Square Root of 202

√202 at a glance

Exact value

√202

Decimal (10 places)

14.2126704036

Rounded

14.2 · 14.21 · 14.213

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.212670

Prime factorization

2 × 101

Cube root

5.867464

How to simplify √202

The prime factorization of 202 is 2 × 101. Every prime appears only once, so there is no pair to bring outside the radical — √202 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 202, 2 and 101 appear an odd number of times, so √202 is irrational and 14.2126704036 is a rounded value.

Where √202 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √202 lies between 14 and 15. 202 is 6 above 196 and 23 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.2069 (0.04% low)
• Tangent from 14, i.e. 14 + 6 ÷ 28: 14.2143 (0.01% high)
• Tangent from 15, i.e. 15 − 23 ÷ 30: 14.2333 (0.15% high)

For √202 the tangent at 14 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 202 is just 6 above 196.

Finding √202 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √202 = 14.2126704036 to every decimal shown.

√202 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √202 the pattern is [14; 4, 1, 2, 2, 1, 4, 28] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √202 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 − 202y² = 1. Its smallest solution in positive whole numbers is x = 19,731,763, y = 1,388,322. Because the period is odd, the equation with −1 on the right also has a solution: 3,141² − 202 × 221² = −1.

√202 in geometry and everyday measurements

• A square patio or deck of 202 square feet is about 14.21 ft (14 ft 3 in) on each side, so edging all the way around takes 4 × √202 ≈ 56.9 ft.
• 202 = 9² + 11², so by the Pythagorean theorem √202 is the diagonal of a 9 × 11 rectangle — and the distance between the points (0, 0) and (9, 11) on a grid.

Square roots near √202 and related roots

• The cube root of 202 is about 5.867464.
• Four times the radicand doubles the root: √808 = 2 × √202 ≈ 28.425341.

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