Square Root Calculator — Find √x Instantly (With Perfect Squares Table)

Square root is one of the most searched math operations online — and for good reason. It comes up in geometry (finding side lengths), finance (calculating volatility), construction (diagonal measurements), and every math course from middle school through college.

The CalcHub Square Root Calculator computes √x for any positive number instantly.

Perfect Squares Table (1–30)

Number	Square (n²)	Square Root (√n²)
1	1	1
2	4	1.414
3	9	1.732
4	16	2
5	25	2.236
6	36	2.449
7	49	2.646
8	64	2.828
9	81	3
10	100	3.162
11	121	3.317
12	144	3.464
13	169	3.606
14	196	3.742
15	225	3.873
16	256	4
20	400	4.472
25	625	5
30	900	5.477
50	2,500	7.071
100	10,000	10

Commonly Searched Square Roots

√	Answer	Context
√2	1.41421	Diagonal of a unit square
√3	1.73205	Height of equilateral triangle (side 2)
√5	2.23607	Appears in golden ratio: (1+√5)/2
√7	2.64575	—
√10	3.16228	Order of magnitude estimation
√50	7.07107	= 5√2
√100	10	Perfect square
√144	12	Inches in a foot, squared
√200	14.14214	= 10√2
√500	22.36068	= 10√5

Where Square Roots Come Up in Real Life

Construction: Diagonal Measurements

The diagonal of a rectangle = √(length² + width²). A room that's 12 feet × 16 feet has a diagonal of √(144 + 256) = √400 = 20 feet. This is the Pythagorean theorem in action — builders use it constantly.

Finance: Volatility and Standard Deviation

In investing, annualized volatility = monthly standard deviation × √12. If a stock's monthly volatility is 5%, its annualized volatility is 5% × 3.46 = 17.3%. Square roots are built into every risk calculation in finance.

Distance Between Two Points

The distance formula: d = √((x₂−x₁)² + (y₂−y₁)²). From point (1, 2) to point (4, 6): d = √(9 + 16) = √25 = 5 units.

Screen Sizes

TV and monitor sizes are measured diagonally. A 16:9 screen that's 40 inches wide: height = 40 × 9/16 = 22.5 inches, diagonal = √(40² + 22.5²) = √(1600 + 506.25) = √2106.25 ≈ 45.9 inches.

How to Estimate Square Roots Without a Calculator

Method 1: Bracket and interpolate To find √50:

• 7² = 49 (too small)
• 8² = 64 (too big)
• √50 is between 7 and 8, closer to 7
• Estimate: 7.07 (actual: 7.071)

Method 2: Divide and average (Babylonian method) To find √20:

1. Guess: 4 (since 4² = 16)
2. Divide: 20 / 4 = 5
3. Average: (4 + 5) / 2 = 4.5
4. Divide: 20 / 4.5 = 4.444
5. Average: (4.5 + 4.444) / 2 = 4.472
6. Actual √20 = 4.472 ✓ (converges in 2–3 steps)

Square Root Properties

Property	Formula	Example
Product rule	√(a×b) = √a × √b	√50 = √25 × √2 = 5√2
Quotient rule	√(a/b) = √a / √b	√(49/4) = 7/2
Square of root	(√a)² = a	(√7)² = 7
Negative numbers	√(−x) = undefined (real numbers)	√(−4) = 2i (complex)

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Can you have a negative square root?

The principal square root (√) is always non-negative. But every positive number has two square roots: +√x and −√x. When we write √25 = 5, we mean the positive root. The equation x² = 25 has solutions x = +5 and x = −5.

What's the square root of zero?

√0 = 0. Zero is the only number that is its own square root (well, 1 is too: √1 = 1).

What about cube roots?

Cube root (∛x) finds the number that, when cubed, gives x. Unlike square roots, cube roots work for negative numbers: ∛−8 = −2 (because (−2)³ = −8). CalcHub also offers a cube root calculator.

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Related Calculators

• Percentage Calculator — percentage operations
• Scientific Calculator — advanced math operations
• Fraction Calculator — fraction arithmetic
• Pythagorean Theorem Calculator — uses square roots