Square Root of 774

√774 at a glance

Exact value

3√86

Decimal (10 places)

27.8208554865

Rounded

27.8 · 27.82 · 27.821

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.820855

Prime factorization

2 × 3² × 43

Cube root

9.181500

How to simplify √774

Look for the largest perfect square that divides 774. Here it is 9 (3²), because 774 = 9 × 86 and 86 has no square factor left:

The prime factorization tells the same story: 774 = 2 × 3² × 43. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 43 stays inside.

Check: (3√86)² = 3² × 86 = 9 × 86 = 774. As a decimal, 3√86 = 3 × 9.2736184955 ≈ 27.8208554865.

Where √774 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √774 lies between 27 and 28. 774 is 45 above 729 and 10 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.8182 (0.01% low)
• Tangent from 27, i.e. 27 + 45 ÷ 54: 27.8333 (0.04% high)
• Tangent from 28, i.e. 28 − 10 ÷ 56: 27.8214 (0% high)

For √774 the tangent at 28 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 774 is just 10 below 784.

Finding √774 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, 28 (28² = 784):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √774 = 27.8208554865 to every decimal shown.

√774 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √774 the pattern is [27; 1, 4, 1, 1, 2, 1, 1, 4, 1, 54] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √774 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 − 774y² = 1. Its smallest solution in positive whole numbers is x = 10,405, y = 374.

√774 in geometry and everyday measurements

• 774 square feet is 71.9 m². Laid out as a square — a small house footprint or a lot — it is about 27.82 ft (27 ft 10 in) on a side.
• 774 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √774 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 17 × 22 box, because 1² + 17² + 22² = 774.
• Since √774 = 3√86, a length of √774 is exactly 3 copies of the length √86 laid end to end.

Square roots near √774 and related roots

• The cube root of 774 is about 9.181500.
• Squaring undoes the root: (√774)² = 774, while 774² = 599,076 — the number whose square root is 774.

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