There are times when a method of calculating the square root of a number is needed in SASS; for example calculating the sides of a right triangle, when the length of the hypotenuse is known, or the other way around, for a triangle mixin.

SASS provides us with some basic number operations, and there are libraries which implement more advanced math functions, but I wanted to experiment.

Using Newton’s method algorithm from Wikipedia, transcribed in SCSS scripting, we get the following:

@function sqrt($r) {
  $x0: 1; // initial value
  $solution: false;

  @for $i from 1 through 10 {
    @if abs(2 * $x0) < 0,00000000000001 { // Don't want to divide by a number smaller than this
      $solution: false;
    }

    $x1: $x0 - ($x0 * $x0 - abs($r)) / (2 * $x0) !global;

    @if ( abs($x1 - $x0) / abs($x1)) < 0,0000001 { // 7 digit accuracy is desired
      $solution: true;
    }

    $x0: $x1;
  }

  @if $solution == true {
    // If $r is negative, then the output will be multiplied with <a href="http://en.wikipedia.org/wiki/Imaginary_number">i = √-1</a>
    // (√xy = √x√y) => √-$r = √-1√$r
    @if $r < 0 {
      @return $x1 #{i};
    }
    @else {
      @return $x1;
    }
  }
  @else {
    @return "No solution";
  }
}

Which works, but looks too complicated.

We can simplify it into,

@function sqrt($r) {
  $x0: 1;
  $x1: $x0;

  @for $i from 1 through 10 {
    $x1: $x0 - ($x0 * $x0 - abs($r)) / (2 * $x0);
    $x0: $x1;
  }

  @return $x1;
}

Note: By default SASS computations are done with a precision of 5 decimals, if you need a higher precision use the “–precision N” (where N is the number of decimals) flag when compiling.