Square Root of 248

√248 at a glance

Exact value

2√62

Decimal (10 places)

15.7480157480

Rounded

15.7 · 15.75 · 15.748

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.748016

Prime factorization

2³ × 31

Cube root

6.282761

How to simplify √248

Look for the largest perfect square that divides 248. Here it is 4 (2²), because 248 = 4 × 62 and 62 has no square factor left:

The prime factorization tells the same story: 248 = 2³ × 31. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 31 stays inside.

Check: (2√62)² = 2² × 62 = 4 × 62 = 248. As a decimal, 2√62 = 2 × 7.874007874 ≈ 15.7480157480.

Where √248 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √248 lies between 15 and 16. 248 is 23 above 225 and 8 below 256, so the root is closer to 16.

• Straight line between 225 and 256: 15.7419 (0.04% low)
• Tangent from 15, i.e. 15 + 23 ÷ 30: 15.7667 (0.12% high)
• Tangent from 16, i.e. 16 − 8 ÷ 32: 15.7500 (0.01% high)

For √248 the tangent at 16 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 248 is just 8 below 256.

Finding √248 with the Babylonian method

Picture a rectangle with an area of 248 and one side x; the other side must be 248 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √248.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √248 = 15.7480157480 to every decimal shown.

√248 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √248 the pattern is [15; 1, 2, 1, 30] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √248 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 − 248y² = 1. Its smallest solution in positive whole numbers is x = 63, y = 4.

√248 in geometry and everyday measurements

• A square patio or deck of 248 square feet is about 15.75 ft (15 ft 9 in) on each side, so edging all the way around takes 4 × √248 ≈ 63 ft.
• 248 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √248 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 10 × 12 box, because 2² + 10² + 12² = 248.
• Since √248 = 2√62, a length of √248 is exactly 2 copies of the length √62 laid end to end.

Square roots near √248 and related roots

• The cube root of 248 is about 6.282761.
• Four times the radicand doubles the root: √992 = 2 × √248 ≈ 31.496031.

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