Square Root of 868

√868 at a glance

Exact value

2√217

Decimal (10 places)

29.4618397253

Rounded

29.5 · 29.46 · 29.462

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.461840

Prime factorization

2² × 7 × 31

Cube root

9.539082

How to simplify √868

Look for the largest perfect square that divides 868. Here it is 4 (2²), because 868 = 4 × 217 and 217 has no square factor left:

The prime factorization tells the same story: 868 = 2² × 7 × 31. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 7 × 31 stays inside.

Check: (2√217)² = 2² × 217 = 4 × 217 = 868. As a decimal, 2√217 = 2 × 14.7309198627 ≈ 29.4618397253.

Where √868 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √868 lies between 29 and 30. 868 is 27 above 841 and 32 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.4576 (0.01% low)
• Tangent from 29, i.e. 29 + 27 ÷ 58: 29.4655 (0.01% high)
• Tangent from 30, i.e. 30 − 32 ÷ 60: 29.4667 (0.02% high)

For √868 the tangent at 29 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 868 is just 27 above 841.

Finding √868 with the Babylonian method

Picture a rectangle with an area of 868 and one side x; the other side must be 868 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √868.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √868 = 29.4618397253 to every decimal shown.

√868 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √868 the pattern is [29; 2, 6, 19, 2, 19, 6, 2, 58] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √868 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 − 868y² = 1. Its smallest solution in positive whole numbers is x = 3,844,063, y = 130,476.

√868 in geometry and everyday measurements

• 868 square feet is 80.6 m². Laid out as a square — a small house footprint or a lot — it is about 29.46 ft (29 ft 6 in) on a side.
• 868 is not a sum of two whole-number squares — the prime factor 7 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √868 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 16 × 24 box, because 6² + 16² + 24² = 868.
• Since √868 = 2√217, a length of √868 is exactly 2 copies of the length √217 laid end to end.

Square roots near √868 and related roots

• The cube root of 868 is about 9.539082.
• Because 868 = 4 × 217, the root is twice √217: 2 × 14.73092 ≈ 29.46184.

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