Square Root of 901

√901 at a glance

Exact value

√901

Decimal (10 places)

30.0166620396

Rounded

30.0 · 30.02 · 30.017

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.016662

Prime factorization

17 × 53

Cube root

9.658468

How to simplify √901

The prime factorization of 901 is 17 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √901 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 901, 17 and 53 appear an odd number of times, so √901 is irrational and 30.0166620396 is a rounded value.

Where √901 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √901 lies between 30 and 31. 901 is 1 above 900 and 60 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.0164 (0% low)
• Tangent from 30, i.e. 30 + 1 ÷ 60: 30.0167 (0% high)
• Tangent from 31, i.e. 31 − 60 ÷ 62: 30.0323 (0.05% high)

For √901 the tangent at 30 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 901 is just 1 above 900.

Finding √901 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 901: following the tangent line down to zero simplifies to averaging x with 901 ÷ x.

Start from the nearest whole number, 30 (30² = 900):

Because the starting guess was already close, two steps are enough to match √901 = 30.0166620396 to every decimal shown.

√901 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √901 the pattern is [30; 60] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 901 is one more than a perfect square (30² + 1). A pattern that never ends is one more proof that √901 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 − 901y² = 1. Its smallest solution in positive whole numbers is x = 1,801, y = 60. Because the period is odd, the equation with −1 on the right also has a solution: 30² − 901 × 1² = −1.

√901 in geometry and everyday measurements

• 901 square feet is 83.7 m². Laid out as a square — a small house footprint or a lot — it is about 30.02 ft (30 ft) on a side.
• 901 = 1² + 30² = 15² + 26², so by the Pythagorean theorem √901 is the diagonal of rectangles measuring 1 × 30 and 15 × 26 — and the distance between the points (0, 0) and (1, 30) on a grid.

Square roots near √901 and related roots

• The cube root of 901 is about 9.658468.
• Squaring undoes the root: (√901)² = 901, while 901² = 811,801 — the number whose square root is 901.

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