Square Root Calculator
Instantly calculate the square root, square, and cube root of any number. Identifies perfect squares, shows nearest perfect squares, and works with decimals and large numbers.
Last updated: March 2026
How to Use the Square Root Calculator
- Enter your number in the input field — any non-negative number works, including decimals like 0.25 or large numbers like 1,000,000.
- Results appear instantly — square root (√n), square (n²), and cube root (∛n) all calculate as you type.
- Check perfect squares — the calculator tells you whether your number is a perfect square and shows the closest perfect squares below and above it.
- Note on negative numbers — the square root of a negative number is imaginary (i). This calculator will display the imaginary form but does not compute complex arithmetic.
What Is a Square Root?
The square root of a number n is a value x such that x × x = n. In mathematical notation, x = √n. For example, √36 = 6 because 6 × 6 = 36. The √ symbol is called the radical sign, and the number beneath it is the radicand.
Every positive number has two square roots: one positive (the principal root) and one negative. √25 = 5 and √25 = -5 are both valid because 5² = 25 and (-5)² = 25. However, when we write √25, we conventionally mean the principal (positive) square root, which is 5. This calculator returns the principal square root.
Zero has only one square root: √0 = 0. Negative numbers have no real square roots — their square roots are imaginary (involving the imaginary unit i, where i = √(-1)). The study of imaginary and complex numbers is a fascinating extension of real-number mathematics, foundational in electrical engineering, quantum physics, and signal processing.
The Radical Symbol and Notation
The radical symbol √ has a long bar (vinculum) over the radicand. When no index is shown, it means the square root. A small number in the crook of the radical indicates a different root: ∛ means cube root (third root), ⁴√ means fourth root, and ⁿ√ means nth root. Alternatively, roots can be expressed as fractional exponents: √n = n^(1/2), ∛n = n^(1/3), ⁴√n = n^(1/4).
Perfect Squares Explained
A perfect square is a non-negative integer that equals another integer squared. The first 20 perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 (1² through 20²). Perfect squares have exact integer square roots — √144 = 12, not an approximation.
How to Test if a Number Is a Perfect Square
Compute the square root and check if it's an integer. If √n gives a whole number with no decimal part, n is a perfect square. For large numbers, check if the integer square root squared equals the original: is floor(√n)² = n? This calculator performs this test automatically.
Pattern of Perfect Squares
Consecutive perfect squares follow a pattern: the gap between n² and (n+1)² is always 2n+1 (an odd number). So: 1, 4 (+3), 9 (+5), 16 (+7), 25 (+9), 36 (+11), 49 (+13)… The differences are 3, 5, 7, 9, 11, 13 — always consecutive odd numbers. This pattern can help you verify or estimate perfect squares mentally.
Perfect Squares in Real Life
- Area: A square room with area 196 sq ft has sides of √196 = 14 ft.
- Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 are classic right triangle side lengths.
- Chessboard and grids: An 8×8 chessboard has 64 squares (8² = 64).
- Computer science: Hash table sizes are often powers of 2; array dimensions in 2D graphics are often square.
Irrational Square Roots and Famous Constants
Most square roots are irrational — their decimal expansions go on forever without repeating or terminating. The ancient Greeks discovered this disturbing fact around 500 BC when Hippasus proved that √2 could not be expressed as a ratio of integers. Legend has it he was drowned for this heretical discovery.
Famous Square Root Values
- √2 ≈ 1.41421356 — diagonal of a unit square; appears in 45-45-90 triangles
- √3 ≈ 1.73205081 — height of an equilateral triangle with side 2; appears in 30-60-90 triangles
- √5 ≈ 2.23606798 — related to the golden ratio φ = (1 + √5) / 2 ≈ 1.618
- √10 ≈ 3.16227766 — a useful approximation: √10 ≈ π (close but not equal; π ≈ 3.14159)
The Babylonian (Heron's) Method for Hand Calculation
Before calculators, the fastest way to approximate a square root was the iterative Babylonian method:
- Make an initial guess g (any positive number close to the answer)
- Improve: g_new = (g + n/g) / 2
- Repeat step 2 until convergence
For √50, starting with g = 7: iteration 1: (7 + 50/7)/2 = (7 + 7.143)/2 = 7.071. Iteration 2: (7.071 + 50/7.071)/2 = (7.071 + 7.071)/2 = 7.071. Converged! √50 ≈ 7.07107.
Square Roots in the Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / 2a uses a square root to solve quadratic equations. The expression under the radical (b² - 4ac) is called the discriminant — if it's positive, two real solutions exist; if zero, one; if negative, two imaginary solutions. Square roots are at the heart of algebra.
Common Square Root Examples
What is the square root of 25?
√25 = 5 exactly. 25 is a perfect square: 5 × 5 = 25.
What is the square root of 2?
√2 ≈ 1.41421356. This is irrational — it cannot be written as a fraction. It appears as the length of the diagonal of a square with side 1.
What is the square root of 100?
√100 = 10 exactly. 100 is a perfect square: 10 × 10 = 100.
What is the square root of 144?
√144 = 12 exactly. 144 is a perfect square: 12 × 12 = 144. It's also 12².
What is the square root of 50?
√50 ≈ 7.07107. Simplified: √50 = √(25 × 2) = 5√2 ≈ 5 × 1.41421 = 7.07107.
What is the square root of 0.25?
√0.25 = 0.5 exactly. Since 0.5 × 0.5 = 0.25. Equivalently, √(1/4) = 1/2 = 0.5.
What is the square root of 1,000,000?
√1,000,000 = 1,000 exactly. 1,000,000 = 10⁶ = (10³)², so √(10⁶) = 10³ = 1000.
What is the square root of 3?
√3 ≈ 1.73205. This is irrational. It appears as the height of an equilateral triangle with side 2 and in the formula for the area of an equilateral triangle: A = (√3/4) × side².
What is the square root of a fraction like 9/16?
√(9/16) = 3/4 = 0.75 exactly. √(a/b) = √a / √b. Since √9 = 3 and √16 = 4, √(9/16) = 3/4.
What is the square root of 200?
√200 ≈ 14.1421. Simplified: √200 = √(100 × 2) = 10√2 ≈ 10 × 1.41421 = 14.1421.
People Also Search For
How to simplify square roots
To simplify √n, factor n into a perfect square times a remaining factor. √72 = √(36 × 2) = 6√2. √48 = √(16 × 3) = 4√3. √180 = √(36 × 5) = 6√5. The goal is to pull out the largest perfect square factor. Use the factors calculator to help find perfect square factors.
How to add and subtract square roots
Like terms only: √2 + 3√2 = 4√2 (same radicand, add coefficients). √2 + √3 cannot be simplified further — they are unlike terms. First simplify each radical, then combine like terms: √8 + √18 = 2√2 + 3√2 = 5√2.
What is the square root of a negative number?
The square root of a negative number is an imaginary number. √(-1) = i, √(-4) = 2i, √(-9) = 3i. Imaginary numbers combine with real numbers to form complex numbers (a + bi). They're used in electrical engineering, quantum mechanics, and signal processing — not just abstract math.
How are square roots used in statistics?
Standard deviation is the square root of variance. σ = √(variance). If variance = 16, standard deviation = 4. The square root brings the measure back to the original units of the data (e.g., if you measured in cm, variance is in cm² but standard deviation is in cm).
Square root vs exponent — how are they related?
Square root is the same as raising to the power of 1/2. √n = n^0.5. ∛n = n^(1/3). ⁴√n = n^(1/4). This means all root calculations can be expressed as fractional exponents — useful in algebra and when using the exponent calculator for root computations.
What is the square root of pi?
√π ≈ 1.7724539. Pi (π ≈ 3.14159) is itself irrational, and so is its square root. √π appears in probability theory in the Gaussian integral: ∫e^(-x²)dx = √π, a fundamental result connecting geometry and probability.
Square Root Calculator — Frequently Asked Questions
What is the square root of 2?
√2 ≈ 1.41421356237. It is irrational — its decimal expansion never terminates or repeats. √2 is the length of the diagonal of a unit square and appears in 45-45-90 right triangles (legs 1, 1, hypotenuse √2).
What is a perfect square?
A perfect square is an integer that equals another integer squared: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144… Perfect squares have exact integer square roots with no decimal part.
What is the square root of a negative number?
The square root of a negative number is imaginary. √(-1) = i (the imaginary unit). √(-16) = 4i. Imaginary numbers extend the real number line into a complex plane and are fundamental to advanced mathematics, physics, and engineering.
How do you calculate a square root by hand?
The Babylonian method: start with a guess g, then repeatedly compute g = (g + n/g) / 2 until stable. For √10, start with g=3: (3 + 10/3)/2 = 3.167; then (3.167 + 10/3.167)/2 ≈ 3.1623. A few iterations gives √10 ≈ 3.16228.
What is the difference between square root and cube root?
Square root (√n) asks "what number times itself equals n?" Cube root (∛n) asks "what number cubed equals n?" √27 ≈ 5.196, but ∛27 = 3. Cube roots of negative numbers are real: ∛(-8) = -2.
What are the first 20 perfect squares?
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. These are 1² through 20².
Is √2 rational or irrational?
√2 is irrational. This was one of the first numbers proven irrational, by ancient Greek mathematicians. It cannot be expressed as p/q (a fraction of two integers) in any form. Its decimal expansion is infinite and non-repeating.
What does the radical sign √ mean?
The radical sign √ means "principal square root" — the non-negative root. √36 = 6 (not -6, even though (-6)² = 36 too). A number in the radical's crook indicates a higher root: ∛ = cube root, ⁴√ = fourth root, ⁿ√ = nth root.
How is the square root related to exponents?
Square root is equivalent to raising to the power 1/2: √n = n^(1/2). Cube root = n^(1/3). Fourth root = n^(1/4). This means square roots obey all the laws of exponents: √(a×b) = √a × √b, √(a/b) = √a / √b.
Does this calculator store my data?
No. All calculations run entirely in your browser. Nothing is transmitted to a server or stored anywhere. Your data is 100% private.
How is the square root used in the Pythagorean theorem?
In a right triangle: c = √(a² + b²). For legs 3 and 4: c = √(9 + 16) = √25 = 5. The square root extracts the hypotenuse length from the sum of squared legs. See our Pythagorean Theorem Calculator for full triangle calculations.