Square Root of 948

√948 at a glance

Exact value

2√237

Decimal (10 places)

30.7896086367

Rounded

30.8 · 30.79 · 30.790

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.789609

Prime factorization

2² × 3 × 79

Cube root

9.823572

How to simplify √948

Look for the largest perfect square that divides 948. Here it is 4 (2²), because 948 = 4 × 237 and 237 has no square factor left:

The prime factorization tells the same story: 948 = 2² × 3 × 79. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 79 stays inside.

Check: (2√237)² = 2² × 237 = 4 × 237 = 948. As a decimal, 2√237 = 2 × 15.3948043183 ≈ 30.7896086367.

Where √948 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √948 lies between 30 and 31. 948 is 48 above 900 and 13 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.7869 (0.01% low)
• Tangent from 30, i.e. 30 + 48 ÷ 60: 30.8000 (0.03% high)
• Tangent from 31, i.e. 31 − 13 ÷ 62: 30.7903 (0% high)

For √948 the tangent at 31 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 948 is just 13 below 961.

Finding √948 with the Babylonian method

Picture a rectangle with an area of 948 and one side x; the other side must be 948 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √948.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √948 = 30.7896086367 to every decimal shown.

√948 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √948 the pattern is [30; 1, 3, 1, 3, 20, 3, 1, 3, 1, 60] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √948 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 − 948y² = 1. Its smallest solution in positive whole numbers is x = 228,151, y = 7,410.

√948 in geometry and everyday measurements

• 948 square feet is 88.1 m². Laid out as a square — a small house footprint or a lot — it is about 30.79 ft (30 ft 9 in) on a side.
• 948 is not a sum of two whole-number squares — the prime factor 3 and 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √948 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 16 × 26 box, because 4² + 16² + 26² = 948.
• Since √948 = 2√237, a length of √948 is exactly 2 copies of the length √237 laid end to end.

Square roots near √948 and related roots

• The cube root of 948 is about 9.823572.
• Because 948 = 4 × 237, the root is twice √237: 2 × 15.394804 ≈ 30.789609.

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