How to estimate square roots in your head

You can find the square root of most numbers to a decimal place or two without a calculator. Bracket it between two perfect squares, then refine it with a single division. Here's how, with worked examples.

Step 1: bracket it between perfect squares

Find the perfect squares either side of your number. For √60: 7² = 49 and 8² = 64, so √60 is between 7 and 8 — and because 60 is much closer to 64, it's nearer 8.

That gets you the whole-number part, and it depends on knowing the squares up to about 30² without thinking. The list of square numbers has them all.

Step 2: divide and average

Take the nearer whole number as your guess. Divide your number by it, then average the guess with the result:

√60 — guess 8 60 ÷ 8 = 7.5 average of 8 and 7.5 = 7.75 true value: 7.746…

Why it works: if the guess is a little too big, dividing by it gives something a little too small, and the true root sits between the two. The average lands very close to it.

The same thing as one formula

Divide-and-average from a guess a works out to exactly a + (n − a²) ÷ 2a. With a the nearest whole number, n − a² is small, which often makes this the easier version to do in your head:

√60 = 8 + (60 − 64) ÷ 16 = 8 − 0.25 = 7.75 √90 = 9 + (90 − 81) ÷ 18 = 9 + 0.5 = 9.5 (true value 9.487…)

How accurate is it?

More than accurate enough. The estimate is always a touch high, never low, and starting from the nearest whole number, one step is within 0.04 of the true square root for every whole number from 10 to 9,999. If you need more, run the step again from the new estimate.

Bigger numbers: pair the digits

Split the number's digits into pairs, starting from the right. The root has one digit before the decimal point for each pair, and the first pair tells you where it starts:

• √4500 → 45 | 00. √45 is about 6.7, so √4500 is about 67. Divide and average: 4500 ÷ 67 ≈ 67.16, and the average of 67 and 67.16 is 67.08. The true value is 67.082…
• √450 → 4 | 50. Two pairs, so two digits before the point, and the first is 2 (because 2² = 4). 21² = 441 is the nearest square, so: 21 + (450 − 441) ÷ 42 = 21 + 9 ÷ 42 ≈ 21.21. The true value is 21.213…

Practise it

R3CK0N's Secret mode has a √n drill. It shows a number, and you type its square root; anything within 0.15 of the true value is marked correct. That's tight enough that the whole-number bracket alone usually fails, and loose enough that one divide-and-average step always passes. Easy uses two-digit numbers, Medium three-digit and Hard four-digit.

Frequently asked questions

Is this the Babylonian method?

Yes. Divide-and-average is usually called the Babylonian method, or Heron's method, and it is the same as one step of Newton's method for square roots.

What if the number is a perfect square?

Then step 1 is already exact: the bracket closes on a whole number, and n − a² is 0.