Square Root of 843

√843 at a glance

Exact value

√843

Decimal (10 places)

29.0344622819

Rounded

29.0 · 29.03 · 29.034

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.034462

Prime factorization

3 × 281

Cube root

9.446607

How to simplify √843

The prime factorization of 843 is 3 × 281. Every prime appears only once, so there is no pair to bring outside the radical — √843 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 843, 3 and 281 appear an odd number of times, so √843 is irrational and 29.0344622819 is a rounded value.

Where √843 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √843 lies between 29 and 30. 843 is 2 above 841 and 57 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.0339 (0% low)
• Tangent from 29, i.e. 29 + 2 ÷ 58: 29.0345 (0% high)
• Tangent from 30, i.e. 30 − 57 ÷ 60: 29.0500 (0.05% high)

For √843 the tangent at 29 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 843 is just 2 above 841.

Finding √843 with the Babylonian method

If a guess is too big, 843 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√843) in one step.

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

Because the starting guess was already close, two steps are enough to match √843 = 29.0344622819 to every decimal shown.

√843 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √843 the pattern is [29; 29, 58] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √843 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 − 843y² = 1. Its smallest solution in positive whole numbers is x = 842, y = 29.

√843 in geometry and everyday measurements

• 843 square feet is 78.3 m². Laid out as a square — a small house footprint or a lot — it is about 29.03 ft (29 ft) on a side.
• 843 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √843 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 29 box, because 1² + 1² + 29² = 843.

Square roots near √843 and related roots

• The cube root of 843 is about 9.446607.
• Squaring undoes the root: (√843)² = 843, while 843² = 710,649 — the number whose square root is 843.

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