Square Root of 853

√853 at a glance

Exact value

√853

Decimal (10 places)

29.2061637330

Rounded

29.2 · 29.21 · 29.206

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.206164

Prime factorization

853

Cube root

9.483814

How to simplify √853

853 is a prime number, so its only factors are 1 and 853. There is no perfect-square factor to pull out, which means √853 is already in its simplest radical form.

The square root of any prime is irrational. If √853 were a fraction a/b in lowest terms, then a² = 853b², so 853 would divide a — and then 853 would divide b too, contradicting “lowest terms.” That is why the decimal 29.2061637330 is only a rounded value.

Where √853 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √853 lies between 29 and 30. 853 is 12 above 841 and 47 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.2034 (0.01% low)
• Tangent from 29, i.e. 29 + 12 ÷ 58: 29.2069 (0% high)
• Tangent from 30, i.e. 30 − 47 ÷ 60: 29.2167 (0.04% high)

For √853 the tangent at 29 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 853 is just 12 above 841.

Finding √853 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 853: following the tangent line down to zero simplifies to averaging x with 853 ÷ x.

Start from the nearest whole number, 29 (29² = 841):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √853 = 29.2061637330 to every decimal shown.

√853 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √853 the pattern is [29; 4, 1, 5, 1, 2, 4, 1, 1, 14, 19, 2, 2, …] with the block of 23 terms after the semicolon repeating forever (only the first 12 of the 23 are shown). A pattern that never ends is one more proof that √853 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 − 853y² = 1. Its smallest solution in positive whole numbers is x = 215,454,135,724,113,414,336,120,649, y = 7,377,009,103,065,498,851,032,020 — 27 digits for x, even though 853 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 10,379,165,085,018² − 853 × 355,375,843,945² = −1.

√853 in geometry and everyday measurements

• 853 square feet is 79.2 m². Laid out as a square — a small house footprint or a lot — it is about 29.21 ft (29 ft 2 in) on a side.
• 853 = 18² + 23², so by the Pythagorean theorem √853 is the diagonal of a 18 × 23 rectangle — and the distance between the points (0, 0) and (18, 23) on a grid.

Square roots near √853 and related roots

• The cube root of 853 is about 9.483814.
• Squaring undoes the root: (√853)² = 853, while 853² = 727,609 — the number whose square root is 853.

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