Free Square Root Calculator Online
This free online square root calculator finds the square root of any number — exact integer for perfect squares, simplified radical form, and decimal result.
Square Root Calculator
Enter any positive or negative number. Negative inputs return imaginary results.
Square Root of N
Enter a number and click Calculate
Perfect Squares 1–144
| Number (N) | Square Root (√N) |
|---|---|
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 49 | 7 |
| 64 | 8 |
| 81 | 9 |
| 100 | 10 |
| 121 | 11 |
| 144 | 12 |
How This Free Square Root Calculator Works
The square root of N is the number that, when multiplied by itself, equals N. Written mathematically: if y = √N, then y × y = N. This free online square root calculator finds that value for any input — calculate online instantly with no setup required.
For perfect squares (1, 4, 9, 16, 25...), the result is an exact integer — √144 = 12 exactly. For other numbers, the square root is irrational: √2 = 1.41421356... going on forever without repeating. The square root calculator shows both the decimal approximation and the simplified radical form.
The simplified radical form factors out any perfect square inside the root. For √72: since 72 = 36 × 2, and √36 = 6, the simplified form is 6√2. This is often the preferred form in algebra and geometry.
Negative numbers don't have real square roots. √(−4) = 2i, where i is the imaginary unit (i² = −1). The square root calculator handles these and shows the imaginary form correctly.
When You Need a Square Root Calculator
Pythagorean Theorem
For a right triangle with legs 3 and 4, the hypotenuse is √(3² + 4²) = √25 = 5. The square root is the final step in the Pythagorean theorem. This free online square root calculator handles this in one step — especially useful when the answer isn't a clean integer.
Statistics
Standard deviation is calculated as the square root of the variance. If variance is 64, the standard deviation is √64 = 8. The square root is built into statistical formulas at every level from high school to research. Use this free online tool to compute square roots instantly.
Area & Dimensions
If a square room has an area of 225 sq ft, its side length is √225 = 15 ft. Finding side lengths from area always requires a free square root calculator — and it comes up constantly in architecture, landscaping, and flooring. Use ours online at no cost.
Square Root Tips & Properties
√(a × b) = √a × √b
The square root of a product equals the product of the square roots. So √72 = √(36×2) = √36 × √2 = 6√2. This property is how the square root calculator simplifies radical expressions.
√N² = |N| (Not Just N)
The square root always returns a non-negative result. So √(−3)² = √9 = 3, not −3. The principal square root is always the positive value. Both +3 and −3 square to 9, but the square root calculator returns the positive one.
Memorize Squares to 15
Knowing that 13² = 169, 14² = 196, 15² = 225 helps you estimate square roots mentally. √200 is between 14 and 15, closer to 14. This mental check is useful before reaching for the square root calculator.
Square Root as a Power
√N = N^(1/2). This notation is useful in algebra and on scientific calculators. It also generalizes: cube root is N^(1/3), and fourth root is N^(1/4). The square root is the most common case.
Frequently Asked Questions
More Math Calculators
Percentage Calculator
Calculate percentages, percent change, and percent of a number.
Percent Change Calculator
Calculate the percentage change between two values — increase or decrease.
Percentage Increase Calculator
Calculate the percentage increase from one value to another.
Fraction Calculator
Add, subtract, multiply, and divide fractions instantly.