Square Root of 724

√724 at a glance

Exact value

2√181

Decimal (10 places)

26.9072480941

Rounded

26.9 · 26.91 · 26.907

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.907248

Prime factorization

2² × 181

Cube root

8.979377

How to simplify √724

Look for the largest perfect square that divides 724. Here it is 4 (2²), because 724 = 4 × 181 and 181 has no square factor left:

The prime factorization tells the same story: 724 = 2² × 181. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 181 stays inside.

Check: (2√181)² = 2² × 181 = 4 × 181 = 724. As a decimal, 2√181 = 2 × 13.4536240471 ≈ 26.9072480941.

Where √724 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √724 lies between 26 and 27. 724 is 48 above 676 and 5 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.9057 (0.01% low)
• Tangent from 26, i.e. 26 + 48 ÷ 52: 26.9231 (0.06% high)
• Tangent from 27, i.e. 27 − 5 ÷ 54: 26.9074 (0% high)

For √724 the tangent at 27 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 724 is just 5 below 729.

Finding √724 with the Babylonian method

Picture a rectangle with an area of 724 and one side x; the other side must be 724 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √724.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √724 = 26.9072480941 to every decimal shown.

√724 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √724 the pattern is [26; 1, 9, 1, 3, 1, 1, 2, 1, 4, 5, 1, 3, …] with the block of 42 terms after the semicolon repeating forever (only the first 12 of the 42 are shown). A pattern that never ends is one more proof that √724 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 − 724y² = 1. Its smallest solution in positive whole numbers is x = 2,469,645,423,824,185,801, y = 91,783,649,341,730,970 — 19 digits for x, even though 724 is small, which is what makes Pell’s equation famous.

√724 in geometry and everyday measurements

• 724 square feet is 67.3 m². Laid out as a square — a small house footprint or a lot — it is about 26.91 ft (26 ft 11 in) on a side.
• 724 = 18² + 20², so by the Pythagorean theorem √724 is the diagonal of a 18 × 20 rectangle — and the distance between the points (0, 0) and (18, 20) on a grid.
• Since √724 = 2√181, a length of √724 is exactly 2 copies of the length √181 laid end to end.

Square roots near √724 and related roots

• The cube root of 724 is about 8.979377.
• Because 724 = 4 × 181, the root is twice √181: 2 × 13.453624 ≈ 26.907248.

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