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Square Root Calculator

Calculate square roots instantly

Square Root

12

Input

144

Perfect Square

Yes

Perfect Squares:

Square Root

12

Perfect Square

√144 = 12

Common Perfect Squares

√11
√42
√93
√164
√255
√366
√497
√648
√819
√10010
√12111
√14412
√16913
√19614
√22515
√40020

Formulas Used

Square Root

result = √n

The square root of n is the value that, when multiplied by itself, equals n.

Where:

n= The input number
result= The square root value (result × result = n)

Perfect Square Check

isPerfect = (round(√n))² === n

A number is a perfect square if the rounded square root squared equals the original number.

Where:

n= The input number
round(√n)= Square root rounded to nearest integer

Example Calculations

1Square Root of a Perfect Square: √144

Inputs

Number144

Result

Square Root12
Perfect SquareYes

√144 = 12. Since 12 × 12 = 144, this is a perfect square. The result displays as 12 with trailing zeros removed.

2Square Root of a Non-Perfect Square: √50

Inputs

Number50

Result

Square Root7.0710678119
Perfect SquareNo

√50 = 7.0710678119 (displayed to 10 decimal places with trailing zeros removed). Since 7² = 49 and 8² = 64, 50 is not a perfect square.

Frequently Asked Questions

Q

What is a square root?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √16 = 4 because 4 × 4 = 16.

  • √16 = 4 because 4 × 4 = 16
  • √100 = 10 because 10 × 10 = 100
  • √2 ≈ 1.41421 (irrational – never terminates or repeats)
  • Every positive number has two square roots: +4 and −4 are both roots of 16
  • The principal (positive) square root is denoted by the √ symbol
Q

What is a perfect square?

A perfect square is a number whose square root is a whole number. Examples: 4 (√4 = 2), 9 (√9 = 3), 16 (√16 = 4), 25 (√25 = 5).

  • First 15 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
  • Perfect squares always end in 0, 1, 4, 5, 6, or 9 (never 2, 3, 7, or 8)
  • The gap between consecutive perfect squares increases: 3, 5, 7, 9, 11...
  • n² – (n−1)² = 2n – 1 (always an odd number)
  • There are 31 perfect squares between 1 and 1,000
NumberSquare RootPerfect Square?
255Yes
507.071...No
648Yes
10010Yes
20014.142...No
22515Yes
Q

Can you take the square root of a negative number?

In real numbers, you cannot take the square root of a negative number. However, in complex numbers, √(−1) = i (imaginary unit).

  • √(−1) = i (the imaginary unit, defined by mathematicians)
  • √(−9) = 3i because (3i) × (3i) = 9i² = 9(−1) = −9
  • Complex numbers have the form a + bi (e.g., 3 + 2i)
  • Used in electrical engineering (AC circuits), quantum physics, and signal processing
  • No real number squared gives a negative result – hence the name "imaginary"
Q

How do you calculate square root manually?

Common methods include prime factorization (for perfect squares), long division method, or estimation. For most purposes, calculators provide instant results.

  • Prime factorization: √144 = √(2⁴ × 3²) = 2² × 3 = 12
  • Estimation: √50 is between √49 (7) and √64 (8), closer to 7
  • Babylonian method: Guess, divide, average, repeat (converges fast)
  • Long division method: Works digit-by-digit, gives exact decimal places
  • Newton’s method: xₙ₊₁ = (xₙ + n/xₙ) / 2 (used by computers)
MethodWorks ForPrecision
Prime factorizationPerfect squares onlyExact
EstimationQuick approximation±0.5
Babylonian/NewtonAny positive numberAny precision
Long divisionAny positive numberDigit by digit

Understanding Square Roots

Square roots are fundamental in mathematics, used in geometry, algebra, and calculus.

Perfect squares are numbers like 4, 9, 16, 25, which have whole number square roots.

Square roots are used to solve quadratic equations and calculate areas.

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Last Updated: Mar 9, 2026

This calculator is provided for informational and educational purposes only. Results are estimates and should not be considered professional financial, medical, legal, or other advice. Always consult a qualified professional before making important decisions. UseCalcPro is not responsible for any actions taken based on calculator results.

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