Square Root of 648

√648 at a glance

Exact value

18√2

Decimal (10 places)

25.4558441227

Rounded

25.5 · 25.46 · 25.456

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.455844

Prime factorization

2³ × 3⁴

Cube root

8.653497

How to simplify √648

Look for the largest perfect square that divides 648. Here it is 324 (18²), because 648 = 324 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 648 = 2³ × 3⁴. Each pair of equal primes leaves the radical as one factor, so 2 × 3² comes out and 2 stays inside.

648 has 5 square factors (4, 9, 36, 81 and 324). Starting with a smaller one still works but takes more rounds: √648 = 2√162, and √162 can be simplified again. Using 324 straight away finishes in one step.

Check: (18√2)² = 18² × 2 = 324 × 2 = 648. As a decimal, 18√2 = 18 × 1.4142135624 ≈ 25.4558441227.

Where √648 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √648 lies between 25 and 26. 648 is 23 above 625 and 28 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.4510 (0.02% low)
• Tangent from 25, i.e. 25 + 23 ÷ 50: 25.4600 (0.02% high)
• Tangent from 26, i.e. 26 − 28 ÷ 52: 25.4615 (0.02% high)

For √648 the tangent at 25 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 648 is just 23 above 625.

Finding √648 with the Babylonian method

Picture a rectangle with an area of 648 and one side x; the other side must be 648 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √648.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √648 = 25.4558441227 to every decimal shown.

√648 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √648 the pattern is [25; 2, 5, 6, 5, 2, 50] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √648 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 − 648y² = 1. Its smallest solution in positive whole numbers is x = 19,601, y = 770.

√648 in geometry and everyday measurements

• A square garage floor of 648 square feet measures about 25.46 ft (25 ft 5 in) per side, and its corner-to-corner diagonal is √1296 ≈ 36 ft.
• 648 = 18² + 18², so by the Pythagorean theorem √648 is the diagonal of a 18 × 18 rectangle — and the distance between the points (0, 0) and (18, 18) on a grid.
• Since √648 = 18√2, a length of √648 is exactly 18 copies of the length √2 laid end to end.

Square roots near √648 and related roots

• The cube root of 648 is about 8.653497.
• Because 648 = 4 × 162, the root is twice √162: 2 × 12.727922 ≈ 25.455844.

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