How do you calculate a square root?

A square root of a number is a value that, when multiplied by itself, equals the original number. For example, √25 = 5 because 5 × 5 = 25. Use our calculator for instant results or learn estimation methods for manual calculation.

What are perfect squares?

Perfect squares are numbers that have whole number square roots. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. These are squares of integers: 1², 2², 3², 4², 5², 6², 7², 8², 9², 10².

What's the square root of a negative number?

The square root of a negative number is not a real number. In advanced mathematics, it's called an imaginary number. For example, √(-1) = i (the imaginary unit). Our calculator handles positive real numbers.

How do you find square roots without a calculator?

Methods include: 1) Memorizing perfect squares, 2) Prime factorization method, 3) Long division method, 4) Estimation by finding the closest perfect squares, 5) Newton's method for approximation.

What are real-world uses of square roots?

Square roots are used in: geometry (Pythagorean theorem), physics (calculating distances and velocities), engineering (signal processing), finance (volatility calculations), statistics (standard deviation), and construction (diagonal measurements).

Free Square Root Calculator

What is a Square Root?

A square root of a number is a value that, when multiplied by itself, equals the original number. The square root symbol is √ (radical sign). For example, √25 = 5 because 5 × 5 = 25.

Key applications: Geometry (Pythagorean theorem), physics calculations, engineering design, financial analysis, statistics, and solving quadratic equations.

Formula: If √x = y, then y × y = x
Examples: √9 = 3 (because 3² = 9), √16 = 4 (because 4² = 16)

Square Root Calculator (√)

Enter Number Enter any positive number to find its square root

Square root calculation will appear here.

Cube Root Calculator (∛)

Find the cube root - a number that when multiplied by itself three times equals the original number.

Enter Number Cube roots work with negative numbers too

Cube root calculation will appear here.

nth Root Calculator

Calculate any root of a number (4th root, 5th root, etc.)

Enter Number

Root Index (n) 2 = square root, 3 = cube root, 4 = fourth root, etc.

nth root calculation will appear here.

Perfect Squares Reference Chart

1² = 1

√1 = 1

2² = 4

√4 = 2

3² = 9

√9 = 3

4² = 16

√16 = 4

5² = 25

√25 = 5

6² = 36

√36 = 6

7² = 49

√49 = 7

8² = 64

√64 = 8

9² = 81

√81 = 9

100

10² = 100

√100 = 10

121

11² = 121

√121 = 11

144

12² = 144

√144 = 12

169

13² = 169

√169 = 13

196

14² = 196

√196 = 14

225

15² = 225

√225 = 15

256

16² = 256

√256 = 16

289

17² = 289

√289 = 17

324

18² = 324

√324 = 18

361

19² = 361

√361 = 19

400

20² = 400

√400 = 20

Square Root Calculator Examples

Example 1: Perfect Square

Problem: Find the square root of 144

Solution: √144 = 12

Verification: 12 × 12 = 144 ✓

Use Case: Finding the side length of a square with area 144 square units

Example 2: Non-Perfect Square

Problem: Find the square root of 50

Solution: √50 ≈ 7.071

Verification: 7.071² ≈ 50

Simplified: √50 = √(25 × 2) = 5√2

Example 3: Pythagorean Theorem

Problem: Find the hypotenuse of a right triangle with sides 3 and 4

Formula: c² = a² + b² = 3² + 4² = 9 + 16 = 25

Solution: c = √25 = 5

Application: Construction, navigation, engineering calculations

Example 4: Cube Root

Problem: Find the cube root of 216

Solution: ∛216 = 6

Verification: 6 × 6 × 6 = 216 ✓

Use Case: Finding the side length of a cube with volume 216 cubic units

How to Calculate Square Roots: Complete Guide

Method 1: Perfect Squares (Memorization)

Memorize common perfect squares for instant recognition:

1-10: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
11-20: 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
Tip: Learn these by heart for quick mental math

Method 2: Estimation Technique

Find nearby perfect squares: For √50, note that 49 < 50 < 64
Identify range: So 7 < √50 < 8
Refine estimate: Since 50 is closer to 49, √50 ≈ 7.1
Check: 7.1² = 50.41 (close enough for estimation)

Method 3: Prime Factorization

Break into prime factors: √72 = √(36 × 2) = √(6² × 2)
Extract perfect squares: = 6√2
Simplify: 6√2 ≈ 6 × 1.414 ≈ 8.485
Verify: Our calculator gives √72 ≈ 8.485

Method 4: Newton's Method (Advanced)

For finding √N using successive approximation:

Start with estimate: x₀ (close to actual answer)
Apply formula: x₁ = (x₀ + N/x₀) ÷ 2
Repeat: Use x₁ as new estimate
Continue: Until desired precision

Real-World Applications of Square Roots

🏗️ Construction & Engineering

Pythagorean Theorem: Calculate diagonal distances
Structural Design: Determine load distributions
Area Calculations: Find side lengths from area
Material Planning: Square footage conversions

📐 Geometry & Mathematics

Distance Formula: Calculate distances between points
Circle Calculations: Radius from area (r = √(A/π))
Quadratic Equations: Finding solutions
Standard Deviation: Statistical calculations

⚡ Physics & Science

Velocity Calculations: v = √(2gh) for falling objects
Wave Mechanics: Frequency and wavelength relationships
Electrical Engineering: RMS values, impedance
Optics: Lens calculations and focal lengths

💰 Finance & Economics

Risk Analysis: Standard deviation of returns
Portfolio Theory: Volatility calculations
Growth Models: Geometric mean calculations
Options Pricing: Black-Scholes formula components

💻 Computer Science

Graphics Programming: 3D distance calculations
Machine Learning: Euclidean distance metrics
Algorithm Analysis: Time complexity calculations
Signal Processing: Fourier transforms

🌱 Everyday Applications

Gardening: Square garden area to side length
Home Improvement: Diagonal measurements
Navigation: Direct distance calculations
Sports: Baseball diamond distances