Square Root of 740

√740 at a glance

Exact value

2√185

Decimal (10 places)

27.2029410175

Rounded

27.2 · 27.20 · 27.203

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.202941

Prime factorization

2² × 5 × 37

Cube root

9.045042

How to simplify √740

Look for the largest perfect square that divides 740. Here it is 4 (2²), because 740 = 4 × 185 and 185 has no square factor left:

The prime factorization tells the same story: 740 = 2² × 5 × 37. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 37 stays inside.

Check: (2√185)² = 2² × 185 = 4 × 185 = 740. As a decimal, 2√185 = 2 × 13.6014705087 ≈ 27.2029410175.

Where √740 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √740 lies between 27 and 28. 740 is 11 above 729 and 44 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.2000 (0.01% low)
• Tangent from 27, i.e. 27 + 11 ÷ 54: 27.2037 (0% high)
• Tangent from 28, i.e. 28 − 44 ÷ 56: 27.2143 (0.04% high)

For √740 the tangent at 27 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 740 is just 11 above 729.

Finding √740 with the Babylonian method

Picture a rectangle with an area of 740 and one side x; the other side must be 740 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √740.

Start from the nearest whole number, 27 (27² = 729):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √740 = 27.2029410175 to every decimal shown.

√740 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √740 the pattern is [27; 4, 1, 12, 1, 4, 54] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √740 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 − 740y² = 1. Its smallest solution in positive whole numbers is x = 9,249, y = 340.

√740 in geometry and everyday measurements

• 740 square feet is 68.7 m². Laid out as a square — a small house footprint or a lot — it is about 27.2 ft (27 ft 2 in) on a side.
• 740 = 8² + 26² = 16² + 22², so by the Pythagorean theorem √740 is the diagonal of rectangles measuring 8 × 26 and 16 × 22 — and the distance between the points (0, 0) and (8, 26) on a grid.
• Since √740 = 2√185, a length of √740 is exactly 2 copies of the length √185 laid end to end.

Square roots near √740 and related roots

• The cube root of 740 is about 9.045042.
• Because 740 = 4 × 185, the root is twice √185: 2 × 13.601471 ≈ 27.202941.

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