36 is both the square of 6 and a triangular number, making it a square triangular number. It is the smallest square triangular number other than 1, and it is also the only triangular number other than 1 whose square root is also a triangular number.
This number is the sum of a twin prime (17 + 19), the sum of the cubes of the first three integers, and the product of the squares of the first three integers.
36 is the number of degrees in the interior angle of each tip of a regular pentagram.
The number of possible outcomes (not summed) in the roll of two distinct dice.
The number of domino tilings of a 4×4 checkerboard is 36.
The number of inches in a yard.
And my favorite: The sum of the integers from 1 to 36 is 666 (see number of the beast).