Square Root of 200

√200 at a glance

Exact value

10√2

Decimal (10 places)

14.1421356237

Rounded

14.1 · 14.14 · 14.142

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.142136

Prime factorization

2³ × 5²

Cube root

5.848035

How to simplify √200

Look for the largest perfect square that divides 200. Here it is 100 (10²), because 200 = 100 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 200 = 2³ × 5². Each pair of equal primes leaves the radical as one factor, so 2 × 5 comes out and 2 stays inside.

200 has 3 square factors (4, 25 and 100). Starting with a smaller one still works but takes more rounds: √200 = 2√50, and √50 can be simplified again. Using 100 straight away finishes in one step.

Check: (10√2)² = 10² × 2 = 100 × 2 = 200. As a decimal, 10√2 = 10 × 1.4142135624 ≈ 14.1421356237.

Where √200 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √200 lies between 14 and 15. 200 is 4 above 196 and 25 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.1379 (0.03% low)
• Tangent from 14, i.e. 14 + 4 ÷ 28: 14.1429 (0.01% high)
• Tangent from 15, i.e. 15 − 25 ÷ 30: 14.1667 (0.17% high)

For √200 the tangent at 14 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 200 is just 4 above 196.

Finding √200 with the Babylonian method

Picture a rectangle with an area of 200 and one side x; the other side must be 200 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √200.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √200 = 14.1421356237 to every decimal shown.

√200 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √200 the pattern is [14; 7, 28] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √200 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 200y² = 1. Its smallest solution in positive whole numbers is x = 99, y = 7.

√200 in geometry and everyday measurements

• A square room or garden bed covering 200 square feet measures about 14.14 ft (14 ft 2 in) along each wall.
• 200 = 2² + 14² = 10² + 10², so by the Pythagorean theorem √200 is the diagonal of rectangles measuring 2 × 14 and 10 × 10 — and the distance between the points (0, 0) and (2, 14) on a grid.
• Since √200 = 10√2, a length of √200 is exactly 10 copies of the length √2 laid end to end.

Square roots near √200 and related roots

• The cube root of 200 is about 5.848035.
• Dividing by 100 divides the root by 10: √2 = √200 ÷ 10 ≈ 1.41421356.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.