² = 361 20² = 400 21² = 441 22² = 484 23² = 529 24² = 576 25² = 625 26² = 676 27² = 729 28² = 784 29² = 841 30² = 900 31² = 961 32² = 1024 33² = 1089 34² = 1156 35² = 1225 36² = 1296 37² = 1369 38² = 1444 39² = 1521 40² = 1600 41² = 1681 42² = 1764 43² = 1849 44² = 1936 45² = 2025 46² = 2116 47² = 2209 48² = 2304 49² = 2401 50² = 2500
For easier memorisation, here are the squares grouped by tens:
1 to 10: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
11 to 20: 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
21 to 30: 441, 484, 529, 576, 625, 676, 729, 784, 841, 900
31 to 40: 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600
41 to 50: 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500
Key squares to remember: • 10² = 100 • 15² = 225 • 20² = 400 • 25² = 625 • 30² = 900 • 40² = 1600 • 50² = 2500
Trick 1: Squaring numbers ending in 5 For any number ending in 5: multiply the tens digit by (tens digit + 1), then append 25. • 15² → 1 × 2 = 2, append 25 → 225 • 25² → 2 × 3 = 6, append 25 → 625 • 35² → 3 × 4 = 12, append 25 → 1225 • 45² → 4 × 5 = 20, append 25 → 2025
Trick 2: Using (a + b)² = a² + 2ab + b² • 23² = (20 + 3)² = 400 + 120 + 9 = 529 • 47² = (50 − 3)² = 2500 − 300 + 9 = 2209
Trick 3: Using the previous square n² = (n−1)² + (n−1) + n • 31² = 30² + 30 + 31 = 900 + 61 = 961 • 41² = 40² + 40 + 41 = 1600 + 81 = 1681
Trick 4: Squaring numbers near 50 n² = (n − 50)² + 100 × (n − 25) • 48² = (−2)² + 100 × 23 = 4 + 2300 = 2304 • 50² = 0² + 100 × 25 = 0 + 2500 = 2500
Pattern 1: Last digits repeat in a cycle Squares can only end in: 0, 1, 4, 5, 6, or 9. A square NEVER ends in 2, 3, 7, or 8. • Numbers ending in 0 → squares end in 0 (10² = 100, 20² = 400) • Numbers ending in 1 or 9 → squares end in 1 (11² = 121, 19² = 361) • Numbers ending in 2 or 8 → squares end in 4 (12² = 144, 18² = 324) • Numbers ending in 3 or 7 → squares end in 9 (13² = 169, 17² = 289) • Numbers ending in 4 or 6 → squares end in 6 (14² = 196, 16² = 256) • Numbers ending in 5 → squares end in 5 (15² = 225, 25² = 625)
Pattern 2: Difference between consecutive squares n² − (n−1)² = 2n − 1 (always an odd number) • 2² − 1² = 4 − 1 = 3 • 3² − 2² = 9 − 4 = 5 • 50² − 49² = 2500 − 2401 = 99
Pattern 3: Sum of first n odd numbers = n² 1 = 1² | 1+3 = 4 = 2² | 1+3+5 = 9 = 3² | 1+3+5+7 = 16 = 4²
All numbers from 1 to 50 produce perfect squares from 1 to 2500. A perfect square is a number that has a whole number as its square root.
List of all perfect squares from 1² to 50²: 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
Total: 50 perfect squares
How to check if a number is a perfect square:
Also: A perfect square always has an even number of prime factors (counting multiplicity).
Here are commonly tested squares alongside their square roots:
n | n² | √n (to 3 decimal places) 1 | 1 | 1.000 2 | 4 | 1.414 3 | 9 | 1.732 4 | 16 | 2.000 5 | 25 | 2.236 6 | 36 | 2.449 7 | 49 | 2.646 8 | 64 | 2.828 9 | 81 | 3.000 10 | 100 | 3.162 15 | 225 | 3.873 20 | 400 | 4.472 25 | 625 | 5.000 30 | 900 | 5.477 36 | 1296 | 6.000 40 | 1600 | 6.325 49 | 2401 | 7.000 50 | 2500 | 7.071
Key relationship: If n² = m, then √m = n. Example: 25² = 625, and √625 = 25.
Example 1: Find 37² without a calculator. Solution: 37² = (40 − 3)² = 40² − 2(40)(3) + 3² = 1600 − 240 + 9 = 1369
Example 2: Find 23² + 27². Solution: 23² = 529, 27² = 729. Sum = 529 + 729 = 1258
Example 3: Is 1444 a perfect square? Solution: √1444 = 38 (since 38² = 1444). Yes, 1444 is a perfect square. ✓
Example 4: Find 50² − 49². Solution: Using n² − (n−1)² = 2n − 1: 50² − 49² = 2(50) − 1 = 99 Verification: 2500 − 2401 = 99 ✓
Example 5: Find the area of a square with side 28 cm. Solution: Area = side² = 28² = 784 cm²
Example 6: What is the sum of first 50 odd numbers? Solution: Sum of first n odd numbers = n². So sum = 50² = 2500
Example 7: Find 45² using the shortcut. Solution: Ends in 5 → 4 × 5 = 20, append 25 → 2025. So 45² = 2025.
Example 8: Simplify √(16² + 12²). Solution: √(256 + 144) = √400 = 20
The squares of numbers from 1 to 50 are: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225, 16² = 256, 17² = 289, 18² = 324, 19² = 361, 20² = 400, 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625, 26² = 676, 27² = 729, 28² = 784, 29² = 841, 30² = 900, 31² = 961, 32² = 1024, 33² = 1089, 34² = 1156, 35² = 1225, 36² = 1296, 37² = 1369, 38² = 1444, 39² = 1521, 40² = 1600, 41² = 1681, 42² = 1764, 43² = 1849, 44² = 1936, 45² = 2025, 46² = 2116, 47² = 2209, 48² = 2304, 49² = 2401, 50² = 2500.
The square of 25 is 625. Calculation: 25 × 25 = 625. Shortcut: For numbers ending in 5, multiply the tens digit by (tens digit + 1) and append 25. So 2 × 3 = 6, append 25 → 625.
The square of 50 is 2500. Calculation: 50 × 50 = 2500. This is the largest square in the 1 to 50 table. Quick method: 5² = 25, then append two zeros → 2500.
Shortcut tricks: (1) Numbers ending in 5: multiply tens digit by (tens+1), append 25 (e.g., 35² → 3×4 = 12, append 25 → 1225). (2) Use (a+b)² formula: 23² = (20+3)² = 400+120+9 = 529. (3) Use previous square: n² = (n−1)² + (n−1) + n (e.g., 31² = 900+30+31 = 961).
A perfect square can only end with the digits 0, 1, 4, 5, 6, or 9. It can NEVER end in 2, 3, 7, or 8. For example: 7² = 49 (ends in 9), 8² = 64 (ends in 4), 10² = 100 (ends in 0). If a number ends in 2, 3, 7, or 8, it is definitely not a perfect square.
The difference between consecutive squares follows the formula: n² − (n−1)² = 2n − 1. This is always an odd number. Examples: 2² − 1² = 3, 10² − 9² = 19, 50² − 49² = 99. This means consecutive squares differ by consecutive odd numbers.
The sum of the first n odd numbers always equals n². Examples: 1 = 1² = 1. 1+3 = 2² = 4. 1+3+5 = 3² = 9. 1+3+5+7 = 4² = 16. So the sum of the first 50 odd numbers (1+3+5+...+99) = 50² = 2500.
The two-digit perfect squares (between 10 and 99) are: 16 (4²), 25 (5²), 36 (6²), 49 (7²), 64 (8²), and 81 (9²). That's 6 two-digit perfect squares. The three-digit perfect squares start from 100 (10²) and go up to 961 (31²).
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