Square Root of 548

√548 at a glance

Exact value

2√137

Decimal (10 places)

23.4093998214

Rounded

23.4 · 23.41 · 23.409

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.409400

Prime factorization

2² × 137

Cube root

8.183269

How to simplify √548

Look for the largest perfect square that divides 548. Here it is 4 (2²), because 548 = 4 × 137 and 137 has no square factor left:

The prime factorization tells the same story: 548 = 2² × 137. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 137 stays inside.

Check: (2√137)² = 2² × 137 = 4 × 137 = 548. As a decimal, 2√137 = 2 × 11.7046999107 ≈ 23.4093998214.

Where √548 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √548 lies between 23 and 24. 548 is 19 above 529 and 28 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.4043 (0.02% low)
• Tangent from 23, i.e. 23 + 19 ÷ 46: 23.4130 (0.02% high)
• Tangent from 24, i.e. 24 − 28 ÷ 48: 23.4167 (0.03% high)

For √548 the tangent at 23 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 548 is just 19 above 529.

Finding √548 with the Babylonian method

Picture a rectangle with an area of 548 and one side x; the other side must be 548 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √548.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √548 = 23.4093998214 to every decimal shown.

√548 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √548 the pattern is [23; 2, 2, 3, 1, 5, 1, 10, 1, 5, 1, 3, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √548 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 − 548y² = 1. Its smallest solution in positive whole numbers is x = 6,083,073, y = 259,856.

√548 in geometry and everyday measurements

• A square garage floor of 548 square feet measures about 23.41 ft (23 ft 5 in) per side, and its corner-to-corner diagonal is √1096 ≈ 33.1 ft.
• 548 = 8² + 22², so by the Pythagorean theorem √548 is the diagonal of a 8 × 22 rectangle — and the distance between the points (0, 0) and (8, 22) on a grid.
• Since √548 = 2√137, a length of √548 is exactly 2 copies of the length √137 laid end to end.

Square roots near √548 and related roots

• The cube root of 548 is about 8.183269.
• Because 548 = 4 × 137, the root is twice √137: 2 × 11.7047 ≈ 23.4094.

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