√180 is irrational — 13.4164 to four decimal places, 13.42 to two
√180 ≈ 13.4164. 180 is not a perfect square, so the root is an irrational number and 13.4164 is a rounded value. Below is where that answer comes from, how to check it in one step, and how to estimate any square root without a calculator.
√180 ≈ 13.4164. Rounded to two decimal places that is 13.42.
180 is not a perfect square, so its root is not a whole number. The nearest whole numbers either side are 13 and 14: 13 × 13 = 169 and 14 × 14 = 196, and 180 sits between those two, so √180 sits between 13 and 14.
The root is irrational, which means the decimal never terminates and never settles into a repeating block. 13.4164 is a rounded value, not the exact one — the exact answer can only be written as √180 itself, or in simplified form as 6√5.
Square your answer and see how close it lands. 13.42 × 13.42 = 180.0964, which is within a whisker of 180 — close enough to confirm the root without a calculator agreeing digit for digit.
Because rounding happens at the end, squaring a rounded root will never give back exactly 180. That gap is the rounding, not an error in the root.
√180 simplifies to 6√5, because 180 = 36 × 5 and 36 is a perfect square. Pulling the 6 out from under the root sign is the standard way to write it in exact form.
Check it numerically: 6 × √5 = 6 × 2.2361 = 13.4164, the same value as √180. Simplified form is the same number written more tidily, not a different one.
Find the two square numbers it falls between and you have already bracketed the answer. 169 and 196 are 13² and 14², and 180 lies between them, so the root is between 13 and 14.
To sharpen the estimate, see how far along the gap the number sits. From 169 to 196 is a jump of 27, and 180 is 11 of the way in, so the root lands roughly 13.41. That linear guess is usually right to within a tenth, which is plenty for a sanity check.
Knowing the square numbers up to 25² by heart is what makes this instant. They are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576 and 625.
Square roots turn up any time an area, a distance or a spread has to be turned back into a length. The diagonal of a rectangle, the standard deviation of a set of numbers, the distance between two points on a grid — all of them finish with a square root.
That is why estimating one quickly is worth more than memorising a decimal. Being able to say "√180 is a bit over 13" in a second is what keeps a longer calculation moving.
The square root of 180 is approximately 13.4164, or 13.42 to two decimal places. It is an irrational number, so the decimal does not end.
No. 180 falls between 169 (13²) and 196 (14²), so its square root is not a whole number.
13 × 13 = 169. Squaring is the inverse of taking a square root, so it is also the way to check a root you have just worked out.
Bracket it between the two nearest square numbers — 169 and 196 — then judge how far along the gap 180 sits. That gets you to roughly 13.41, which is close enough for almost any mental check.
√180 = 6√5, because 180 = 36 × 5 and 36 is a perfect square.
The square numbers up to 25² each have a page here, and the square numbers index lists them all in order.